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# A criterion for estimating the largest linear homoscedastic region in Gaussian data

Jean-Marc Bardet

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Jean-Marc Bardet. A criterion for estimating the largest linear homoscedastic region in Gaussian data. 2023. hal-04062059

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HAL
open science

# A criterion for estimating the largest linear homoscedastic region in Gaussian data

Jean-Marc Bardet

<figure><img src="image_2.png" /></figure>

## To cite this version:

Jean-Marc Bardet. A criterion for estimating the largest linear homoscedastic region in Gaussian data. 2023. hal-04062059v2

HAL Id: hal-04062059

https://hal.science/hal-04062059v2

Preprint submitted on 7 Aug 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
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# A criterion for estimating the largest linear homoscedastic region in Gaussian data

Jean-Marc Bardet¹

bardet@univ-paris1.fr

¹ SAMM, Université Paris 1, 90 rue de Tolbiac, 75013 Paris, FRANCE

April 7, 2023

## Abstract

We construct a criterion to identify the largest homoscedastic region in a data set. This can be reduced to a one-sided non-parametric break detection, knowing that up to a certain index the output is governed by a linear homoscedastic model, while after this index it is different (e.g. a different model, different variables, different volatility, ...). We show the convergence of the estimator of this index, with asymptotic concentration inequalities that can be exponential. Monte Carlo experiments will also confirm its very good numerical performance.

*Keywords:* Change detection; Gaussian linear model; Model selection; Hydrography

## 1 Introduction

The idea of this work has its roots in a numerical work done in physical geography, but which could have been done in other fields of application. The aim was to compute the fractal dimension of networks of ravines and gullies in the French Alps (see Figure 1). For this purpose, we chose to use the correlation dimension introduced by Grassberger and Proccacia (see [8]), which consists, for a "small" radius $r$, in calculating the log proportion $\log(C(r))$ of pairs of points $(X_i, X_j)$ such that the distance between $X_i$ and $X_j$ is less than $r$. And it is known that if $\lim_{r\to 0} \log(C(r))/\log(1/r) = D$ then $D$ is the correlation dimension, which is a fractal dimension often equal to the Hausdorff dimension. To estimate such a fractal dimension, a log-log regression is usually performed by considering a cloud of points $(\log(r_i), \log(C(r_i)))_{1\le i\le n}$ (see Figure 1):

Consider this point cloud. It is clear that if $r$ is too small (typically smaller than the smallest of the interdistances between the $X_i$) the fraction $C(r)$ tends to 0 and if $r$ is too large the fraction $C(r)$ tends to 1 by jumping in steps. The idea to estimate $D$ will be to consider a "good" linearity zone, for example the one that at first sight would give for $\log(r) \in [2, 4]$ (we will find $[2.3, 3.4]$ after applying our criterion). The question is: how to find this optimal linearity zone with statistical criteria?
# A criterion for estimating the largest linear homoscedastic zone in Gaussian data

Jean-Marc Bardet¹

bardet@univ-paris1.fr

¹ SAMM, Université Paris 1, 90 rue de Tolbiac, 75013 Paris, FRANCE

July 16, 2024

## Abstract

A criterion is constructed to identify the largest homoscedastic region in a Gaussian dataset. This can be reduced to a one-sided non-parametric break detection, knowing that up to a certain index the output is governed by a linear homoscedastic model, while after this index it is different (e.g. a different model, different variables, different volatility, ...). We show the convergence of the estimator of this index, with asymptotic concentration inequalities that can be exponential. A criterion and convergence results are derived when the linear homoscedastic zone is bounded by two breaks on both sides. Additionally, a criterion for choosing between zero, one, or two breaks is proposed. Monte Carlo experiments will also confirm its very good numerical performance.

*Keywords:* Change detection; Gaussian linear model; Model selection; Hydrography

## 1 Introduction

The idea of this work has its roots in a numerical work done in physical geography, but which could have been done in other fields of application. The aim was to compute the fractal dimension of networks of ravines and gullies in the French Alps (see Figure 1). For this purpose, we chose to use the correlation dimension introduced by Grassberger and Proccacia (see [8]), which consists, for a "small" radius $r$, in calculating the proportion $C(r)$ of pairs of points $(X_i, X_j)$ such that the distance between $X_i$ and $X_j$ is less than $r$, where $r > 0$, i.e.

$$C(r) = \frac{2}{N(N-1)} \sum_{1 \le i < j \le N} \mathbf{1}_{\|X_i - X_j\| \le r} \quad \text{for } (X_1, \dots, X_N) \text{ observed,}$$

with $\mathbf{1}_A = 1$ if the event $A$ occurs and $\mathbf{1}_A = 0$ otherwise. And it is known that if $\lim_{r \to 0} \frac{\log(C(r))}{\log(1/r)} = D$ then $D$ is the correlation dimension, which is a fractal dimension often equal to the Hausdorff dimension. To estimate such a fractal dimension, a log-log regression is usually performed by considering
<figure><img src="image_2.png" /><figcaption>Figure 1: Network of ravines and gullies in the French Alps (left) and plot of log($C(r_i)$) versus log($r_i$) where $1 \le r_i \le 250$ in meters (right).</figcaption></figure>

This question of the largest linearity zone is, of course, close to the problem of break detection. This problem can be simplified as follows: the outputs $Y_i$ (real-valued) depend linearly on $p$ exogenous variables $Z_i^{(j)}$, $j = 1, \dots, p$ with homoscedastic errors, for an index $i \le n$, where $n$ is unknown, while for $i \ge n+1$ this linear model is no longer valid. Much has been written about detecting a break in a linear model. The first thing that comes to mind is a Fisher test or what economists call a Chow test (see [6] or more generally in [3]). This consists of setting up a first linear model up to a certain index (e.g. individual or time) and then a second linear model after this index. The detection of the break index is done by minimizing the double sum of residual squares, which is also the Gaussian maximum likelihood estimator. This can also be done by minimizing the sum of the absolute residuals, as in [2].

But this framework is much more restrictive than the one we were considering. We would rather have a non-parametric breakthrough detection. We can then refer to work such as that of Carlstein [5] or Ben Hariz et al. [4]. Another method of non-parametric detection is to construct a criterion based on kernel density estimation, as in Arlot et al. [1]. However, these works assume an identical distribution before and after the break, which is not at all the case we are interested in (even in the case of $i \le t$, which is the case of the linear model).

We have therefore constructed an ad hoc criterion for detecting this larger linear region, which applies both when there is one break (typically something other than the linear model after an $n$ index) and when there are two breaks (other than the linear model before an $n_1$ index and after an $n_2$ index). The assumptions made outside the linear homoscedastic zone are completely nonparametric and are equally suited to an abrupt change in the model as a function of the exogenous variables or an abrupt change in the homoscedasticity of the model.

The criterion, whose precise definition is given in (3.7) in the case of one break and in (3.13) in the case of two breaks, is constructed from a minimisation of the residual sum of squares, carefully renormalised so as to choose the largest possible homoscedastic linear region, while increasing as soon as the linear region is exceeded. Under rather general assumptions, to which is added that of Gaussianity, asymptotic concentration inequalities at the breakpoint are obtained and the asymptotic normality of the least squares estimators of the linear model is also proved.
<figure><img src="image_3.png" /><figcaption>Figure 1: Network of canyons and gullies in the French Alps (left) and plot of log($C(r_i)$) against log($r_i$) where $1 \le r_i \le 250$ in m (right).</figcaption></figure>

a cloud of points ($\log(r_i)$, $\log(C(r_i))$)$_{1\le i\le n}$ (see Figure 1):

Consider this point cloud. It is clear that if $r$ is too small (typically smaller than the smallest of the interdistances between the $X_i$) the proportion $C(r)$ tends to 0 and if $r$ is too large the proportion $C(r)$ tends to 1 by jumping in steps. The idea to estimate $D$ will be to consider a "good" linearity zone, for example the one that at first sight would give for $\log(r) \in [2, 4]$ (we will find [2.3, 3.4] after applying our criterion). The question is: how to find this optimal linearity zone with statistical criteria?

This question of the largest linearity zone is of course close to the problem of break detection. This problem can be simplified as follows: the outputs $Y_i$ (real valued) depend linearly of the $p$ exogenous variables $Z_i^{(j)}$, $j = 1, \dots, p$ with homoscedastic errors, for an index $i \le n$, where $n$ is unknown, whereas for $i \ge n+1$ this linear model is no longer valid. Many works have been written about the detection of a break in a linear model. The first thing that comes to mind is a Fisher test or what economists call a Chow test (see [6] or more generally in [3]). This consists in setting up a first linear model up to a certain index (individual or time for example) then a second linear model after this index. The detection of the break index is done by minimizing the double sum of residual squares, which also corresponds to the Gaussian maximum likelihood estimator. This can also be done by minimizing the double sum of the absolute residual values, as for example in [2].

But this framework is much more restrictive than the one we considered. We would rather have a nonparametric breakthrough detection. We can then refer to work like that of Carlstein [5] or Ben Hariz et al. [4]. Another method of nonparametric detection is to construct a criterion based on kernel density estimation, as in Arlot et al. [1]. However, these works assume an identical distribution of the marginals before and after the break, which is not at all the case we are interested in.

We have developed an ad hoc criterion to detect a larger linear range. This criterion is applicable in cases where there is one break (usually a nonlinear model after an index $n$) or two breaks (a nonlinear model before an index $n_1$ and after an index $n_2$). The assumptions made beyond the linear homoscedastic zone are nonparametric and suitable for an abrupt change in the model due to exogenous variables or a sudden change in the homoscedasticity of the model.
Finally, Monte Carlo experiments are carried out to demonstrate the accuracy of the new criterion. We first consider the case of a single break and, after optimizing the meta-parameters of the criterion, we place ourselves in different frameworks of changes in linearity or changes in volatility. The results are very convincing and the convergence as $n$ increases towards the breakpoint is clear. By moving to the framework of a zone of homoscedastic linearity between 2 breakpoints $n_1$ and $n_2$, the results of the simulations show the high quality of the performance of the criterion. Finally, the application to the data set of Figure 1 allows to identify a very convincing zone of homoscedastic linearity and to obtain a fractal dimension of 1.74.

The following section 2 is devoted to the assumptions. The definitions of the criteria and the main results are given in section 3. Numerical experiments are proposed in section 4 and proofs are established in section 5.

## 2 Assumptions

Let $(\varepsilon_i)_{i \in \mathbb{N}}$ be a sequence of random variables such as $\mathbb{E}(\varepsilon_i) = 0$ and $\mathbb{E}(\varepsilon_i^2) = \sigma^2 < \infty$ for all $i \in \mathbb{N}$. Let $p \in \mathbb{N}^*$, known, and $(Z_i^{(j)})_{1 \le j \le p, i \in \mathbb{N}}$ a family of known reals numbers. Let $N \in \mathbb{N}^*$, $N \ge p + 1$ such as $\text{rank}((Z_i^{(j)})_{1 \le j \le p, 1 \le i \le N}) = p$. Let $(Y_i)_{i \in \mathbb{N}}$ be a sequence of random variables and assume that $(Y_1, \dots, Y_N)$ is observed. In the sequel we will assume:

**Assumption A:** there exists $1 \le n \le N - 1$, an unknown integer number such as

$$Y_i = \sum_{j=1}^{p} \theta_j Z_i^{(j)} + \varepsilon_i \quad \text{for all } i \in \{1, \dots, n\}, \qquad (2.1)$$

where $\theta = (\theta_1, \dots, \theta_p)' \in \mathbb{R}^p$ is unknown.

The aim of your study is the estimation of $n$ and $\theta$. For $p \le t \le N$, we define the ordinary least squares estimator (OLSE) $\hat{\theta}_t$ of $\theta$

$$\hat{\theta}_t := (Z(t)' Z(t))^{-1} Z(t)' Y(t) \qquad (2.2)$$

where, for $p \le t \le N$,

* $Z(t) := (Z_i^{(j)})_{1 \le i \le t, 1 \le j \le p}$ and we assume that $Z(t)$ is a regular matrix (i.e. its rank is $p$);

* $Y(t) := (Y_i)_{1 \le i \le t}$ and $\varepsilon(t) := (\varepsilon_i)_{1 \le i \le t}$.

Even if the following definitions of the criteria are valid for any $p \le n \le N$, an additional and usual assumption will be added for obtaining asymptotic results:

**Assumption B:** there exists $0 < \tau < 1$, an unknown real numbers such as for any $N \in \mathbb{N}^*$,

$$n = [\tau N].$$

In the sequel, we will consider a general scenario of change for the time series $(Y_i)_{1 \le i \le N}$, which satisfies Assumption A. Indeed, we assume
The criterion, whose exact definition is given in (3.8) in the case of one break and in (3.14) in the case of two breaks, is constructed from a minimization of the residual sum of squares, carefully renormalized so as to choose the largest possible linear zone, while increasing as soon as the linear zone is exceeded. Under rather general assumptions, to which is added that of Gaussianity, asymptotic concentration inequalities at the breakpoint are obtained and the asymptotic normality of the linearity parameter is also proved. Monte Carlo simulations are performed to demonstrate the quality of the estimate even in the case of very smooth breaks.

Finally, a penalized version of the criterion is proposed if the number of breaks is unknown. This allows choosing between 0, 1 or 2 breaks in the series. Numerical experiments demonstrate the good asymptotic performance of this model selection procedure.

Section 2 presents the assumptions and general framework for detecting linearity zones, while Section 3 covers the construction of a criterion and the convergence results for linearity zone detection. Section 4 is dedicated to numerical applications, including Monte-Carlo experiments and real datasets, and Section 5 contains the proofs.

## 2 Assumptions

Let $(\varepsilon_i)_{i \in \mathbb{N}}$ be a sequence of random variables such that $\mathbb{E}(\varepsilon_i) = 0$ and $\mathbb{E}(\varepsilon_i^2) = \sigma^2 < \infty$ for all $i \in \mathbb{N}$. Let $p \in \mathbb{N}^*$, known, and $(Z_i^{(j)})_{1 \le j \le p, i \in \mathbb{N}}$ a family of known real numbers (deterministic design) such that $\mathrm{rank}((Z_i^{(j)})_{1 \le j \le p, 1 \le i \le p}) = p$.

Let $N \ge p$ such that a sample $(Y_1^{(N)}, \dots, Y_N^{(N)})$ of a triangular $(Y_i^{(N)})_{1 \le i \le N, p \le N}$ is observed. In the sequel we will also assume:

**Assumption A:** For all $i \in \{1, \dots, n\}$,

$$Y_i^{(N)} = \sum_{j=1}^{p} \theta_j Z_i^{(j)} + \varepsilon_i, \quad (2.1)$$

where $\theta = (\theta_1, \dots, \theta_p)'$ is an unknown vector of $\mathbb{R}^p$ and $n$ is an unknown integer number, which may depend on $N$ and such that $1 \le n \le N - 1$.

The aim of your study is the estimation of $n$ and $\theta$. Note that when $n$ is known, a common estimator of $\theta$ is the ordinary least squares estimator (OLSE) defined by

$$\hat{\theta}_n := (Z(n)'Z(n))^{-1}Z(n)'Y(n) \quad (2.2)$$

where, for $p \le t \le N$,

* $Z(t) := (Z_i^{(j)})_{1 \le i \le t, 1 \le j \le p}$;

* $Y^{(N)}(t) := (Y_i^{(N)})_{1 \le i \le t}$ and $\varepsilon(t) := (\varepsilon_i)_{1 \le i \le t}$.

Even if the following procedures are valid for any $p \le n \le N$, an additional and usual (in the framework of change detection) assumption will be added for obtaining asymptotic results:
**Assumption C:** There exists a sequence of random variables $(\delta_t)_{t \in \mathbb{N}}$, independent to $(\varepsilon_t)_{t \in \mathbb{N}}$ and satisfying

$$Y_i = \sum_{j=1}^{p} \theta_j Z_i^{(j)} + \varepsilon_i + \delta_i \quad \text{for all } i \in \{n+1, \dots, N\} \qquad (2.3)$$

and there exists $C > 0$ and a positive random variable $k_0$ such that for any $k \ge k_0$

$$\Delta_k := \sum_{i=n+1}^{n+k} \delta_i^2 \ge Ck \quad \text{and} \quad \max_{n+1 \le i \le n+k} \delta_i^2 \le \frac{C}{2p}k \quad \text{almost surely.} \qquad (2.4)$$

The framework of (2.3) and more particularly of (2.4) is very general and may correspond to:

* A non-parametric change of the trend of $(Y_i)$. A typical non-parametric example is the following: $Y_i = \theta_0 + \theta_1 x_i + \varepsilon_i$ for $i = 1, \dots, n$ and $Y_i = g(i) + \varepsilon_i$ for $i = n+1, \dots, N$, with $|g(n+k) - (\theta_0 + \theta_1 x_{n+k})| \ge C > 0$ for $k \in \mathbb{N}^*$.

* A non-parametric change of variance of $(Y_i)_{1 \le i \le N}$, which corresponds for instance to an increasing of the volatility. A typical non-parametric example is the following: $Y_i = \theta_0 + \theta_1 x_i + \varepsilon_i$ for $i = 1, \dots, n$ and $Y_i = \theta_0 + \theta_1 x_i + \xi_i$ for $i = n+1, \dots, N$, with $\mathbb{E}(\xi_i) = 0$ and $\sum_{i=n+1}^{n+k} \text{Var}(\xi_i) > k \sigma^2$ for $k$ large enough.

# 3 Detecting change with a criterion of selection of individuals

## 3.1 Definition of a criterion of individual selection

First, define for $k \in \{p, \dots, N\}$,

* for any $u \in \mathbb{R}^k$, $\|u\|^2 = u'u$ where $u'$ denotes the transpose of $u$.

* $[Z]_k := \{Z(k) \theta, \theta \in \mathbb{R}^p\}$ which is a linear subspace of $\mathbb{R}^k$;

* $H(k) := Z(k)(Z'(k)Z(k))^{-1}Z'(k)$ which is the orthogonal projection matrix on $[Z]_k$;

For estimating $n$ under the previous assumptions, we consider a new criterion, $\widehat{Cr}_a$:

**Definition of $\widehat{Cr}_a$:** Let $a: \mathbb{N} \mapsto ]0, \infty[$ be a decreasing function and for $p+1 \le t \le N$ define

$$\widehat{Cr}_a(t) = \frac{a(t)}{t-p} \|Y(t) - H(t)Y(t)\|^2. \qquad (3.1)$$

It is well known that under Assumption A and for $t \le n$,

$$\widehat{Cr}_a(t) = a(t) \hat{\sigma}^2(t) \quad \text{where} \quad \hat{\sigma}^2(t) := \frac{1}{t-p} \|Y(t) - \hat{Y}(t)\|^2 \quad \text{and} \quad \hat{Y}(t) = H(t) Y(t),$$

the usual estimated variance and predicted vector obtained from ordinary least square estimation. From this definition, we define the following estimators:

$$\hat{n} := \arg \min_{p+1 \le t \le N} \widehat{Cr}(t) \quad \text{and} \quad \hat{\tau} := \frac{\hat{n}}{N}, \qquad (3.2)$$
**Assumption B:** there exists $0 < \tau < 1$, an unknown real numbers such that for any $N \in \mathbb{N}^*$,

$$n = [\tau N].$$

Therefore, $n$ is supposed to depend on $N$ and this explains why we consider $(Y_i^{(N)})$ as a triangular array rather than a sequence of random variables.

In the sequel, we will consider a general scenario of change for the triangular array $((Y_i^{(N)})_{1\le i\le N, N\ge p})$ satisfying Assumption A. Indeed, we also assume:

**Assumption C:** There exists a sequence of random variables $(\delta_t)_{t\in\mathbb{N}^*}$, independent to $(\varepsilon_t)_{t\in\mathbb{N}}$, such that

$$Y_{n+k}^{(N)} = \sum_{j=1}^{p} \theta_j Z_{n+k}^{(j)} + \varepsilon_{n+k} + \delta_k \quad \text{for all } k \in \{1, \dots, N-n\}. \qquad (2.3)$$

Moreover there exists $C > 0$ and a positive integer random variable $k_0$, which do not depend on $N$, such that for any $k \ge k_0$

$$\Delta_k := \sum_{i=1}^{k} \delta_i^2 \ge C k \quad \text{and} \quad \max_{1 \le i \le k} \delta_i^2 \le \frac{1}{2p} \Delta_k \quad \text{almost surely.} \qquad (2.4)$$

The framework of (2.3) coupled with (2.4) is very general and may correspond to:

* A non-parametric change of the trend of $(Y_i^{(N)})_{1\le i\le N, N\in\mathbb{N}^*}$. A typical non-parametric example is as follows:

$$\left\{ \begin{array}{ll} Y_i^{(N)} = \theta_0 + \theta_1 x_i + \varepsilon_i & \text{for } i = 1, \dots, n \\ Y_{n+k}^{(N)} = \theta_0 + \theta_1 x_{n+k} + \varepsilon_{n+k} + g(k) & \text{for } k = 1, \dots, N-n \end{array} \right. $$

where the function $g$ is such that there exists $C > 0$ satisfying $g^2(k) \ge C$ for any $k \in \mathbb{N}^*$. Then, $\delta_k^2 = g^2(k)$ and the condition $\Delta_k \ge Ck$ is satisfied for any $k \in \mathbb{N}^*$. We also add the assumption $\max_{1\le i\le k} g^2(i) \le \frac{1}{2p} \Delta_k$ for $k$ large enough. So Assumption C is checked.

Basic examples are the special cases $x_i = i$ and $g(k) = \theta_2 k^2$ or $g(k) = \theta_2 \sqrt{k}$. However the change can also be very smooth and this will be the case of the model (M1) used in numerical applications, where $g(k) = \log^2(1 + \frac{k}{10})$.

* A non-parametric change of variance of $(Y_i^{(N)})_{1\le i\le N, N\in\mathbb{N}^*}$, which corresponds for instance to an increasing of the volatility. A typical non-parametric example is the following:

$$\left\{ \begin{array}{ll} Y_i^{(N)} = \theta_0 + \theta_1 x_i + \varepsilon_i & \text{for } i = 1, \dots, n \\ Y_{n+k}^{(N)} = \theta_0 + \theta_1 x_{n+k} + \varepsilon_{n+k} + \xi_k & \text{for } k = 1, \dots, N-n \end{array} \right. $$

where $(\xi_i)_{i\in\mathbb{N}}$ is a sequence of independent Gaussian centered random variables such that $\text{Var}(\xi_i) = \gamma^2$ with $\gamma^2 > 0$. Indeed, in such a case, $\delta_k = \xi_k$ and from the law of large number $\frac{1}{k} \Delta_k \xrightarrow{k\to\infty} \gamma^2$, implying that there exists a random variable $k_1$ such that $\Delta_k \ge \frac{1}{2} \gamma^2 k$ for any $k \ge k_1$. Moreover, since we assume that $\xi_i$ are Gaussian random variables, we know that $\max_{1\le i\le k} \xi_i^2 \le \frac{\text{max}_{1\le i\le k} \xi_i^2}{2\log k} \xrightarrow{k\to\infty} 1$ and therefore there exists a random variable $k_2$ such that $\max_{1\le i\le k} \xi_i^2 \le \frac{1}{2p} \Delta_k$ for any $k \ge k_2$. Then, set $k_0 = \max(k_1, k_1)$, $C = \frac{1}{2} \gamma^2$ and (2.4) is satisfied.
since we chose the criterion to be minimized.

As a consequence the function $a(\cdot)$ should be a decreasing function because we would like to select the "largest" acceptable linear zona $\{1, \dots, t\}$. Indeed, if $t \le n$, we would like that $\widehat{Cr}_a(t+k) < \widehat{Cr}_a(t)$ for any $k \in \{1, \dots, n-t\}$ with high probability. But it is well known that under Assumption A, $\widehat{Cr}_a(t) \sim a(t) \sigma^2$ almost surely when $n \to \infty$ and $t \to \infty$ with $t \le n$ and this implies that $a(\cdot)$ has to be a decreasing function.

However we can not chose any decreasing function $a(\cdot)$ since Assumption C holds. Indeed, for $t \ge n+1$, we expect that the probability that $\widehat{Cr}_a(t) \le \widehat{Cr}_a(n)$ becomes small when $t-n$ increases. A first answer to this question can be given by considering the following computation. Indeed, we have

$$ \bullet \text{ for } t \le n, \quad \widehat{Cr}_a(t) = \frac{a(t)}{t-p} \|(I(t) - H(t)) \varepsilon(t)\|^2 \quad (3.3) $$

$$ \bullet \text{ for } t \ge n+1, \quad \widehat{Cr}_a(t) = \frac{a(t)}{t-p} \|(I(t) - H(t)) \varepsilon(t) + (I(t) - H(t)) D(t)\|^2 \quad (3.4) $$

where $I(t)$ is the identity matrix of size $t$, and

$$ D(t) = (0, \dots, 0, \delta_{n+1}, \dots, \delta_t)' \quad (3.5) $$

**Lemma 1.** Using Assumptions A, B and C, there exists a positive random variable $n_0$ such as for any $t \ge n+1$ and $n \ge n_0$,

$$ \|(I(t) - H(t)) D(t)\|^2 \ge \frac{C}{2}(t-n). $$

A first condition on the function $a(\cdot)$ can be established by considering the expectation of the criterion. Indeed, for detecting the change, we expect: $\mathbb{E}(\widehat{Cr}(n+k+1)) > \mathbb{E}(\widehat{Cr}_a(n+k))$ for any $k \ge 0$.

Thus, with $F_k = \mathbb{E}(\|(I(n+k) - H(n+k)) D(n+k)\|^2)$, this condition implies:

$$ \begin{aligned} a(n+k+1) \left( \sigma^2 + \frac{F_{k+1}}{n+k+1-p} \right) &> a(n+k) \left( \sigma^2 + \frac{F_k}{n+k-p} \right) \\ \implies a(n+k+1) &> a(n+k) \left( 1 + \frac{F_k}{\sigma^2(n+k-p)} \right) \left( 1 + \frac{F_{k+1}}{\sigma^2(n+k-p)} \right)^{-1} \end{aligned} $$

Using approximations, we deduce that there exists $c_k$ a bounded sequence such as

$$ a(n+k+1) \ge a(n+k) \left(1 - \frac{c_k}{(n+k-p)}\right). \quad (3.6) $$

Hence we chose a decreasing sequence $(a(t))_{t \in \mathbb{N}^*}$ satisfying (3.6) and our choice is $(a_{\alpha, \beta}(t))_{t \ge p+1}$ where $\beta > 1$, $\beta - \frac{1}{2} < \alpha < \beta$ defined by

$$ \begin{aligned} a_{\alpha, \beta}(t) &:= 1 + \frac{N^\alpha}{(t-p)^\beta} \quad \text{for any } t \in \{p, \dots, N\} \\ &\implies \widehat{Cr}_{\alpha, \beta}(t) := \frac{1}{t-p} \left(1 + \frac{N^\alpha}{(t-p)^\beta}\right) \|Y(t) - H(t)Y(t)\|^2. \quad (3.7) \end{aligned} $$

The term $N^\alpha$ and the relations $\beta > 1$ and $\frac{1}{2} < \beta - \frac{1}{2} < \alpha < \beta$ will be explained with details by the proof of the forthcoming Theorem 3.1. Roughly speaking, we could say that $t \mapsto (t-p)^{-\beta}$
**Remark 1.** The two conditions in Assumption C are, to the best of my knowledge, new and have not previously been stated in the literature. Instead, we can refer to the conditions given in Bai and Perron (1998) [3], where in the case of a single break, which is the case we are examining here, the condition would be:

$$Y_{n+k}^{(N)} = \sum_{j=1}^{p} \theta'_{j} Z_{n+k}^{(j)} + \varepsilon_{n+k} \quad \text{for all } k \in \{1, \dots, N-n\} \quad \text{where} \quad \sum_{j=1}^{p} (\theta'_{j} - \theta_{j})^{2} > 0. \quad (2.5)$$

The two examples mentioned above can be written in this form (in the case of a homoscedasticity break, it suffices to consider $Z_i^{(p)} = \delta_i$ and $\theta_p = 0$ whereas $\theta'_p = 1$). Clearly, this assumption leads to faster convergence rates of the estimator than the one we are about to obtain. This is also shown by simulations where such a framework is proposed with the model (M0), Section 4. However, the framework we are considering is more general and non-parametric, since we do not assume in advance that data of indices $n+1$ to $N$ still follow a linear model from $Z^{(1)}$ to $Z^{(p)}$ (this could be functions of these variables or anything else). Assumption C thus gives us greater flexibility in describing the model for these indices.

**Remark 2.** The second condition of Assumption C, i.e. there exists a positive r.v. $k_0$, which do not depend on $N$, such that for any $k \ge k_0$,

$$\max_{1 \le i \le k} \delta_i^2 \le \frac{1}{2p} \Delta_k \quad \text{almost surely}$$

is problematic in that it may require $k_0$, which is necessarily larger than $2p$, to be very large, which could prevent detection when $N - n \le k_0$. But it is required that $k_0$ depends neither on $n$ nor on $N$, since $\delta_i$ does not itself depend on $n$ and $N$. So $k_0$ can be very large, as in any case we are only considering asymptotic results for $N \to \infty$, so for $N - n \sim (1 - \tau)N \to \infty$ (due to Assumption B) implying $k_0 \le N - n$.

# 3 Detecting change with a criterion of selection of individuals

## 3.1 Definition of a criterion of individual selection

First, define for $k \in \{p, \dots, N\}$,

* for any $u \in \mathbb{R}^k$, $\|u\|^2 := u'u$ where $u'$ denotes the transpose of $u$.

* $[Z]_k := \{Z(k) \theta, \theta \in \mathbb{R}^p\}$ which is a linear subspace of $\mathbb{R}^k$;

* $H(k) := Z(k)(Z'(k)Z(k))^{-1}Z'(k)$ which is the orthogonal projection matrix on $[Z]_k$;

For estimating $n$ under the previous assumptions, we consider a new criterion, $\widehat{Cr}_a$:

**Definition of $\widehat{Cr}_a$:** Let $a: \mathbb{N} \mapsto ]0, \infty[$ be a decreasing function and for $p+1 \le t \le N$ define

$$\widehat{Cr}_a(t) = \frac{a(t)}{t-p} \|Y^{(N)}(t) - H(t) Y^{(N)}(t)\|^2. \quad (3.1)$$

It is well known that under Assumption A and for $t \le n$,

$$\widehat{Cr}_a(t) = a(t) \hat{\sigma}^2(t) \quad \text{where} \quad \hat{\sigma}^2(t) := \frac{1}{t-p} \|Y^{(N)}(t) - \hat{Y}^{(N)}(t)\|^2 \quad \text{and} \quad \hat{Y}^{(N)}(t) = H(t) Y^{(N)}(t),$$
is the decreasing function and $N^\alpha$ with $\alpha > 0$ insures that the probability to select a "small" $\hat{n}$ decreases to 0 when $N \to \infty$ while if $\alpha = 0$ this is possible. Moreover, after applying usual Taylor expansions, we have for $t \to \infty$ such as $t^\beta N^{-\alpha} \to \infty$ (which is implied by $t \ge n$ and $n = [\tau N]$ from Assumption B),

$$ \frac{a_{\alpha,\beta}(t+1)}{a_{\alpha,\beta}(t)} = 1 - \beta \frac{N^\alpha}{(t-p)^{\beta+1}} (1+o(1)). $$

and thus (3.6) is satisfied.

**Remark 1.** Another possible choice of a decreasing sequence $(a(t))_{t \in \mathbb{N}^*}$ satisfying (3.6) could be $a(t) = \exp(c N^\alpha t^{-\beta})$ with $c > 0$. From the proof of Theorem 3.1, such a choice makes quite impossible a selection of a "small" $\hat{n}$. However such a choice could imply some numerical difficulties since the computation of $\exp(N^\alpha)$ with $\alpha > 1/2$ is explosive when $N$ increases.

## 3.2 Consistency of the estimators

Now, the consistency of the criterion can be established:

**Theorem 3.1.** Let $\hat{n}$, $\hat{\tau}$ obtained in (3.2) from the criterion $\widehat{Cr}_{\alpha,\beta}$ defined in (3.7) with $1 < \alpha < \beta < \alpha + \frac{\alpha}{2\beta} < \alpha + \frac{1}{2}$. Then, under Assumptions A, B and C and if $(\varepsilon_t)_t$ is a Gaussian sequence, for any $0 < \gamma < 1 - 2(\beta - \alpha)$,

$$ \mathbb{P}(|\hat{\tau} - \tau| \ge N^{-\gamma}) \xrightarrow{N \to \infty} 0. \qquad (3.8) $$

From this theorem, we deduce that the smaller $\beta - \alpha$ the larger $\gamma$ and the more accurate the estimation $\hat{n}$. Moreover, intermediate relationships (5.4), (5.7) and (5.8) imply that the larger $\alpha$ and $\beta$ the more accurate the estimation $\hat{n}$. However, simulations in Section 4 will temperate these purposes.

**Corollary 1.** Under the assumptions of Theorem 3.1, if we consider $\tilde{n} = \arg \min_{N^\varepsilon \le t \le N} \widehat{Cr}_{\alpha,\beta}(t)$ with $0 < \varepsilon < 1$, there exists $C > 0$ and $\rho > 0$ such as

$$ \mathbb{P}(|\tilde{n} - n| \ge N^{1-\gamma}) \le e^{-C N^{\rho}}. \qquad (3.9) $$

**Corollary 2.** Define $\tilde{\theta}_N = \hat{\theta}_{\hat{n}-N^\delta}$ with $1 - 2(\beta - \alpha) < \delta < 1$, where $\hat{\theta}_t$ is defined in (2.2). Under the assumptions of Theorem 3.1, then:

$$ \tilde{\theta}_N \xrightarrow[N \to \infty]{P} \theta \quad \text{and} \quad (Z(\hat{n} - N^\delta)' Z(\hat{n} - N^\delta))^{1/2} (\tilde{\theta}_N - \theta) \xrightarrow[N \to \infty]{L} \mathcal{N}(0, \sigma^2). \qquad (3.10) $$

The choice of $\delta$ when $1 - 2(\beta - \alpha) < \delta < 1$ in the previous corollary is asymptotically not crucial since it does not modify the convergence rate of the estimator of $\theta$ which is almost given by $(Z(n)' Z(n))^{1/2}$. A possible choice is $\delta = 1 - (\beta - \alpha)$.

## 3.3 Application to the detection of the largest homoscedastic linear zone in a sequence of data indexed by time

We now consider the following problem:
the usual estimated variance and predicted vector obtained from ordinary least square estimation.
From this definition, we define the following estimators:

$$ \hat{n} := \arg \min_{p+1 \le t \le N} \widehat{Cr}_a(t) \quad \text{and} \quad \widehat{\tau} := \frac{\hat{n}}{N}, \qquad (3.2) $$

since we chose the criterion to be minimized.

As a consequence the function $a(\cdot)$ should be a decreasing function because we would like to select the "largest" acceptable linear zona $\{1, \dots, t\}$. Indeed, if $t \le n$, we would like that $\widehat{Cr}_a(t) < \widehat{Cr}_a(t)$ for any $k \in \{1, \dots, n-t\}$ with high probability. But it is well known that under Assumption A, $\widehat{Cr}_a(t) \sim a(t) \sigma^2$ almost surely when $n \to \infty$ and $t \to \infty$ with $t \le n$.

This implies that $t \in \mathbb{N}^* \mapsto a(t)$ has to be a **decreasing function**.

However we can not chose any decreasing function $a(\cdot)$ since Assumption C holds. Indeed, we also expect that the probability that $\widehat{Cr}_a(t) \le \widehat{Cr}_a(n)$ becomes small for $t \ge n+1$ and when $t-n$ increases. To understand what this implies, we need to take into account Assumption C and what it implies about the fact that for $t > n$ the asymptotic equivalence $\widehat{Cr}_a(t) \sim a(t) \sigma^2$ almost surely is no longer valid and that we instead have $\widehat{Cr}_a(t) > a(t) \sigma^2$ almost surely. Since $a(\cdot)$ is necessarily decreasing, the decrease must not be too rapid so that we can have $\widehat{Cr}_a(t) > a(n) \sigma^2$ when $t > n$ with a high probability.

We therefore need to know a little more about the behavior of $\widehat{Cr}_a(t)$ when $t \ge n$. This can be obtained from the following notations and calculations. Indeed, we have

$$ \bullet \text{ for } t \le n, \quad \widehat{Cr}_a(t) = \frac{a(t)}{t-p} \left\| (I(t) - H(t)) \varepsilon(t) \right\|^2 \qquad (3.3) $$

$$ \bullet \text{ for } t \ge n+1, \quad \widehat{Cr}_a(t) = \frac{a(t)}{t-p} \left\| (I(t) - H(t)) \varepsilon(t) + (I(t) - H(t)) D^{(N)}(t) \right\|^2 \qquad (3.4) $$

where $I(t)$ is the identity matrix of size $t$, and for $t \ge n+1$

$$ D^{(N)}(t) = (0, \dots, 0, \delta_1, \dots, \delta_{t-n})', \qquad (3.5) $$

where $\delta_i$ is defined in Assumption C.

We then show (the proof of this result, like that of all the others, is detailed in Section 5):

**Lemma 1.** Using Assumptions A, B and C, there exists a positive random variable $k_0$ such that for any $t \ge n+k_0$,

$$ \|(I(t) - H(t)) D^{(N)}(t)\|^2 \ge \frac{C}{2} (t - n) \quad \text{almost surely,} \qquad (3.6) $$

where the constant $C > 0$ is defined in Assumption C.

Referring to the expressions (3.3) and (3.4), this lemma provides a better measure of the gap between $\widehat{Cr}_a(n+k)$ and $\widehat{Cr}_a(n)$. Indeed, roughly speaking (Theorem 3.1 and its proof will give more rigor to this reasoning), we can say that for $n$ sufficiently large and $k \ge k_0$,

$$ \widehat{Cr}_a(n) \approx a(n) \sigma^2 \quad \text{and} \quad \widehat{Cr}_a(n+k) \ge a(n+k) \left( \sigma^2 + \frac{C}{2} \frac{k}{n+k-p} \right). $$
**Assumption A':** there exist two unknown change instants $n_1$ and $n_2$ in the law of $(Y_1, \dots, Y_n)$ and a known integer number $n_0$ such as $n_0 \in \{n_1, \dots, n_2 - 1\}$ and

$$Y_i = \sum_{j=1}^{p} \theta_j Z_i^{(j)} + \varepsilon_i \quad \text{for all } i \in \{n_1, n_1+1, \dots, n_2-1\},$$

where $\theta = (\theta_1, \dots, \theta_p)' \in \mathbb{R}^p$ is unknown.

**Remark:** Knowing such $n_0$ is crucial to avoid selecting a potential linear region contained in $\{1, \dots, n_1 - 1\}$ or in $\{n_2, \dots, N\}$.

The previous Assumptions B and C can be transform according to this new framework. Thus we suppose:

**Assumption B':** there exists $0 < \tau_1 < \tau_2 < 1$, two unknown real numbers, and $\tau_0$ a known real number such as $\tau_1 < \tau_0 < \tau_2$, satisfying for any $N \in \mathbb{N}^*$,

$$n_1 = [\tau_1 N], n_0 = [\tau_1 N] \quad \text{and} \quad n_2 = [\tau_2 N].$$

**Assumption C':** there exists two sequences of random variables $(\delta_t^{(1)})_{t \in \mathbb{N}}$ and $(\delta_t^{(2)})_{t \in \mathbb{N}}$, independent to $(\varepsilon_t)_{t \in \mathbb{N}}$ and satisfying

$$\left\{ \begin{array}{ll} Y_i = \sum_{j=1}^{p} \theta_j Z_i^{(j)} + \varepsilon_i + \delta_i^{(1)} & \text{for all } i \in \{1, \dots, n_1 - 1\} \\ Y_i = \sum_{j=1}^{p} \theta_j Z_i^{(j)} + \varepsilon_i + \delta_i^{(2)} & \text{for all } i \in \{n_2, \dots, N\} \end{array} \right. \qquad (3.11)$$

and there exists $C > 0$ and a positive random variable $k_0$ such that for any $k \ge k_0$

$$\sum_{i=1}^{k} (\delta_{n_1-i}^{(1)})^2 \ge Ck \quad \text{and} \quad \sum_{i=1}^{k} (\delta_{n_2+i}^{(2)})^2 \ge Ck \quad \text{almost surely.} \qquad (3.12)$$

Then, under those assumptions, it is possible to prove that an appropriated version of the new criterion $\widehat{Cr}'_{\alpha,\beta}$ is able to determinate the "largest linear zone" (in time) in $\{1, \dots, N\}$.

**Theorem 3.2.** Under Assumptions A', B' and C', define for $1 \le t_1 \le n_0 \le t_2 - p \le N$:

$$\widehat{Cr}'(t_1, t_2) := \frac{a_{\alpha,\beta}(t_2 - t_1)}{t_2 - t_1 - p} \|Y(t_1, t_2) - H(t_1, t_2) Y(t_1, t_2)\|^2, \qquad (3.13)$$

where $Y(t_1, t_2) := (Y_i)_{t_1 \le i \le t_2}$, $H(t_1, t_2) := Z(t_1, t_2)(Z'(t_1, t_2) Z(t_1, t_2))^{-1} Z'(t_1, t_2)$ and $Z(t_1, t_2) := (Z_i^{(j)})_{t_1 \le i \le t_2, 1 \le j \le p}$ (assume that $Z(t_1, t_2)$ is a regular matrix for any $t_1 < t_2 + p$). Then, define

$$(\hat{n}_1, \hat{n}_2) := \operatorname*{arg\,min}_{1 \le t_1 \le n_0 \le t_2 - p \le N} \widehat{Cr}'(t_1, t_2). \qquad (3.14)$$

Then for any $0 < \gamma < 1 - 2(\beta - \alpha)$,

$$\mathbb{P}(|\hat{n}_1 - n_1| + |\hat{n}_2 - n_2| \ge N^{1-\gamma}) \xrightarrow{N \to \infty} 0. \qquad (3.15)$$

Now, as in the previous subsection, we could add two corollaries to this theorem:
So for the probability of $\widehat{Cr}_a(n) > \widehat{Cr}_a(n + k)$ to be small, we would want at least that:

$$a(n) \sigma^2 < a(n + k) \left(\sigma^2 + \frac{C}{2} \frac{k}{n + k - p}\right) \quad \text{or} \quad a(n + k) > a(n) \left(1 - \kappa \frac{k}{n + k - p}\right), \quad (3.7)$$

with $\kappa$, which does not depend on $n$ (unknown) and $k$ but may depend on $N$.

To achieve such behavior, we chose a decreasing sequence $(a_{\alpha,\beta}(t))_{t \ge p+1}$ where $\beta > 1$, $\beta - \frac{1}{2} < \alpha < \beta$ defined by

$$a_{\alpha,\beta}(t) := 1 + \frac{N^{\alpha}}{(t-p)^{\beta}} \quad \text{for any } t \in \{p+1, \dots, N\} \\ \implies \widehat{Cr}_{\alpha,\beta}(t) := \frac{1}{t-p} \left(1 + \frac{N^{\alpha}}{(t-p)^{\beta}}\right) \|Y^{(N)}(t) - H(t) Y^{(N)}(t)\|^2. \quad (3.8)$$

Indeed, for such a choice, using Taylor expansions and thanks to the fact that we have $\alpha < \beta$ and $n \simeq \tau N$, we obtain $a_{\alpha,\beta}(n+k) \ge a_{\alpha,\beta}(n) \left(1 - \frac{N^{\alpha}}{(n-p)^{\beta}} \frac{\beta k}{n+k-p}\right)$ and the property (3.7) is verified since $0 < \frac{N^{\alpha}}{(n-p)^{\beta}} < \tau^{-\beta}$.

The term $N^\alpha$ in the expression of $a_{\alpha,\beta}$ and the relations $\beta > 1$ and $\frac{1}{2} < \beta - \frac{1}{2} < \alpha < \beta$ will be explained with details by the proof of the forthcoming Theorem 3.1. Roughly speaking, we could say that $t \mapsto (t-p)^{-\beta}$ is the decreasing function and $N^\alpha$ with $\alpha > 0$ insures that the probability to select a "small" $\hat{n}$ decreases to 0 when $N \to \infty$ while if $\alpha = 0$ this is possible.

**Remark 3.** Clearly, we could also have taken as criterion

$$\frac{1}{t-p} \left(1 + M \frac{N^{\alpha}}{(t-p)^{\beta}}\right) \|Y^{(N)}(t) - H(t) Y^{(N)}(t)\|^2,$$

where $M$ is a positive real constant. This does not change the asymptotic behavior of the break detection, but it can be interesting numerically to improve performance. In particular, in Section 4 devoted to numerical applications, we will choose $M = 1/10$ after numerous simulations. To avoid overloading the proofs, we have chosen to work with the criterion without $M$, i.e. the one defined in (3.8).

**Remark 4.** Another possible choice of a decreasing sequence $(a(t))_{t \in \mathbb{N}^*}$ satisfying (3.7) could be $a(t) = \exp(c N^\alpha t^{-\beta})$ with $c > 0$. From the proof of Theorem 3.1, such a choice makes quite impossible a selection of a "small" $\hat{n}$. However such a choice could imply some numerical difficulties since the computation of $\exp(N^\alpha)$ with $\alpha > 1/2$ is explosive when $N$ increases.

## 3.2 Consistency of the estimators

Now, the consistency of the estimators obtain from the criterion can be established:

**Theorem 3.1.** Let $\hat{n}$, $\hat{\tau}$ and $\tilde{\theta}$ defined from (3.2) where the criterion $\widehat{Cr}_{\alpha,\beta}$ is defined in (3.8) and $1 < \alpha < \beta < \alpha + \frac{\alpha}{2\beta} < \alpha + \frac{1}{2}$. Then, under Assumptions A, B and C and if $(\varepsilon_t)_t$ is a Gaussian sequence, for any $0 < \gamma < 1 - 2(\beta - \alpha)$,

$$\mathbb{P}(|\hat{\tau} - \tau| \ge N^{-\gamma}) \underset{N \to \infty}{\longrightarrow} 0. \quad (3.9)$$
**Corollary 3.** Under the assumptions of Theorem 3.2, if we consider $(\tilde{n}_1, \tilde{n}_2) := \arg \min_{N^e \le t_1 \le n_0 \le t_2 - p \le N - N^e} \widehat{Cr}'(t_1)$ with $0 < \varepsilon < 1$, there exists $C > 0$ and $\rho > 0$ such as

$$ \mathbb{P}(|\tilde{n}_1 - n_1| + |\tilde{n}_2 - n_2| \ge N^{1-\gamma}) \le e^{-C N^\rho}. \quad (3.16) $$

**Corollary 4.** For any $\delta$ such as $1 - 2(\beta - \alpha) < \delta < 1$, define

$$ \tilde{\theta}_N = (Z(\hat{n}_1 + N^\delta, \hat{n}_2 - N^\delta)' Z(\hat{n}_1 + N^\delta, \hat{n}_2 - N^\delta))^{-1} Z(\hat{n}_1 + N^\delta, \hat{n}_2 - N^\delta)' Y(\hat{n}_1 + N^\delta, \hat{n}_2 - N^\delta). $$

Then, under the assumptions of Theorem 3.1, then:

$$ \tilde{\theta}_N \xrightarrow[N \to \infty]{P} \theta \quad \text{and} \quad (Z(\hat{n}_1 + N^\delta, \hat{n}_2 - N^\delta)' Z(\hat{n}_1 + N^\delta, \hat{n}_2 - N^\delta))^{1/2} (\tilde{\theta}_N - \theta) \xrightarrow[N \to \infty]{L} \mathcal{N}(0, \sigma^2). $$

# 4 Numerical applications

## 4.1 Case of a linear homoscedastic zone with break on one side

In this first subsection, we consider the following framework: $(Y_1, \dots, Y_N)$ is observed, and there exists $n$, unknown integer number, such as

$$ Y_i = \begin{cases} \sum_{j=1}^{p} \theta_j Z_i^{(j)} + \varepsilon_i & \text{for all } i \in \{1, \dots, n\}; \\ \sum_{j=1}^{p} \theta_j Z_i^{(j)} + \varepsilon_i + \delta_i & \text{for all } i \in \{n+1, \dots, N\}. \end{cases} $$

where $\theta = (\theta_1, \dots, \theta_p)' \in \mathbb{R}^p$ is unknown. More precisely, we will consider the following particular cases:

* The size $N$ of the observed sample is such as: $N = 100, 200, 400, 800$ and $1600$.

* The error $(\varepsilon_i)$ of the homoscedastic linear model is a sequence of Gaussian centered i.i.d.r.v. with $\sigma^2 = \text{Var}(\varepsilon_0)$ is such as $\sigma = 1$ and $\sigma = 2$.

* The unknown change date $n$ is such as $n = [0.4 \times N]$.

* Three distinct frameworks are considered:

1. (M1) Trend break in a time series:

$$ Y_i = \begin{cases} 3 + 0.1 \times i + \varepsilon_i & \text{for all } i \in \{1, \dots, n\}; \\ 3 + 0.1 \times i + (2\sqrt{3 + (i-n)} - 4) + \varepsilon_i & \text{for all } i \in \{n+1, \dots, N\}. \end{cases} $$

2. (M2) Time series volatility break:

$$ Y_i = \begin{cases} 3 + 0.1 \times i + \varepsilon_i & \text{for all } i \in \{1, \dots, n\}; \\ 3 + 0.1 \times i + 2 * \varepsilon_i & \text{for all } i \in \{n+1, \dots, N\}. \end{cases} $$

3. (M3) Model change for an homoscedastic linear model:

$$ Y_i = \begin{cases} -4 + 2 Z_i^{(1)} - 5 Z_i^{(2)} + 3 Z_i^{(3)} + \varepsilon_i & \text{for all } i \in \{1, \dots, n\}; \\ -8 - Z_i^{(1)} + 3 \log(|Z_i^{(3)}|) + \varepsilon_i & \text{for all } i \in \{n+1, \dots, N\}. \end{cases} $$

where $(Z_i^{(1)})$ and $(Z_i^{(3)})$ are observed i.i.d.r.v. following respectively a uniform distribution $U([-1, 1])$ and a Student distribution $t(4)$, and $Z_i^{(2)} = (Z_i^{(1)})^2$.
From this theorem, we deduce that the smaller $\beta - \alpha$ the larger $\gamma$ and the more accurate the estimation $\hat{n}$. Moreover, intermediate relationships (5.4), (5.7) and (5.8) imply that the larger $\alpha$ and $\beta$ the more accurate the estimation $\hat{n}$. However, simulations in Section 4 will temperate these purposes.

**Corollary 1.** Under the assumptions of Theorem 3.1, if we consider $\tilde{n} = \arg \min_{N^\epsilon \le t \le N} \widehat{Cr}_{\alpha,\beta}(t)$ with $0 < \epsilon < 1$, there exists $C > 0$ and $\rho > 0$ such that

$$ \mathbb{P}(|\tilde{n} - n| \ge N^{1-\gamma}) \le e^{-C N^{\rho}}. \quad (3.10) $$

**Corollary 2.** Define $\tilde{\theta}_N = \hat{\theta}_{\hat{n}-N^\delta}$ with $1 - 2(\beta - \alpha) < \delta < 1$. Under the assumptions of Theorem 3.1, then:

$$ (Z'(\hat{n} - N^\delta) Z(\hat{n} - N^\delta))^{1/2} (\tilde{\theta}_N - \theta) \xrightarrow[N \to \infty]{} \mathcal{N}(0, \sigma^2). \quad (3.11) $$

The choice of $\delta$ when $1 - 2(\beta - \alpha) < \delta < 1$ in the previous corollary is asymptotically not crucial since it does not modify the convergence rate of the estimator of $\theta$ which is almost given by $(Z'(n)Z(n))^{1/2}$. A possible choice is $\delta = 1 - (\beta - \alpha)$.

### 3.3 Application to the detection of the largest homoscedastic linear zone in a sequence of data indexed by time

We now consider the following problem:

**Assumption A':** there exist two unknown change instants $n_1$ and $n_2$ in the law of $(Y_1^{(N)}, \dots, Y_n^{(N)})$ and a known integer number $n_0$ such that $n_0 \in \{n_1, \dots, n_2\}$ and

$$ Y_i^{(N)} = \sum_{j=1}^{p} \theta_j Z_i^{(j)} + \varepsilon_i \quad \text{for all } i \in \{n_1, n_1+1, \dots, n_2\}, $$

where $\theta = (\theta_1, \dots, \theta_p)' \in \mathbb{R}^p$ is unknown.

**Remark:** Knowing such $n_0$ is crucial to avoid selecting a potential linear region contained in $\{1, \dots, n_1 - 1\}$ or in $\{n_2 + 1, \dots, N\}$.

The previous Assumptions B and C can be transform according to this new framework. Thus we suppose:

**Assumption B':** there exists $0 < \tau_1 < \tau_2 < 1$, two unknown real numbers, and $\tau_0$ a known real number such that $\tau_1 < \tau_0 < \tau_2$, satisfying for any $N \in \mathbb{N}^*$,

$$ n_1 = [\tau_1 N], n_0 = [\tau_1 N] \quad \text{and} \quad n_2 = [\tau_2 N]. $$

**Assumption C':** there exists two sequences of random variables $(\delta_t^{(1)})_{t \in \mathbb{N}}$ and $(\delta_t^{(2)})_{t \in \mathbb{N}}$, independent to $(\varepsilon_t)_{t \in \mathbb{N}}$ and satisfying

$$ \left\{ \begin{array}{ll} Y_{n_1-k}^{(N)} = \sum_{j=1}^{p} \theta_j Z_{n_1-k}^{(j)} + \varepsilon_{n_1-k} + \delta_k^{(1)} & \text{for all } k \in \{1, \dots, n_1-1\} \\ Y_{n_2+k}^{(N)} = \sum_{j=1}^{p} \theta_j Z_{n_2+k}^{(j)} + \varepsilon_{n_2+k} + \delta_k^{(2)} & \text{for all } k \in \{1, \dots, N-n_2\} \end{array} \right. \quad (3.12) $$
<table><caption>Table 1: Square roots of the MSE for the estimators $\hat{\tau}$ minimizing the criterion $\hat{Cr}$ for the three different models (M1), (M2) and (M3), and the different values of parameters $\sigma$ and $N$.</caption><thead><tr><th></th><th>N</th><th>100</th><th>200</th><th>400</th><th>800</th><th>1600</th></tr></thead><tbody><tr><td rowspan="2">(M1)</td><td>σ = 1</td><td>0.1026</td><td>0.0695</td><td>0.0472</td><td>0.0311</td><td>0.0202</td></tr><tr><td>σ = 2</td><td>0.1851</td><td>0.0728</td><td>0.0480</td><td>0.0313</td><td>0.0213</td></tr><tr><td rowspan="2">(M2)</td><td>σ = 1</td><td>0.1037</td><td>0.0672</td><td>0.0515</td><td>0.0341</td><td>0.0213</td></tr><tr><td>σ = 2</td><td>0.1018</td><td>0.0702</td><td>0.0486</td><td>0.0325</td><td>0.0233</td></tr><tr><td rowspan="2">(M3)</td><td>σ = 1</td><td>0.0972</td><td>0.0665</td><td>0.0456</td><td>0.0330</td><td>0.0211</td></tr><tr><td>σ = 2</td><td>0.0975</td><td>0.0710</td><td>0.0423</td><td>0.0308</td><td>0.0184</td></tr></tbody></table>

We applied the $\hat{Cr}_{\alpha,\beta}$ criterion defined in (3.7). After an important empirical study not reproduced here, we have decided on the parameters $\alpha = 2$ and $\beta = 2.1$, and we have added a constant to compensate for the fact that $n$ is not very large (which does not change the asymptotic results of the previous section). Hence the chosen criterion is:

$$ \hat{Cr}(t) = \frac{1}{t-p} \left( 1 + \frac{N^2}{10(t-p)^{2.1}} \right) \|Y(t) - H(t)Y(t)\|^2. $$

After performing 1000 independent replications of the scatterplot for the different models and parameters, and applying the largest linear area detection criterion $\hat{Cr}$, we obtained the results recorded in Table 1:

**Conclusions of Monte Carlo experiments:** We first observe the consistency of the estimator $\hat{\tau}$ as $N$ becomes large. And the rate of convergence is close to $\sqrt{N}$. We also note two important points:

* The variance $\sigma^2$ of the noise has very little effect on the rate of convergence of the estimator, except in one case;

* For a fixed $N$, the square root of the MSE is almost the same whatever model is chosen. And for all the other different models we tried, we also find almost the same values!

## 4.2 Case of a linear homoscedastic zone with breaks on both sides

Now we consider the following framework: $(Y_1, \dots, Y_N)$ is observed, and there exists $n_1$ and $n_2$, unknown integer numbers, such as

$$ Y_i = \begin{cases} \sum_{j=1}^{p} \theta_j Z_i^{(j)} + \varepsilon_i & \text{for all } i \in \{n_1, \dots, n_2 - 1\}; \\ \sum_{j=1}^{p} \theta_j Z_i^{(j)} + \varepsilon_i + \delta_i^{(1)} & \text{for all } i \in \{1, \dots, n_1 - 1\} \\ \sum_{j=1}^{p} \theta_j Z_i^{(j)} + \varepsilon_i + \delta_i^{(2)} & \text{for all } i \in \{n_2, \dots, N\}. \end{cases} $$

where $\theta = (\theta_1, \dots, \theta_p)' \in \mathbb{R}^p$ is unknown. More precisely, we will consider the following particular cases:
and there exists $C > 0$ and a positive random variable $k_0$, which do not depend on $N$, such that for any $k \ge k_0$

$$ \sum_{i=1}^{k} (\delta_{i}^{(1)})^{2} \ge Ck \quad \text{and} \quad \sum_{i=1}^{k} (\delta_{i}^{(2)})^{2} \ge Ck \quad \text{almost surely.} \qquad (3.13) $$

Then, under those assumptions, it is possible to prove that an appropriated version of the new criterion $\widehat{Cr}'_{\alpha,\beta}$ is able to determinate the "largest linear zone" (in time) in $\{1, \dots, N\}$.

**Theorem 3.2.** Under Assumptions A', B' and C', define for $1 \le t_1 \le n_0 \le t_2 - p \le N$:

$$ \widehat{Cr}'(t_1, t_2) := \frac{a_{\alpha, \beta}(t_2 - t_1)}{t_2 - t_1 - p} \| Y^{(N)}(t_1, t_2) - H(t_1, t_2) Y^{(N)}(t_1, t_2) \|^2, \qquad (3.14) $$

where $Y^{(N)}(t_1, t_2) := (Y_i^{(N)})_{t_1 \le i \le t_2}$, $H(t_1, t_2) := Z(t_1, t_2)(Z'(t_1, t_2) Z(t_1, t_2))^{-1} Z'(t_1, t_2)$ and $Z(t_1, t_2) := (Z_i^{(j)})_{t_1 \le i \le t_2, 1 \le j \le p}$ (assume that $Z(t_1, t_2)$ is a regular matrix for any $t_1 < t_2 + p$). Then, define

$$ (\hat{n}_1, \hat{n}_2) := \arg \min_{1 \le t_1 \le n_0 \le t_2 - p \le N} \widehat{Cr}'(t_1, t_2). \qquad (3.15) $$

Then for any $0 < \gamma < 1 - 2(\beta - \alpha)$,

$$ \mathbb{P}(|\hat{n}_1 - n_1| + |\hat{n}_2 - n_2| \ge N^{1-\gamma}) \xrightarrow{N \to \infty} 0. \qquad (3.16) $$

## 3.4 And when the number of breaks is unknown?

In the previous sub-sections 3.1 and 3.3, it was assumed that the number of breaks was known. However, in cases where this information is absent, it is interesting to consider the three possible scenarios:

* (B0) no break, in which case the entire point cloud can be considered linear with respect to the exogenous variables;

* (B1) one break, in which case there are two possibilities - either the linear part is present between indices 1 and $n$, or it occurs between indices $n$ and $N$;

* (B2) two breaks, in which case the framework of the previous subsection 3.3 applies.

A first idea might be to use the criterion $\widehat{Cr}'(t_1, t_2)$ defined in (3.14) and apply it in the case (B0) by setting $t_1 = 1$ and $t_2 = N$, and in the case (B1) by setting $t_1 = 1$ and minimising along $t_2$, or by setting $t_2 = N$ and minimising along $t_1$. Obviously, this idea will almost always lead to select 2 breaks, i.e. (B2).

It is therefore necessary to penalize this criterion according to the number of breaks considered. Thus, by analogy with the well-known adjusted R-squared criterion classically used in regression, we propose to consider the criterion $\widehat{Cr}'(t_1, t_2)$ penalized. This criterion, denoted $\widehat{C}_b(t_1, t_2)$, will be defined by:

$$ \widehat{C}_b(t_1, t_2) := \frac{N}{N - b p u_N} \frac{a_{\alpha, \beta}(t_2 - t_1)}{t_2 - t_1 - p} \| Y^{(N)}(t_1, t_2) - H(t_1, t_2) Y^{(N)}(t_1, t_2) \|^2, \qquad (3.17) $$
<table><caption>Table 2: Square roots of the MSE for the estimators $\hat{\tau}_1$ and $\hat{\tau}_2$ minimizing the criterion $\widehat{Cr}'$ for the two different models (MM1) and (MM2), and the different values of parameters $\sigma$ and $N$.</caption><thead><tr><th rowspan="2"></th><th rowspan="2">N</th><th colspan="2">100</th><th colspan="2">200</th><th colspan="2">400</th><th colspan="2">800</th></tr><tr><th>τ̂<sub>1</sub></th><th>τ̂<sub>2</sub></th><th>τ̂<sub>1</sub></th><th>τ̂<sub>2</sub></th><th>τ̂<sub>1</sub></th><th>τ̂<sub>2</sub></th><th>τ̂<sub>1</sub></th><th>τ̂<sub>2</sub></th></tr></thead><tbody><tr><td rowspan="2">(MM1)</td><td>σ = 1</td><td>0.0696</td><td>0.1107</td><td>0.0563</td><td>0.0930</td><td>0.0315</td><td>0.0555</td><td>0.0220</td><td>0.0405</td></tr><tr><td>σ = 2</td><td>0.0735</td><td>0.1463</td><td>0.0516</td><td>0.1279</td><td>0.0382</td><td>0.1020</td><td>0.0258</td><td>0.0775</td></tr><tr><td rowspan="2">(MM2)</td><td>σ = 1</td><td>0.0492</td><td>0.0671</td><td>0.0309</td><td>0.0383</td><td>0.0146</td><td>0.0140</td><td>0.0112</td><td>0.0057</td></tr><tr><td>σ = 2</td><td>0.0574</td><td>0.1550</td><td>0.0362</td><td>0.0811</td><td>0.0185</td><td>0.0183</td><td>0.0121</td><td>0.0080</td></tr></tbody></table>

* The size $N$ of the observed sample is such as: $N = 100, 200, 400$ and $800$.

* The error $(\varepsilon_i)$ of the homoscedastic linear model is a sequence of Gaussian centered i.i.d.r.v. with $\sigma^2 = \text{Var}(\varepsilon_0)$ is such as $\sigma = 1$ and $\sigma = 2$.

* The unknown change dates $n_1$ and $n_2$ are such as $n_1 = [0.4 \times N]$ and $n_2 = [0.7 \times N]$, and $n_0 = [0.5 \times N]$ is known.

* Two distinct frameworks are considered:

1. (MM1) Trend and volatility breaks in a time series:

$$Y_i = \begin{cases} 3 + 0.1 \times i + 2 * \varepsilon_i & \text{for all } i \in \{1, \dots, n_1 - 1\}; \\ 3 + 0.1 \times i + \varepsilon_i & \text{for all } i \in \{n_1, \dots, n_2 - 1\}; \\ 3 + 0.1 \times i + (2\sqrt{3} + (i - n) - 4) + \varepsilon_i & \text{for all } i \in \{n_2, \dots, N\}. \end{cases}$$

2. (MM2) Model changes for homoscedastic linear model:

$$Y_i = \begin{cases} -4 + 2 Z_i^{(1)} - 5 Z_i^{(2)} + 3 Z_i^{(3)} + \varepsilon_i & \text{for all } i \in \{1, \dots, n_1 - 1\}; \\ -8 - Z_i^{(1)} + 3 \log(|Z_i^{(3)}|) + \varepsilon_i & \text{for all } i \in \{n_1, \dots, n_2 - 1\}; \\ -4 + 2 Z_i^{(1)} - 5 Z_i^{(2)} + 3 Z_i^{(3)} + 2 * \varepsilon_i & \text{for all } i \in \{n_2, \dots, N\}. \end{cases}$$

where $(Z_i^{(1)})$ and $(Z_i^{(3)})$ are observed i.i.d.r.v. following respectively a uniform distribution $U([-1, 1])$ and a Student distribution $t(4)$, and $Z_i^{(2)} = (Z_i^{(1)})^2$.

We applied the $\widehat{Cr}'(t_1, t_2)$ criterion defined in (3.13) with the same data-driven choice of parameters obtained in the previous one-sided case:

$$\widehat{Cr}'(t_1, t_2) = \frac{1}{t_2 - t_1 - p} \left( 1 + \frac{N^2}{10 (t_2 - t_1 - p)^{2.1}} \right) \|Y(t_2 - t_1) - H(t_2 - t_1) Y(t_2 - t_1)\|^2. \quad (4.1)$$

Thus, the criterion $\widehat{Cr}'$ is applied and Table 2 is obtained after performing 1000 independent replications.

**Conclusions of Monte Carlo experiments:** As in the previous case, the convergence rates of the estimators seem to be close to $\sqrt{N}$. However, in contrast to the one-sided break detection, the variance $\sigma^2$ of the noise has a non-negligible influence on the convergence rate of the estimator as well as on the chosen model.
where $b = 0, 1, 2$ is the considered number of breaks, where $(u_n)$ is such that $u_n \xrightarrow{n \to \infty} \infty$. In the section devoted to numerical applications, we will use $u_n = \sqrt{n}$. As a consequence, the estimated number $\hat{b}$ of breaks will be:

$$ \hat{b} = \arg \min_{b=0,1,2} \left\{ \hat{C}_0(1,N), \min_{1+p \le t \le N-1} \{\hat{C}_1(1,t)\}, \min_{2 \le t \le N-p} \{\hat{C}_1(t,N)\}, \min_{2 \le t_1 \le t_2 \le p \le N-1} \{\hat{C}_2(t_1,t_2)\} \right\}. \quad (3.18) $$

Note here that we only calculate the minimum of $\hat{C}_1(1,t)$ for $t \le N-1$ and not for $t = N$, otherwise we would be in the case (B0) and calculating $\hat{C}_0(1,N)$. The section below on numerical applications demonstrates the effectiveness of this method of selecting the number of breaks.

# 4 Numerical applications

## 4.1 Case of a linear homoscedastic zone with breaks on one side

In this first subsection, we consider the following framework: $(Y_1^{(N)}, \dots, Y_N^{(N)})$ is observed, and there exists $n$, unknown integer number, such that

$$ Y_i^{(N)} = \begin{cases} \sum_{j=1}^{p} \theta_j Z_i^{(j)} + \varepsilon_i & \text{for all } i \in \{1, \dots, n\}; \\ \sum_{j=1}^{p} \theta_j Z_i^{(j)} + \varepsilon_i + \delta_{i-n} & \text{for all } i \in \{n+1, \dots, N\}. \end{cases} $$

where $\theta = (\theta_1, \dots, \theta_p)' \in \mathbb{R}^p$ is unknown. More precisely, we will consider the following particular cases:

* The size $N$ of the observed sample is such that: $N = 100, 200, 400, 800$ and $1600$.

* The error $(\varepsilon_i)$ is sequence of Gaussian centered i.i.d.r.v. with $\sigma^2 = \text{Var}(\varepsilon_0)$ is such that $\sigma = 1$ and $\sigma = 2$.

* The unknown change date $n$ is such that $n = [0.4 \times N]$.

* Three distinct frameworks are considered:

1. (M0) Linear change in the trend of a time series:

$$ Y_i^{(N)} = \begin{cases} 3 + 0.1 \times i + \varepsilon_i & \text{for all } i \in \{1, \dots, n\}; \\ 3 + 0.1 \times i - \frac{1}{30}(i - n) + \varepsilon_i & \text{for all } i \in \{n + 1, \dots, N\}. \end{cases} $$

2. (M1) Trend break in a time series with smooth change:

$$ Y_i^{(N)} = \begin{cases} 3 + 0.1 \times i + \varepsilon_i & \text{for all } i \in \{1, \dots, n\}; \\ 3 + 0.1 \times i + \log^2\left(1 + \frac{1}{10}(i - n)\right) + \varepsilon_i & \text{for all } i \in \{n + 1, \dots, N\}. \end{cases} $$

3. (M2) Time series volatility break:

$$ Y_i^{(N)} = \begin{cases} 3 + 0.1 \times i + \varepsilon_i & \text{for all } i \in \{1, \dots, n\}; \\ 3 + 0.1 \times i + 2\varepsilon_i & \text{for all } i \in \{n + 1, \dots, N\}. \end{cases} $$
<figure><img src="image_3.png" /><figcaption>Figure 2: Correlation dimension of a network of ravines and gullies in the French Alps: selection of the largest homoscedastic linear zone of the log-log plot $(X_i, Y_i)$ where $Y_i = \log(C(r_i))$, $X_i = \log(r_i)$ and $1 \le r_i \le 250$ in meters (right).</figcaption></figure>

## 4.3 Application to the estimation of the correlation dimension of a network of talwegs and gullies

Let us return to the real data that motivated this work and that were presented in the introduction. It concerns the estimation of a fractal dimension, the correlation dimension, in a network of ravines and gullies in the French Alps. We then apply the criterion $\hat{Cr}'$ defined in (4.1) to this data set composed of $(X_i, Y_i)_{1 \le i \le N}$ where $Y_i = \log(C(r_i))$, $X_i = \log(r_i)$ and $1 \le r_i \le 250$, and $N = 73$, the aim being to find the largest zone of homoscedastic linearity to best estimate the slope, which is the estimator of the correlation dimension of the network. Figure 2 shows the point cloud and the optimal linear portion obtained by the $\hat{Cr}'$ criterion. The results of the log-log regression are very convincing: the multiple R-squared coefficient is now 0.9998, the p-value of the Fisher test is $2 \times 10^{-16}$ and finally the estimator of the slope, which is also the estimator $\hat{D}$ of the correlation dimension, is almost 1.741.

## 5 Proofs

*Proof of Theorem 3.1.* For ease of writing, denote now:

$$n_p := n - p \quad \text{and} \quad t_p := t - p.$$

We are going to study successively the cases $t < n$ and $t > n$. Indeed, we have the property:

$$ \begin{aligned} \mathbb{P}(|\hat{\tau} - \tau| \ge N^{-\gamma}) &\le \mathbb{P}\left(\max\{\widehat{Cr}(t), t \text{ such as } |t-n| \ge N^{1-\gamma}\} < \widehat{Cr}(n)\right) \\ &\le \sum_{t=p+1}^{n-N^{1-\gamma}} \mathbb{P}(\widehat{Cr}(t) < \widehat{Cr}(n)) + \sum_{t=n+N^{1-\gamma}}^{N} \mathbb{P}(\widehat{Cr}(t) < \widehat{Cr}(n)). \end{aligned} \quad (5.1) $$
4. (M3) Model change for an homoscedastic linear model:

$$Y_i^{(N)} = \begin{cases} -4 + 2 Z_i^{(1)} - 5 Z_i^{(2)} + 3 Z_i^{(3)} + \varepsilon_i & \text{for all } i \in \{1, \dots, n\}; \\ -5 - 3 Z_i^{(1)} + 2 Z_i^{(1)} (Z_i^{(2)})^2 + \varepsilon_i & \text{for all } i \in \{n+1, \dots, N\}. \end{cases}$$

where $(Z_i^{(1)})$ and $(Z_i^{(3)})$ are observed i.i.d.r.v. following respectively a uniform distribution $U([-1, 1])$ and a Student distribution $t(4)$, and $Z_i^{(2)} = (Z_i^{(1)})^2$.

We applied the $\widehat{Cr}_{\alpha, \beta}$ criterion defined in (3.8). After an important empirical study not reproduced here, we have decided on the parameters $\alpha = 2$ and $\beta = 2.1$. Moreover, instead of using a penalty term $\frac{N^\alpha}{(t-p)^\beta}$ in the contrast definition, we observed that it is more efficient to use the penalty term $\frac{N^\alpha}{10(t-p)^\beta}$, which does not change the asymptotic results as already explained in Remark 3. Hence the chosen criterion is:

$$\widehat{Cr}(t) = \frac{1}{t-p} \left(1 + \frac{N^2}{10(t-p)^{2.1}}\right) \|Y^{(N)}(t) - H(t) Y^{(N)}(t)\|^2.$$

**Comparison with a LS criterion:** For comparison, we have also applied the parametric breakpoint estimator by minimization of a double sum of least squares, an estimator for which a detailed asymptotic study can be found in Bai and Perron 1998 [2]. As with the $\widehat{Cr}(t)$ criterion, this estimator requires the knowledge of $H(1, t)$ for any $t \in \{1, \dots, N-p\}$, but also the knowledge of another orthogonal projection matrix $H'(t, N)$ for any $t \in \{p+1, \dots, N\}$, i.e. we assume another linear model satisfied by $Y_i^{(N)}$ for $i \ge n+1$, for which we know the exogenous variables (see for example model (M0)). This is a fairly restrictive assumption, which is not required by the non-parametric $\widehat{Cr}(t)$ criterion. Generally speaking, this LSE of the breakpoint time obtained $\hat{n}_{LS}$, which is also the one obtained by maximum likelihood in our framework of $(\varepsilon_i)_i$ errors that are independent and identically distributed Gaussian random variables, is defined by:

$$\begin{align*} \hat{n}_{LS} &= \underset{p \le t \le N-p}{\arg\min} \left\{ \begin{aligned} &\|Y^{(N)}(1, t) - H(1, t) Y^{(N)}(1, t)\|^2 \\ &+ \|Y^{(N)}(t+1, N) - H'(t+1, N) Y^{(N)}(t+1, N)\|^2 \end{aligned} \right\} \\ &\implies \hat{\tau}_{LS} = \frac{\hat{n}_{LS}}{N}. \end{align*}$$

For processes (M0), (M1), (M2), and (M3), we will use this criterion and a matrix $H' = H$. This assumes a break in the coefficients of the linear models.

**Remark 5.** We have chosen a model (M1) with a very smooth break, even though it satisfies the conditions of Assumption C. In Figure 2, two trajectory plots with two N values are proposed. This model choice also allows least-squares break detection to be competitive with our nonparametric estimator of larger homoscedastic linearity zone.
1. Let $t < n$ such as $|t - n| \ge N^{1-\gamma}$, i.e. $p + 1 \le t \le [\tau N] - N^{1-\gamma}$.

Denote $J_{t,n}$ the subspace of $\mathbb{R}^n$ such as $J_{t,n} = \{(x_1, \dots, x_t, 0, \dots, 0), (x_1, \dots, x_t) \in \mathbb{R}^t\}$ and $P_{J_{t,n}}$ the orthogonal projection on $J_{t,n}$ in $\mathbb{R}^n$. Since $I(n) - H(n) = P_{[Z]_n^\perp}$, matrix of the orthogonal projection on $[Z]_n^\perp$ in $\mathbb{R}^n$, then $V_t = (I(t) - H(t))\varepsilon(t) = P_{[Z]_n^\perp} P_{J_{t,n}} \varepsilon(n)$. But we have $P_{[Z]_n^\perp} \varepsilon(n) = P_{[Z]_n^\perp} P_{J_{t,n}} \varepsilon(n) + P_{[Z]_n^\perp} P_{J_{t,n}^\perp} \varepsilon(n)$, with $W_{n-t} = P_{[Z]_n^\perp} P_{J_{t,n}} \varepsilon(n) = P_{[Z]_n^\perp \cap J_{t,n}} \varepsilon(n)$. Therefore, using Cochran' Theorem, $\|V_t\|^2$ follows a chi-square distribution with $t_p = (t-p)$ degrees of freedom and $\|W_{n-t}\|^2$ follows a chi-square distribution with $(n-t)$ degrees of freedom, where $V_t$ and $W_{n-t}$ are independent Gaussian variables.

But, since $t \le n$:

$$ 
\begin{aligned}
\mathbb{P}(\widehat{Cr}(t) < \widehat{Cr}(n)) &= \mathbb{P}\left(\frac{a_{\alpha,\beta}(t)}{t_p} \|(I(t) - H(t))\varepsilon(t)\|^2 < \frac{a_{\alpha,\beta}(n)}{n_p} \|(I(n) - H(n))\varepsilon(n)\|^2\right) \\
&= \mathbb{P}\left(\frac{a_{\alpha,\beta}(t)}{t_p} \|V_t\|^2 < \frac{a_{\alpha,\beta}(n)}{n_p} \left(\|V_t\|^2 + \|W_{n-t}\|^2\right)\right) \\
&= \mathbb{P}\left(\|W_{n-t}\|^2 - A \|V_t\|^2 > 0\right) \quad \text{with } A := \frac{n_p a_{\alpha,\beta}(t)}{t_p a_{\alpha,\beta}(n)} - 1.
\end{aligned}
$$

For $U$ a random variable such as there exists $\ell > 0$ satisfying $\mathbb{E}(e^{\ell U}) < \infty$, we have:

$$ \mathbb{P}(U > 0) \le \exp \left[ \inf_{x \ge 0} \left\{ \log \left( \mathbb{E}(e^{xU}) \right) \right\} \right]. \quad (5.2) $$

For $\xi$ a chi-square variable with $m$ freedom degrees, $\mathbb{E}(e^{x\xi}) = (1 - 2x)^{-m/2}$. Therefore, with $U = \|W_{n-t}\|^2 - A \|V_t\|^2$ and the independence between $V_t$ and $W_{n-t}$, we obtain from (5.2),

$$ 
\begin{aligned}
\mathbb{P}(\widehat{Cr}(t) > \widehat{Cr}(n)) &\le \exp\left(\inf_{x \ge 0} \left\{ \log \left[ \frac{1}{(1 + 2x A)^{t_p/2}} \times \frac{1}{(1 - 2x)^{(n_p - t_p)/2}} \right] \right\}\right) \\
&\le \exp\left(-\frac{1}{2} \sup_{0 \le x \le 1/2} \{f(x)\}\right) \\
&\quad \text{with } f(x) := t_p \log(1 + 2x A) + (n_p - t_p) \log(1 - 2x).
\end{aligned}
$$

Classical computations show that the supremum of $f$ is obtained for

$$ x_0 := \frac{t_p A - (n_p - t_p)}{2 n_p A} = \frac{1}{2} \frac{t_p (a_{\alpha,\beta}(t) - a_{\alpha,\beta}(n))}{(n_p a_{\alpha,\beta}(t) - t_p a_{\alpha,\beta}(n))}. $$

and straightforward computations show that $0 < x_0 < 1/2$. As a consequence, we obtain:

$$ 
\begin{aligned}
f(x_0) &= t_p \log \left( \frac{a_{\alpha,\beta}(t)}{a_{\alpha,\beta}(n)} \right) + (n_p - t_p) \log \left( \frac{(n_p - t_p)a_{\alpha,\beta}(t)}{n_p a_{\alpha,\beta}(t) - t_p a_{\alpha,\beta}(n)} \right) \\
&= n_p \log (1 + b_{\alpha,\beta}(n,t)) - (n_p - t_p) \log \left( 1 + \frac{n_p}{n_p - t_p} b_{\alpha,\beta}(n,t) \right) \\
&\qquad \text{with } b_{\alpha,\beta}(n,t) := \frac{a_{\alpha,\beta}(t)}{a_{\alpha,\beta}(n)} - 1 \ge 0.
\end{aligned}
$$

The function $t \in \{p+1, \dots, n\} \mapsto b_{\alpha,\beta}(n,t) = a_{\alpha,\beta}^{-1}(n)\left(1 + \frac{N^\alpha}{(t-p)^\beta}\right) - 1$ is a decreasing function. Moreover, since $\alpha < \beta$, if $t$ (or $t_p = t - p$) is negligible with respect to $N^{\alpha/\beta}$ then with $b_{\alpha,\beta}(n,t) \underset{N \to \infty}{\sim} N^\alpha t_p^{-\beta} \underset{N \to \infty}{\longrightarrow} \infty$. And if $N^{\alpha/\beta}$ is negligible with respect to $t_p$, then $b_{\alpha,\beta}(n,t) \underset{N \to \infty}{\longrightarrow} 0$.

Thus, we will now consider 3 cases inducing 3 different behaviors of $b_{\alpha,\beta}(n,t)$ and therefore of $x_0$ and
<figure><img src="image_4.png" /><figcaption>Figure 2: Two observed trajectories of the model (M1) for $N = 100$ (left) and $N = 200$ (right).</figcaption></figure>

After performing 1000 independent replications of the scatterplot for the different models and parameters, and applying the largest linear area detection criterion $\hat{Cr}$ and the LS break detector, we obtained the results recorded in Table 1:

<table><caption>Table 1: Square roots of the MSE for the estimators $\hat{\tau}$ and $\hat{\tau}_{LS}$ for the four different models (M0), (M1), (M2) and (M3), and the different values of parameters $\sigma$ and $N$.</caption><thead><tr><th></th><th></th><th>N</th><th>100</th><th>200</th><th>400</th><th>800</th></tr></thead><tbody><tr><td rowspan="4">(M0)</td><td rowspan="2">σ = 1</td><td>τ̂</td><td>0.380</td><td>0.241</td><td>0.080</td><td>0.050</td></tr><tr><td>τ̂<sub>LS</sub></td><td>0.278</td><td>0.129</td><td>0.050</td><td>0.022</td></tr><tr><td rowspan="2">σ = 2</td><td>τ̂</td><td>0.406</td><td>0.387</td><td>0.197</td><td>0.063</td></tr><tr><td>τ̂<sub>LS</sub></td><td>0.319</td><td>0.234</td><td>0.091</td><td>0.039</td></tr><tr><td rowspan="4">(M1)</td><td rowspan="2">σ = 1</td><td>τ̂</td><td>0.258</td><td>0.099</td><td>0.061</td><td>0.036</td></tr><tr><td>τ̂<sub>LS</sub></td><td>0.164</td><td>0.082</td><td>0.058</td><td>0.047</td></tr><tr><td rowspan="2">σ = 2</td><td>τ̂</td><td>0.370</td><td>0.285</td><td>0.093</td><td>0.044</td></tr><tr><td>τ̂<sub>LS</sub></td><td>0.272</td><td>0.138</td><td>0.068</td><td>0.049</td></tr><tr><td rowspan="4">(M2)</td><td rowspan="2">σ = 1</td><td>τ̂</td><td>0.102</td><td>0.064</td><td>0.042</td><td>0.038</td></tr><tr><td>τ̂<sub>LS</sub></td><td>0.410</td><td>0.438</td><td>0.462</td><td>0.474</td></tr><tr><td rowspan="2">σ = 2</td><td>τ̂</td><td>0.105</td><td>0.077</td><td>0.041</td><td>0.029</td></tr><tr><td>τ̂<sub>LS</sub></td><td>0.407</td><td>0.426</td><td>0.443</td><td>0.470</td></tr><tr><td rowspan="4">(M3)</td><td rowspan="2">σ = 1</td><td>τ̂</td><td>0.092</td><td>0.059</td><td>0.041</td><td>0.034</td></tr><tr><td>τ̂<sub>LS</sub></td><td>0.150</td><td>0.144</td><td>0.142</td><td>0.115</td></tr><tr><td rowspan="2">σ = 2</td><td>τ̂</td><td>0.087</td><td>0.071</td><td>0.054</td><td>0.037</td></tr><tr><td>τ̂<sub>LS</sub></td><td>0.145</td><td>0.146</td><td>0.144</td><td>0.129</td></tr></tbody></table>

**Conclusions of Monte Carlo experiments:** We first observe the convergence of the estimator $\hat{\tau}$ when $N$ becomes large and its rate of convergence is close to be $\sqrt{N}$. We also notice three important points:
$f(x_0)$: $t_p \in \{[n_p/2], [n_p/2]+1, \dots, n_p - [N^{1-\gamma}]\}$, $t_p \in \{[(3N^\alpha)^{1/\beta}], [(3N^\alpha)^{1/\beta}]+1, \dots, [n_p/2]-1\}$ and $t_p \in \{1, 2, \dots, [(3N^\alpha)^{1/\beta}] - 1\}$.

Since $\alpha < \beta$, we will also use a Taylor-Lagrange expansion of $b_{\alpha,\beta}(n,t)$, which implies that for any $1 \le t_p \le n_p$, there exists $\theta_{t_p,n_p} \in [t_p, n_p]$ such as

$$b_{\alpha,\beta}(n,t) \underset{N \to \infty}{\sim} N^{\alpha} (t_p^{-\beta} - n_p^{-\beta}) \underset{N \to \infty}{\sim} \beta \frac{N^{\alpha} \theta_{t_p, n_p}^{\beta-1}}{n_p^{\beta} t_p^{\beta}} (n_p - t_p). \quad (5.3)$$

i. Let $t_p \in \{[n_p/2], [n_p/2]+1, \dots, n_p - [N^{1-\gamma}]\}$. Then $b_{\alpha,\beta}(n,t) \le (3^\beta - 1) N^\alpha (\tau N)^{-\beta} \underset{N \to \infty}{\longrightarrow} 0$ and from (5.3), $n_p(n_p - t_p)^{-1} b_{\alpha,\beta}(n,t) \underset{N \to \infty}{\longrightarrow} 0$. Therefore with Taylor expansions, for $N$ large enough,

$$\begin{align*}
f(x_0) &\underset{N \to \infty}{\sim} \frac{1}{2} \frac{t_p n_p}{n_p - t_p} b_{\alpha, \beta}^2(n, t) \\
&\underset{N \to \infty}{\sim} \frac{\beta^2}{2} \frac{N^{2\alpha} \theta_{t_p, n_p}^{2\beta-2}}{n_p^{2\beta-1} t_p^{2\beta-1}} (n_p - t_p) \\
&\ge \frac{\beta^2}{2^\beta} N^{2\alpha} n_p^{2\beta} (n_p - t_p) \\
&\ge \frac{\beta^2}{2^\beta \tau^{2\beta}} N^{2\alpha-2\beta} N^{1-\gamma}.
\end{align*}$$

Then for $t_p \in \{[n_p/2], [n_p/2] + 1, \dots, n_p - [N^{1-\gamma}]\}$ and $N$ large enough, using the assumption $1 - \gamma - 2(\beta - \alpha) > 0$,

$$\mathbb{P}(\widehat{Cr}(t) > \widehat{Cr}(n)) \le \exp\left(-\frac{\beta^2}{2^{\beta+1}\tau^{2\beta}} N^{1-\gamma-2(\beta-\alpha)}\right). \quad (5.4)$$

ii. Let $t_p \in \{[(3N^\alpha)^{1/\beta}], [(3N^\alpha)^{1/\beta}] + 1, \dots, [n_p/2] - 1\}$. Simple computations imply that $0 \le b_{\alpha,\beta}(n,t) < \frac{1}{2}$ when $N$ is large enough. From the definition of $f(x_0)$, we obtain:

$$\begin{align}
f(x_0) &= t_p \log (1 + b_{\alpha,\beta}(n,t)) - (n_p - t_p) \log \left(1 + \frac{t_p}{n_p - t_p} \frac{b_{\alpha,\beta}(n,t)}{1 + b_{\alpha,\beta}(n,t)}\right) \nonumber \\
&\ge t_p \left(\log (1 + b_{\alpha,\beta}(n,t)) - \frac{b_{\alpha,\beta}(n,t)}{1 + b_{\alpha,\beta}(n,t)}\right), \tag{5.5}
\end{align}$$

using $\log(1+u) \le u$ for $u > -1$. Using also $\log(1+u) \ge u - \frac{1}{2}u^2$ for $u \ge 0$, we obtain from (5.5),

$$\begin{align}
f(x_0) &\ge t_p \left( b_{\alpha,\beta}(n,t) - \frac{1}{2} b_{\alpha,\beta}^2(n,t) - \frac{b_{\alpha,\beta}(n,t)}{1 + b_{\alpha,\beta}(n,t)} \right) \nonumber \\
&\ge \frac{1}{2} t_p \frac{(1 - b_{\alpha,\beta}(n,t)) b_{\alpha,\beta}^2(n,t)}{(1 + b_{\alpha,\beta}(n,t))} \nonumber \\
&\ge \frac{1}{6} t_p b_{\alpha,\beta}^2(n,t). \tag{5.6}
\end{align}$$

Using (5.3), $\beta > 1$ and $t_p \in \{[(3N^\alpha)^{1/\beta}], [(3N^\alpha)^{1/\beta}] + 1, \dots, [n_p/2] - 1\}$, we deduce for $N$ large enough,

$$b_{\alpha,\beta}(n,t) \ge \frac{1}{2} \beta \frac{N^\alpha}{n_p^\beta t_p} (n_p - t_p) \ge \frac{\beta}{2 \tau^\beta} N^{\alpha-\beta}.$$
* The variance $\sigma^2$ of the noise has only a small influence on the speed of convergence of the estimator when $N$ becomes large enough, but it often has a strong impact when $N$ is 100 or 200;

* A fixed $N$, the root of the MSE is almost the same whatever the chosen model, except for model (M0).

* if we had considered a pure linear model change where the variables involved in the model are known a priori, i.e. a parametric break detection, then the least squares estimator $\hat{\tau}_{LS}$, which is defined for such a framework, gives better convergence results than $\hat{\tau}$ and this is such the case for the model (M0). Otherwise, $\hat{\tau}_{LS}$ is used in misspecified frameworks, since there is a model change after the break that is not known in advance. However, it provides competitive results with the $\hat{\tau}$ estimator when $N$ is small for the (M1) and (M3) models. However, it is not expected to converge. For the (M2) model, where there is a change in volatility, it is quite clear that the $\hat{\tau}_{LS}$ estimator is not at all capable of detecting a change, and we cannot be surprised by the disastrous results obtained.

## 4.2 Case of a linear homoscedastic zone with breaks on both sides

Now we consider the following framework: $(Y_1^{(N)}, \dots, Y_N^{(N)})$ is observed, and there exists $n_1$ and $n_2$, unknown integer numbers, such that

$$ Y_i^{(N)} = \begin{cases} \sum_{j=1}^{p} \theta_j Z_i^{(j)} + \varepsilon_i & \text{for all } i \in \{n_1, \dots, n_2\}; \\ \sum_{j=1}^{p} \theta_j Z_i^{(j)} + \varepsilon_i + \delta_{n_1-i}^{(1)} & \text{for all } i \in \{1, \dots, n_1-1\}; \\ \sum_{j=1}^{p} \theta_j Z_i^{(j)} + \varepsilon_i + \delta_{i-n_2}^{(2)} & \text{for all } i \in \{n_2+1, \dots, N\}. \end{cases} $$

where $\theta = (\theta_1, \dots, \theta_p)' \in \mathbb{R}^p$ is unknown. More precisely, we will consider the following particular cases:

* The size $N$ of the observed sample is such that: $N = 100, 200, 400$ and $800$.

* The error $(\varepsilon_i)$ is sequence of Gaussian centered i.i.d.r.v. with $\sigma^2 = \text{Var}(\varepsilon_0)$ is such that $\sigma = 1$ and $\sigma = 2$.

* The unknown change dates $n_1$ and $n_2$ are such that $n_1 = [0.4 \times N]$ and $n_2 = [0.7 \times N]$, and $n_0 = [0.5 \times N]$ is known.

* Two distinct frameworks are considered:

1. (MM1) Trend and volatility breaks in a time series:

$$ Y_i^{(N)} = \begin{cases} 3 + 0.1 \times i + 2 * \varepsilon_i & \text{for all } i \in \{1, \dots, n_1 - 1\}; \\ 3 + 0.1 \times i + \varepsilon_i & \text{for all } i \in \{n_1, \dots, n_2\}; \\ 3 + 0.1 \times i + (2\sqrt{3 + (i - n_2) - 4}) + \varepsilon_i & \text{for all } i \in \{n_2 + 1, \dots, N\}. \end{cases} $$

2. (MM2) Model changes for homoscedastic linear model:

$$ Y_i^{(N)} = \begin{cases} -4 + 2 Z_i^{(1)} - 5 Z_i^{(2)} + 3 Z_i^{(3)} + \varepsilon_i & \text{for all } i \in \{1, \dots, n_1 - 1\}; \\ -8 - Z_i^{(1)} + 3 \log(|Z_i^{(3)}|) + \varepsilon_i & \text{for all } i \in \{n_1, \dots, n_2\}; \\ -4 + 2 Z_i^{(1)} - 5 Z_i^{(2)} + 3 Z_i^{(3)} + 2 * \varepsilon_i & \text{for all } i \in \{n_2 + 1, \dots, N\}. \end{cases} $$
Therefore, from (5.6) and always with $N$ large enough:

$$ 
\begin{aligned}
f(x_0) &\geq \frac{1}{6} (3 N^{\alpha})^{1/\beta} \left(\frac{\beta}{2 \tau^{\beta}} N^{\alpha-\beta}\right)^2 \\
&\geq \frac{3^{1/\beta} \beta^2}{24 \tau^{2\beta}} N^{\alpha/\beta-2(\beta-\alpha)}.
\end{aligned}
$$

Then if $t_p \in \{[(3 N^{\alpha})^{1/\beta}], [(3 N^{\alpha})^{1/\beta}] + 1, \dots, [n_p/2] - 1\}$ and for $N$ large enough,

$$ \mathbb{P}(\widehat{Cr}(t) > \widehat{Cr}(n)) \leq \exp\left(-\frac{3^{1/\beta} \beta^2}{48 \tau^{2\beta}} N^{\alpha/\beta - 2(\beta-\alpha)}\right), \quad (5.7) $$

which explains the condition $\frac{\alpha}{2\beta} > \beta - \alpha$.

iii. Let $t_p \in \{1, 2, \dots, [(3 N^{\alpha})^{1/\beta}] - 1\}$. Simple computations imply that $b_{\alpha,\beta}(n,t) > \frac{1}{4}$ when $N$ is large enough. But it can be easily proved that $\log(1+u) - u(1+u)^{-1} \ge \frac{1}{10} \log(1+u)$ for all $u > 1/4$. Therefore, from (5.5),

$$ f(x_0) \geq \frac{1}{10} t_p \log (1 + b_{\alpha,\beta}(n,t)). $$

But for $t_p \in \{1, 2, \dots, [(3 N^{\alpha})^{1/\beta}] - 1\}$ and when $N$ is large enough, $b_{\alpha,\beta}(n,t) \ge \frac{1}{2} \frac{N^{\alpha}}{t_p^{\beta}}$ and thus

$$ f(x_0) \geq \frac{1}{10} t_p \log \left(1 + \frac{1}{2} \frac{N^{\alpha}}{t_p^{\beta}}\right). $$

As a consequence, $t_p \in \{1, 2, \dots, [(3 N^{\alpha})^{1/\beta}] - 1\}$ and when $N$ is large enough,

$$ \mathbb{P}(\widehat{Cr}(t) > \widehat{Cr}(n)) \leq \left(1 + \frac{1}{2} \frac{N^{\alpha}}{t_p^{\beta}}\right)^{-t_p/20}. \quad (5.8) $$

2. Let $t > n$ such as $|t - n| > N^{1-\gamma}$, i.e. $[\tau N] + N^{1-\gamma} \le t \le N$. Then:

$$ 
\begin{aligned}
\mathbb{P}(\widehat{Cr}(t) < \widehat{Cr}(n)) &= \mathbb{P}\left(\frac{a_{\alpha,\beta}(t)}{t_p} \left\|(I(t) - H(t))(\varepsilon(t) + D(t))\right\|^2 < \frac{a_{\alpha,\beta}(n)}{n_p} \left\|(I(n) - H(n))\varepsilon(n)\right\|^2\right) \\
&= \mathbb{P}\left(\frac{a_{\alpha,\beta}(t)}{t_p} \left\|(V_n + W_{t-n}) + P_{[Z]_t^\perp} D(t)\right\|^2 < \frac{a_{\alpha,\beta}(n)}{n_p} \left\|V_n\right\|^2\right) \\
&= \mathbb{P}\left(\frac{a_{\alpha,\beta}(t)}{t_p} \left(\left\|V_n + P_{[Z]_t^\perp} D(t)\right\|^2 + \left\|W_{t-n}\right\|^2\right) < \frac{a_{\alpha,\beta}(n)}{n_p} \left\|V_n\right\|^2\right) \\
&= \mathbb{P}\left(\left(\left\|V_n - B P_{[Z]_t^\perp} D(t)\right\|^2 - B(B+1) \left\|P_{[Z]_t^\perp} D(t)\right\|^2 - B \left\|W_{t-n}\right\|^2\right) > 0\right),
\end{aligned}
$$

where $B = n_p a_{\alpha,\beta}(t) (t_p a_{\alpha,\beta}(n) - n_p a_{\alpha,\beta}(t))^{-1}$, and $V_n$ and $W_{t-n}$ defined as in the first part of the proof and $D(t)$ defined in (3.5). The conditional probability given $D(t)$ can be bounded using again (5.2). Indeed, given $D(t)$, $\sigma^{-2}\|V_n - B P_{[Z]_t^\perp} D(t)\|^2$ follows a noncentral chi-squared distribution with $n_p$ degrees of freedom, independent to $\sigma^{-2}\|W_{t-n}\|^2$ which follows a chi-squared distribution with $(t_p - n_p)$ degrees of freedom. Moreover, given $D(t)$, the moment generating function of $\sigma^{-2}\|V_n - B P_{[Z]_t^\perp} D(t)\|^2$ is given by $(1-2x)^{n_p/2} \exp(\lambda x (1-2x)^{-1})$ where $\lambda = \sigma^{-2}B^2 \|P_{[Z]_t^\perp} D(t)\|^2$.
where $(Z_i^{(1)})$ and $(Z_i^{(3)})$ are observed i.i.d.r.v. following respectively a uniform distribution $U([-1, 1])$ and a Student distribution $t(4)$, and $Z_i^{(2)} = (Z_i^{(1)})^2$.

We applied the $\widehat{Cr}'(t_1, t_2)$ criterion defined in (3.14) with the same data-driven choice of parameters obtained in the previous one-sided case:

$$ \widehat{Cr}'(t_1, t_2) = \frac{1}{t_2 - t_1 - p} \left(1 + \frac{N^2}{10(t_2 - t_1 - p)^{2.1}}\right) \|Y^{(N)}(t_2 - t_1) - H(t_2 - t_1) Y^{(N)}(t_2 - t_1)\|^2. \quad (4.1) $$

Thus, the criterion $\widehat{Cr}'$ is applied and Table 2 is obtained after performing 1000 independent replications.

<table><caption>Table 2: Square roots of the MSE for the estimators $\hat{\tau}_1$ and $\hat{\tau}_2$ minimizing the criterion $\widehat{Cr}'$ for the two different models (MM1) and (MM2), and the different values of parameters $\sigma$ and $N$.</caption><thead><tr><th rowspan="2"></th><th rowspan="2">N</th><th colspan="2">100</th><th colspan="2">200</th><th colspan="2">400</th><th colspan="2">800</th></tr><tr><th>τ̂<sub>1</sub></th><th>τ̂<sub>2</sub></th><th>τ̂<sub>1</sub></th><th>τ̂<sub>2</sub></th><th>τ̂<sub>1</sub></th><th>τ̂<sub>2</sub></th><th>τ̂<sub>1</sub></th><th>τ̂<sub>2</sub></th></tr></thead><tbody><tr><td rowspan="2">(MM1)</td><td>σ = 1</td><td>0.0696</td><td>0.1107</td><td>0.0563</td><td>0.0930</td><td>0.0315</td><td>0.0555</td><td>0.0220</td><td>0.0405</td></tr><tr><td>σ = 2</td><td>0.0735</td><td>0.1463</td><td>0.0516</td><td>0.1279</td><td>0.0382</td><td>0.1020</td><td>0.0258</td><td>0.0775</td></tr><tr><td rowspan="2">(MM2)</td><td>σ = 1</td><td>0.0492</td><td>0.0671</td><td>0.0309</td><td>0.0383</td><td>0.0146</td><td>0.0140</td><td>0.0112</td><td>0.0057</td></tr><tr><td>σ = 2</td><td>0.0574</td><td>0.1550</td><td>0.0362</td><td>0.0811</td><td>0.0185</td><td>0.0183</td><td>0.0121</td><td>0.0080</td></tr></tbody></table>

**Conclusions of Monte Carlo experiments:** We first observe the convergence of the estimators $\hat{\tau}_1$ and $\hat{\tau}_2$ as $N$ becomes large. We can also see that the variance of the noise has a greater effect on the estimation quality of the breakpoints than in the case of a single break. This is not surprising since we have mixed model changes and volatility changes in the models.

## 4.3 Case of a linear homoscedastic zone with unknown number of breaks

Here we are going to use some of the previously simulated time series, choosing that 0, 1 or 2 breaks are present, but without knowing in advance the number of these breaks. We will then apply a criterion $\widehat{C}_b$ defined as in (3.17). More precisely we used:

$$ \widehat{C}_b(t_1, t_2) := \frac{N}{N - b p \sqrt{N}} \frac{1}{(t_2 - t_1 - p)} \left(1 + \frac{N^2}{10(t_2 - t_1 - p)^{2.1}}\right) \|Y^{(N)}(t_2 - t_1) - H(t_2 - t_1) Y^{(N)}(t_2 - t_1)\|^2, $$

where $b = 0, 1, 2$ is the considered number of breaks. The estimator $\hat{b}$ if the number of breaks is defined as in (3.18).

We will therefore consider the following 3 frameworks of observed trajectory $(Y_1^{(N)}, \dots, Y_N^{(N)})$:

* Case B0: There is no break and

$$ Y_i^{(N)} = 3 + 0.1 \times i + \varepsilon_i \quad \text{for all } i \in \{1, \dots, N\} $$

* Case B1: There is 1 break and

$$ Y_i^{(N)} = \begin{cases} 3 + 0.1 \times i + \varepsilon_i & \text{for all } i \in \{1, \dots, n\}; \\ 3 + 0.1 \times i + (2\sqrt{3 + (i - n)} - 4) + \varepsilon_i & \text{for all } i \in \{n + 1, \dots, N\}. \end{cases} $$
Thus,

$$
\begin{align*}
\mathbb{P}(\widehat{Cr}(t) < \widehat{Cr}(n) | D(t)) &\le \exp\left(\inf_{x \ge 0} \left\{ \log \left[ \frac{\exp\left(\frac{B^2 \|P_{[Z]_t}^{\perp} D(t)\|^2 x}{\sigma^2(1-2x)}\right)}{(1-2x)^{n_p/2}} \times \frac{\exp\left(-x \frac{B(B+1) \|P_{[Z]_t}^{\perp} D(t)\|^2}{\sigma^2}\right)}{(1+2Bx)^{(t_p-n_p)/2}} \right] \right\}) \\
&\le \exp\left(-\frac{1}{2} \sup_{0 \le x \le 1/2} \{g(x)\}\right)
\end{align*}
$$

$$
\text{with } g(x) := n_p \log(1-2x) + (t_p - n_p) \log(1+2Bx) + \frac{2B \|P_{[Z]_t}^{\perp} D(t)\|^2 x}{\sigma^2} \frac{(1-2(B+1)x)}{1-2x}.
$$

Using $n_p \sim \tau N$ and $t_p > n_p$, we obtain the following expansion:

$$
B = \frac{n_p}{t_p - n_p} \left( 1 - \beta \frac{N^\alpha}{n_p^\beta} (1 + o(1)) \right).
$$

As a consequence we have $B \sim \frac{n_p}{t_p - n_p}$ and also $B \ge \frac{\tau}{2(1-\tau)}$ for $N$ large enough. Moreover, for any $u \in [-1/2, 1]$, $\log(1+u) \ge u - u^2$. Therefore, for $x$ such as $x \in [0, \max(1/4; 1/2B)]$ and $N$ large enough,

$$
\begin{align*}
g(x) &\ge n_p (-2x - 4x^2) + (t_p - n_p)(2Bx - 4B^2x^2) + \frac{2B \|P_{[Z]_t}^{\perp} D(t)\|^2 x}{\sigma^2} (1 - 2(B+1)x) \\
&\ge -4\beta \frac{N^\alpha}{n_p^\beta} (t_p - n_p)x - 4 \frac{t_p n_p}{t_p - n_p} x^2 + \frac{C}{\sigma^2} (t_p - n_p) B x (1 - 2(B+1)x),
\end{align*}
$$

using Assumption C and Lemma 1, implying $\|P_{[Z]_t}^{\perp} D(t)\|^2 \ge \frac{1}{2} C (t_p - n_p)$ for $N$ large enough. For
a given $\varepsilon > 0$ such as $\varepsilon \le \max(\frac{\tau}{8(1-\tau)}; \frac{1}{2})$, for $N$ large enough,

$$
\begin{align*}
g(\varepsilon/B) &\geq -4\beta \frac{N^\alpha}{n_p^\beta} \frac{(t_p - n_p)^2}{n_p} \varepsilon - 4 \frac{t_p}{n_p} (t_p - n_p) \varepsilon^2 + \frac{C}{\sigma^2} \left(1 - 2\varepsilon - 4 \frac{1 - \tau}{\tau} \varepsilon\right) (t_p - n_p) \varepsilon \\
&\geq -4\beta \frac{1 - \tau}{\tau} \frac{N^\alpha}{n_p^\beta} (t_p - n_p) \varepsilon - \frac{4}{\tau} (t_p - n_p) \varepsilon^2 + \frac{C}{\sigma^2} \left(1 - \frac{4}{\tau} \varepsilon\right) (t_p - n_p) \varepsilon \\
&\geq (t_p - n_p) \varepsilon \left(-\frac{5}{\tau} \varepsilon + \frac{C}{\sigma^2} \left(1 - \frac{4}{\tau} \varepsilon\right)\right).
\end{align*}
$$

Therefore, for $\varepsilon < C \tau (4C + 5\sigma^2)^{-1}$, for instance $\varepsilon = \frac{C}{2} \tau (4C + 5\sigma^2)^{-1}$, we obtain:

$$
\sup_{0 \le x \le 1/2} \{g(x)\} \ge g(\varepsilon/B) \ge \frac{C^2 \tau}{4\sigma^2 (4C + 5\sigma^2)} (t_p - n_p).
$$

Then, for $[\tau N] + N^{1-\gamma} \le t \le N$ and $N$ large enough,

$$
\begin{align*}
\mathbb{P}(\widehat{Cr}(t) < \widehat{Cr}(n) | D(t)) &\le \exp\left(-\frac{C^2 \tau}{8\sigma^2 (4C + 5\sigma^2)} (t_p - n_p)\right) \\
\implies \mathbb{P}(\widehat{Cr}(t) < \widehat{Cr}(n)) &\le \exp\left(-\frac{C^2 \tau}{8\sigma^2 (4C + 5\sigma^2)} N^{1-\gamma}\right). \tag{5.9}
\end{align*}
$$

Finally, using (5.1), (5.4), (5.7), (5.8) and (5.9) we deduce

$$
\begin{align*}
\mathbb{P}(|\hat{\tau} - \tau| \ge N^{-\gamma}) &\le \sum_{t_p=1}^{[(3N^\alpha)^{1/\beta}]-1} \left(1 + \frac{1}{2} \frac{N^\alpha}{t_p^\beta}\right)^{-t_p/20} \\
&\quad + \sum_{t_p=[(3N^\alpha)^{1/\beta}]}^{[n_p/2]-1} \exp\left(-\frac{3^{1/\beta} \beta^2}{48 \tau^{2\beta}} N^{\alpha/\beta - 2(\beta-\alpha)}\right) \\
&\quad + \sum_{t_p=[n_p/2]}^{n_p - [N^{1-\gamma}]} \exp\left(-\frac{\beta^2}{8\tau^{2\beta}} N^{1-\gamma - 2(\beta-\alpha)}\right) \\
&\quad + \sum_{t_p=n_p + [N^{1-\gamma}]}^{N-p} \exp\left(-\frac{C^2 \tau}{8\sigma^2 (4C + 5\sigma^2)} N^{1-\gamma}\right) \\
&\le C N^{-\alpha/20} \quad \underset{N \to \infty}{\longrightarrow} \quad 0, \tag{5.10}
\end{align*}
$$
• Case B2: There is 2 breaks and

$$
Y_i^{(N)} = \begin{cases} 3 + 0.1 \times i + 2 \times \varepsilon_i & \text{for all } i \in \{1, \dots, n_1\} \\ 3 + 0.1 \times i + \varepsilon_i & \text{for all } i \in \{n_1 + 1, \dots, n_2\}; \\ 3 + 0.1 \times i + (2\sqrt{3 + (i - n_2)} - 4) + \varepsilon_i & \text{for all } i \in \{n_2 + 1, \dots, N\}. \end{cases}
$$

Thus, the criterion $\widehat{Cr}_b$ is applied for $b = 0, 1$ and $2$ and the estimator $\widehat{b}$ is obtained from the minimum as in in (3.18). Table 3 is obtained after performing 1000 independent replications.

<table><caption>Table 3: Relative frequencies $\hat{f}_b$ of obtaining $b$ breaks for $b = 0, 1$ and $2$, in the cases of the models B0, B1, and B2 and different values of $N$.</caption>
  <thead>
    <tr>
      <th rowspan="2">N</th>
      <th colspan="3">100</th>
      <th colspan="3">200</th>
      <th colspan="3">400</th>
      <th colspan="3">800</th>
    </tr>
    <tr>
      <th>f&#x0302;<sub>0</sub></th>
      <th>f&#x0302;<sub>1</sub></th>
      <th>f&#x0302;<sub>2</sub></th>
      <th>f&#x0302;<sub>0</sub></th>
      <th>f&#x0302;<sub>1</sub></th>
      <th>f&#x0302;<sub>2</sub></th>
      <th>f&#x0302;<sub>0</sub></th>
      <th>f&#x0302;<sub>1</sub></th>
      <th>f&#x0302;<sub>2</sub></th>
      <th>f&#x0302;<sub>0</sub></th>
      <th>f&#x0302;<sub>1</sub></th>
      <th>f&#x0302;<sub>2</sub></th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>B0</td>
      <td>0.504</td>
      <td>0.316</td>
      <td>0.180</td>
      <td>0.715</td>
      <td>0.245</td>
      <td>0.040</td>
      <td>0.815</td>
      <td>0.154</td>
      <td>0.031</td>
      <td>0.899</td>
      <td>0.101</td>
      <td>0</td>
    </tr>
    <tr>
      <td>B1</td>
      <td>0</td>
      <td>0.714</td>
      <td>0.286</td>
      <td>0</td>
      <td>0.835</td>
      <td>0.165</td>
      <td>0</td>
      <td>0.915</td>
      <td>0.085</td>
      <td>0</td>
      <td>0.971</td>
      <td>0.029</td>
    </tr>
    <tr>
      <td>B2</td>
      <td>0</td>
      <td>0.488</td>
      <td>0.512</td>
      <td>0</td>
      <td>0.497</td>
      <td>0.503</td>
      <td>0</td>
      <td>0.269</td>
      <td>0.731</td>
      <td>0</td>
      <td>0.106</td>
      <td>0.894</td>
    </tr>
  </tbody>
</table>

**Conclusions of Monte Carlo experiments:** Note that as $N$ increases, the estimator $\hat{b}$ seems to converge to the true value of $b$, i.e., the true number of breaks: the procedure for choosing the number of breaks seems to be consistent.

4.4 Application to real datasets

4.4.1 Application to the estimation of the correlation dimension of a network of talwegs and gullies

Let us return to the real data that motivated this work and that were presented in the introduction.
It concerns the estimation of a fractal dimension, the correlation dimension, in a network of ravines
and gullies in the French Alps. We then apply the criterion $\widehat{Cr}'$ defined in (4.1) to this data set
composed of $(X_i, Y_i)_{1\le i\le N}$ where $Y_i = \log(C(r_i))$, $X_i = \log(r_i)$ and $1 \le r_i \le 250$, and $N = 73$, the
aim being to find the largest zone of homoscedastic linearity to best estimate the slope, which is
the estimator of the correlation dimension of the network. Figure 3 shows the point cloud and the
optimal linear portion obtained by the criterion $\widehat{Cr}'$.
with $C > 0$.

<figure><img src="image_4.png" /></figure>

*Proof of Corollary 1.* If $0 < \varepsilon < \alpha/\beta$, then using the previous proof we have for $N$ large enough,

$$
\begin{align*}
\mathbb{P}(|\tilde{n} - n| \ge N^{1-\gamma}) &\le \sum_{t_p=N^\varepsilon}^{[(3N^\alpha)^{1/\beta}]-1} \left(1 + \frac{1}{2} \frac{N^\alpha}{t_p^\beta}\right)^{-t_p/20} + \sum_{t_p=[(3N^\alpha)^{1/\beta}]}^{[n_p/2]-1} \exp\left(-\frac{3^{1/\beta} \beta^2}{48 \tau^{2\beta}} N^{\alpha/\beta - 2(\beta-\alpha)}\right) \\
&\quad + \sum_{t_p=[n_p/2]}^{n_p - [N^{1-\gamma}]} \exp\left(-\frac{\beta^2}{8\tau^{2\beta}} N^{1-\gamma - 2(\beta-\alpha)}\right) + \sum_{t_p=n_p + [N^{1-\gamma}]}^{N-p} \exp\left(-\frac{C^2 \tau}{8\sigma^2 (4C + 5\sigma^2)} N^{1-\gamma}\right) \\
&\le N \left[e^{-C_1 N^\varepsilon} + e^{-C_2 N^{\alpha/\beta - 2(\beta-\alpha)}} + e^{-C_3 N^{1-\gamma - 2(\beta-\alpha)}} + e^{-C_4 N^{1-\gamma}}\right] \\
&\le e^{-C N^\rho},
\end{align*}
$$

where $C_1, C_2, C_3, C_4$ and $C$ are positive real numbers and $0 < \rho < \min(\varepsilon, 1 - \gamma - 2(\beta - \alpha))$.
The case $\varepsilon \ge \alpha/\beta$ can be studied identically except that the first sum is omitted.

<figure><img src="image_5.png" /></figure>

*Proof of Corollary 2.* Let $x \in \mathbb{R}$ and denote $W_N = (Z'(\hat{n} - N^\delta) Z(\hat{n} - N^\delta))^{1/2} (\tilde{\theta}_N - \theta)$. Then:

$$
\mathbb{P}(W_N \le x) = \mathbb{P}(W_N \le x \mid \hat{n} - N^{\delta} \le n) \mathbb{P}(\hat{n} - N^{\delta} \le n) + \mathbb{P}(W_N \le x \cap \hat{n} - N^{\delta} \le n).
$$

But given $\hat{n} - N^{\delta} \le n$, under Assumption D, it is well known (see for instance Huber, 1981) that $\tilde{\theta}_N$ is asymptotically Gaussian and therefore $W_N \xrightarrow[N \to \infty]{} \mathcal{N}(0, \sigma^2)$. Moreover, from the proof of Theorem 3.1, $\mathbb{P}(\hat{n} \ge n + N^{1-\gamma}) \le N \exp(-C_4 N^{1-\gamma})$ with $C_4 > 0$ and therefore

$$
\mathbb{P}(\hat{n} - N^{\delta} \le n) = 1 - \mathbb{P}(\hat{n} \le n + N^{\delta}) \xrightarrow{N \to \infty} 1.
$$

Moreover $\mathbb{P}(W_N \le x \cap \hat{n} - N^{\delta} \le n) \le \mathbb{P}(\hat{n} - N^{\delta} \le n) \xrightarrow{N \to \infty} 0$. Thus, with $F_W$ the cumulative distribution function of $W \sim \mathcal{N}'0, \sigma^2)$, we have

$$
\mathbb{P}(W_N \le x) \xrightarrow{N \to \infty} F_W(x),
$$

and the asymptotic normality of $\tilde{\theta}_N$ can be deduced. The convergence $\tilde{\theta}_N \xrightarrow{N \to \infty} \theta$ is just a consequence of the asymptotic normality with Assumption D.

<figure><img src="image_6.png" /></figure>

*Proof of Theorem 3.2.* From Assumption B', we deduce that we can consider $N$ large enough, such as $n_0 \ge n_1 + N^{1-\gamma}$ and $n_0 + p \le n_2 - N^{1-\gamma}$. Now, using the notation $c(t_1, t_2) = \mathbb{P}(\hat{Cr}'(t_1, t_2) < \hat{Cr}'(n_1, n_2))$ and as it was already established in the proof of Theorem (3.1), we have

$$
\begin{align*}
& \mathbb{P}(|\hat{n}_1 - n_1| + |\hat{n}_2 - n_2| \ge N^{1-\gamma}) \\
&\le \mathbb{P}\left(\max\left\{\hat{Cr}'(t_1, t_2), t_1 \le n_0 \le t_2 - p, |t_1 - n_1| \ge N^{1-\gamma} \text{ and } |t_2 - n_2| \ge N^{1-\gamma}\right\} < \hat{Cr}'(n_1, n_2)\right) \\
&\le \sum_{t_1=p}^{n_1-p-N^{1-\gamma}} \left\{ \sum_{t_2=n_0+p}^{n_2-N^{1-\gamma}} c(t_1, t_2) + \sum_{t_2=n_2+N^{1-\gamma}}^{N} c(t_1, t_2) \right\} \\
&\quad + \sum_{t_1=n_1+N^{1-\gamma}}^{n_0} \left\{ \sum_{t_2=n_0+p}^{n_2-N^{1-\gamma}} c(t_1, t_2) + \sum_{t_2=n_2+N^{1-\gamma}}^{N} c(t_1, t_2) \right\} \\
&\le I_{11} + I_{12} + I_{21} + I_{22}.
\end{align*}
$$
<figure><img src="image_5.png" /><figcaption>Figure 3: Correlation dimension of a network of ravines and gullies in the French Alps: selection of the largest homoscedastic linear zone of the log-log plot (Xᵢ, Yᵢ) where Yᵢ = log(C(rᵢ)), Xᵢ = log(rᵢ) and 1 ≤ rᵢ ≤ 250 in meters (right).</figcaption></figure>

The result of the log-log regression are extremely convincing: the Multiple R-squared coefficient is 0.9998, the p-value of the Fisher test is 2·10<sup>-16</sup> and finally the estimator of the slope, which is also the estimator D̂ of the correlation dimension is almost 1.741.

4.4.2 Application to the study of the average global temperature on Earth since 1880

We are going to apply our linear longest range detector to a well-known data series, that of average annual temperatures at the Earth’s surface, considered between 1880 and 2015. Figure 4 shows the evolution of this annual temperature.

We will try to detect the largest linear range, considering it to be relative to a linear trend. So we will use the criterion defined in (3.17) with $v_n = \sqrt{n}$ to determine whether it is better to choose no break (i.e. a single linear trend from 1880 to 2015), a single break (i.e., a linear trend from 1880 to a certain date before 2015, or a linear trend from a certain date after 1880 to 2015) or 2 breaks (i.e. a linear trend from a date after 1880 to another date before 2015).

The result obtained is that of a single break with a linear trend from 1964 to 2015. We can compare this result with that obtained using a single break between 2 linear trends following the least-squares residual minimization method (see Bai and Perron [3]): the result is an abrupt break in 1962 and the p.value of the Chow test (H₀: "no break" versus H₁: "one break") is then 1.449285e-12, leading to the rejection of the hypothesis H₀. These results are shown in Figure 4.
From a time translation, the cases of $t_1$ and $t_2$ defined in $I_{11}$, $I_{12}$, $I_{21}$ and $I_{22}$ are similar to the one described in Theorem 3.1, i.e. at least $t_1$ or $t_2$ are in the "linear" zone. Therefore, these sums all satisfy $I_1 \underset{N \to \infty}{\longrightarrow} 0$.

We deduce that:

$$ \mathbb{P}(|\hat{n}_1 - n_1| + |\hat{n}_2 - n_2| \ge N^{1-\gamma}) \underset{N \to \infty}{\longrightarrow} 0. $$

<figure><img src="image_7.png" /></figure>

*Proofs of Corollaries 3 and 4.* See the proofs of Corollaries 1 and 2.

<figure><img src="image_8.png" /></figure>

## References

[1] Arlot, A., Celisse, A. and Harchaoui, Z. (2019). A Kernel Multiple Change-point Algorithm via Model Selection. *Journal of Machine Learning Research*, 20, 1-56.

[2] Bai, J. (1998). Estimation of multiple-regime regressions with least absolutes deviation, *Journal of Statistical Planning and Inference*, **74**, 103-134.

[3] Bai J. and Perron P. (1998). Estimating and testing linear models with multiple structural changes. *Econometrica* **66**, 47-78.

[4] Ben Hariz, S., Wylie J. and Zhang Q. (2007). Optimal rate of convergence for nonparametric change-point estimators for nonstationary sequences. *The Annals of Statistics* **35**, 1802-1826.

[5] Carlstein, E. (1988). Nonparametric change-point estimation. *The Annals of Statistics* **16**, 188-197.

[6] Chow, G. (1960). Tests of Equality Between Sets of Coefficients in Two Linear Regressions. *Econometrica*, **28**, p. 591-605.

[7] Drygas, H. (1976). Weak and strong consistency of the least squares estimators in regression models. *Z. Wahrscheinlichkeitstheorie Verw. Geb.* **34**, 119-127.

[8] Grassberger, P. and Proccacia, I. (1983). Measuring the strangeness of strange attractors. *Physica D* **9**, 189-208.

[9] Huber, P.J. (1981). *Robust statistics*, Wiley Series in Probability and Mathematical Statistics, Wiley & Sons Inc., New York.

[10] Lai T.L., Robbins, H. and Wei C.Z. (1978). Strong consistency of least squares estimates in multiple regression. *Proc. Natl. Acad. Sci. USA* **75**, 3034-3036.

[11] Lavielle, M. (1999). Detection of multiple changes in a sequence of dependent variables. *Stoc. Proc. Appl.* **83**, 79-102.

[12] Lavielle, M. and Moulines, E. (2000). Least-squares estimation of an unknown number of shifts in a time series. *J. of Time Series Anal.* **21**, 33-59.
<figure><img src="image_6.png" /><figcaption>Figure 4: Global annual Earth temperature data: temperature plot (black), linear trend obtained with the $\hat{C}_b$ criterion (red) and two linear trends with break at 1962 obtained with the least-squares criterion (blue).</figcaption></figure>

The results obtained are therefore very similar, except that the method using our criterion $\hat{C}_b$ does not make the assumption that the area between 1880 and 1964 has a linear trend, which is invalidated by the observation of residuals as a function of predicted values (see Figure 5, left graph). On the other hand, the linear trend after 1964 seems quite acceptable (see Figure 5, right graph), the multiple R-squared is 0.897 (it was 0.358 for a linear trend between 1880 and 1964) and the p-value of the Fisher model validity test is $2.2e-16$.
[13] Thommeret, N., Bailly, J.-S., Bardet, J.-M., Kaiser, B. and Puech, C. (2014). Dimensions fractales de réseaux vectoriels: méthodes d'estimation et robustesse des résultats. *Cybergeo: European Journal of Geography*
<figure><img src="image_7.png" /><figcaption>Figure 5: Global annual Earth temperature data: Plots of residuals versus predicted values, in the case of a linear trend between 1880 and 1964 (left) and in the case of a linear trend between 1964 and 2015 (left).</figcaption></figure>

# 5 Proofs

*Proof of Lemma 1.* We first have $||(I(t) - H(t))D^{(N)}(t)||^2 = ||D^{(N)}(t)||^2 - ||H(t) D^{(N)}(t)||^2$ from Pythagoras' Theorem for $t \ge n + 1$.

We can write $H(t) = Q(t) J_p(t) Q(t)'$ where $Q$ is an orthogonal matrix and $J_p(t) = \begin{pmatrix} I(p) & 0_{p,t-p} \\ 0_{t-p,p} & 0_{t-p,t-p} \end{pmatrix}$ since $H(t)$ is an orthogonal projection matrix on a $p$-dimension space. As a consequence, for $t \ge n + 1$,

$$ \begin{aligned} ||H(t) D^{(N)}(t)||^2 &= (D^{(N)}(t))' H(t) D^{(N)}(t) \\ &= (Q(t) D^{(N)}(t))' J_p(t) (Q(t) D^{(N)}(t)) \\ &= \sum_{i=1}^{p} (Q(t) D^{(N)}(t))_i^2 \\ &\le p \max_{1 \le j \le t-n} \delta_j^2. \end{aligned} $$

Then from Assumption C and condition 2.4, we deduce that for $t \ge n + k_0$ and almost surely,

$$ ||H(t) D^{(N)}(t)||^2 \le \frac{1}{2} \Delta_{t-n} \implies ||(I(t) - H(t))D^{(N)}(t)||^2 \ge \frac{1}{2} \Delta_{t-n}. $$

Finally, using the first condition 2.4, i.e. $\sum_{i=1}^k \delta_i^2 \ge Ck$, we deduce that for $t$ large enough, (3.6) holds. $\square$

*Proof of Theorem 3.1.* For ease of writing, denote now:

$$ n_p := n - p \quad \text{and} \quad t_p := t - p. $$
We are going to study successively the cases $t < n$ and $t > n$. Indeed, we have the property:

$$ 
\begin{aligned}
\mathbb{P}(|\hat{\tau} - \tau| \ge N^{-\gamma}) & \le \mathbb{P}\left(\max \{\widehat{Cr}(t), t \text{ such that } |t-n| \ge N^{1-\gamma}\} < \widehat{Cr}(n)\right) \\
& \le \sum_{t=p+1}^{n-N^{1-\gamma}} \mathbb{P}(\widehat{Cr}(t) < \widehat{Cr}(n)) + \sum_{t=n+N^{1-\gamma}}^{N} \mathbb{P}(\widehat{Cr}(t) < \widehat{Cr}(n)).
\end{aligned} 
\quad (5.1) $$

1. Let $t < n$ such that $|t - n| \ge N^{1-\gamma}$, i.e. $p + 1 \le t \le [\tau N] - N^{1-\gamma}$.

Denote $J_{t,n}$ the subspace of $\mathbb{R}^n$ such that $J_{t,n} = \{(x_1, \dots, x_t, 0, \dots, 0), (x_1, \dots, x_t) \in \mathbb{R}^t\}$ and $P_{J_{t,n}}$ the orthogonal projection on $J_{t,n}$ in $\mathbb{R}^n$. Since $I(n) - H(n) = P_{[Z]_n^\perp}$, matrix of the orthogonal projection on $[Z]_n^\perp$ in $\mathbb{R}^n$, then $V_t = (I(t) - H(t))\varepsilon(t) = P_{[Z]_n^\perp} P_{J_{t,n}} \varepsilon(n)$. But we have $P_{[Z]_n^\perp} \varepsilon(n) = P_{[Z]_n^\perp} P_{J_{t,n}} \varepsilon(n) + P_{[Z]_n^\perp} P_{J_{t,n}^\perp} \varepsilon(n)$, with $W_{n-t} = P_{[Z]_n^\perp} P_{J_{t,n}} \varepsilon(n) = P_{[Z]_n^\perp \cap J_{t,n}} \varepsilon(n)$. Therefore, using Cochran' Theorem, $\|V_t\|^2$ follows a chi-square distribution with $t_p = (t-p)$ degrees of freedom and $\|W_{n-t}\|^2$ follows a chi-square distribution with $(n-t)$ degrees of freedom, where $V_t$ and $W_{n-t}$ are independent Gaussian variables.

But, since $t \le n$:

$$ 
\begin{aligned}
\mathbb{P}(\widehat{Cr}(t) < \widehat{Cr}(n)) &= \mathbb{P}\left(\frac{a_{\alpha,\beta}(t)}{t_p} \|(I(t) - H(t))\varepsilon(t)\|^2 < \frac{a_{\alpha,\beta}(n)}{n_p} \|(I(n) - H(n))\varepsilon(n)\|^2\right) \\
&= \mathbb{P}\left(\frac{a_{\alpha,\beta}(t)}{t_p} \|V_t\|^2 < \frac{a_{\alpha,\beta}(n)}{n_p} \left(\|V_t\|^2 + \|W_{n-t}\|^2\right)\right) \\
&= \mathbb{P}\left(\|W_{n-t}\|^2 - A \|V_t\|^2 > 0\right) \quad \text{with } A := \frac{n_p a_{\alpha,\beta}(t)}{t_p a_{\alpha,\beta}(n)} - 1.
\end{aligned} 
$$

For $U$ a random variable such that there exists $\ell > 0$ satisfying $\mathbb{E}(e^{\ell U}) < \infty$, we have:

$$ \mathbb{P}(U > 0) \le \exp\left[\inf_{x \ge 0} \left\{\log\left(\mathbb{E}(e^{xU})\right)\right\}\right]. \quad (5.2) $$

For $\xi$ a chi-square variable with $m$ freedom degrees, $\mathbb{E}(e^{x\xi}) = (1 - 2x)^{-m/2}$. Therefore, with $U = \|W_{n-t}\|^2 - A \|V_t\|^2$ and the independence between $V_t$ and $W_{n-t}$, we obtain from (5.2),

$$ 
\begin{aligned}
\mathbb{P}(\widehat{Cr}(t) > \widehat{Cr}(n)) & \le \exp\left(\inf_{x \ge 0} \left\{\log\left[\frac{1}{(1 + 2x A)^{t_p/2}} \times \frac{1}{(1 - 2x)^{(n_p - t_p)/2}}\right]\right\}\right) \\
& \le \exp\left(-\frac{1}{2} \sup_{0 \le x \le 1/2} \{f(x)\}\right) \\
& \quad \text{with } f(x) := t_p \log(1 + 2x A) + (n_p - t_p) \log(1 - 2x).
\end{aligned} 
$$

Classical computations show that the supremum of $f$ is obtained for

$$ x_0 := \frac{t_p A - (n_p - t_p)}{2 n_p A} = \frac{1}{2} \frac{t_p (a_{\alpha,\beta}(t) - a_{\alpha,\beta}(n))}{(n_p a_{\alpha,\beta}(t) - t_p a_{\alpha,\beta}(n))}. $$

and straightforward computations show that $0 < x_0 < 1/2$. As a consequence, we obtain:

$$ 
\begin{aligned}
f(x_0) &= t_p \log \left( \frac{a_{\alpha,\beta}(t)}{a_{\alpha,\beta}(n)} \right) + (n_p - t_p) \log \left( \frac{(n_p - t_p) a_{\alpha,\beta}(t)}{n_p a_{\alpha,\beta}(t) - t_p a_{\alpha,\beta}(n)} \right) \\
&= n_p \log (1 + b_{\alpha,\beta}(n,t)) - (n_p - t_p) \log \left( 1 + \frac{n_p}{n_p - t_p} b_{\alpha,\beta}(n,t) \right) \\
& \quad \text{with } b_{\alpha,\beta}(n,t) := \frac{a_{\alpha,\beta}(t)}{a_{\alpha,\beta}(n)} - 1 \ge 0.
\end{aligned} 
$$
The function $t \in \{p+1, \dots, n\} \mapsto b_{\alpha,\beta}(n,t) = a_{\alpha,\beta}^{-1}(n)\left(1 + \frac{N^\alpha}{(t-p)^\beta}\right) - 1$ is a decreasing function. Moreover, since $\alpha < \beta$, if $t$ (or $t_p = t-p$) is negligible with respect to $N^{\alpha/\beta}$ then with $b_{\alpha,\beta}(n,t) \sim N^\alpha t_p^{-\beta} \underset{N \to \infty}{\longrightarrow} \infty$. And if $N^{\alpha/\beta}$ is negligible with respect to $t_p$, then $b_{\alpha,\beta}(n,t) \underset{N \to \infty}{\longrightarrow} 0$.

Thus, we will now consider 3 cases inducing 3 different behaviors of $b_{\alpha,\beta}(n,t)$ and therefore of $x_0$ and $f(x_0)$: $t_p \in \{[n_p/2], [n_p/2]+1, \dots, n_p - [N^{1-\gamma}]\}$, $t_p \in \{[(3N^\alpha)^{1/\beta}], [(3N^\alpha)^{1/\beta}]+1, \dots, [n_p/2]-1\}$ and $t_p \in \{1, 2, \dots, [(3N^\alpha)^{1/\beta}] - 1\}$.

Since $\alpha < \beta$, we will also use a Taylor-Lagrange expansion of $b_{\alpha,\beta}(n,t)$, which implies that for any $1 \le t_p \le n_p$, there exists $\theta_{t_p,n_p} \in [t_p, n_p]$ such that

$$b_{\alpha,\beta}(n,t) \underset{N \to \infty}{\sim} N^{\alpha} (t_p^{-\beta} - n_p^{-\beta}) \underset{N \to \infty}{\sim} \beta \frac{N^{\alpha} \theta_{t_p, n_p}^{\beta-1}}{n_p^{\beta} t_p^{\beta}} (n_p - t_p). \quad (5.3)$$

i. Let $t_p \in \{[n_p/2], [n_p/2]+1, \dots, n_p - [N^{1-\gamma}]\}$. Then $b_{\alpha,\beta}(n,t) \le (3^\beta - 1) N^\alpha (\tau N)^{-\beta} \underset{N \to \infty}{\longrightarrow} 0$ and from (5.3), $n_p(n_p - t_p)^{-1} b_{\alpha,\beta}(n,t) \underset{N \to \infty}{\longrightarrow} 0$. Therefore with Taylor expansions, for $N$ large enough,

$$ \begin{align*} f(x_0) & \underset{N \to \infty}{\sim} \frac{1}{2} \frac{t_p n_p}{n_p - t_p} b_{\alpha, \beta}^2(n,t) \\ & \underset{N \to \infty}{\sim} \frac{\beta^2}{2} \frac{N^{2\alpha} \theta_{t_p, n_p}^{2\beta-2}}{n_p^{2\beta-1} t_p^{2\beta-1}} (n_p - t_p) \\ & \ge \frac{\beta^2}{2^\beta} N^{2\alpha} n_p^{2\beta} (n_p - t_p) \\ & \ge \frac{\beta^2}{2^\beta \tau^{2\beta}} N^{2\alpha-2\beta} N^{1-\gamma}. \end{align*} $$

Then for $t_p \in \{[n_p/2], [n_p/2] + 1, \dots, n_p - [N^{1-\gamma}]\}$ and $N$ large enough, using the assumption $1 - \gamma - 2(\beta - \alpha) > 0$,

$$\mathbb{P}(\widehat{Cr}(t) > \widehat{Cr}(n)) \le \exp\left(-\frac{\beta^2}{2^{\beta+1}\tau^{2\beta}} N^{1-\gamma-2(\beta-\alpha)}\right). \quad (5.4)$$

ii. Let $t_p \in \{[(3N^\alpha)^{1/\beta}], [(3N^\alpha)^{1/\beta}] + 1, \dots, [n_p/2] - 1\}$. Simple computations imply that $0 \le b_{\alpha,\beta}(n,t) < \frac{1}{2}$ when $N$ is large enough. From the definition of $f(x_0)$, we obtain:

$$ \begin{align} f(x_0) &= t_p \log (1 + b_{\alpha,\beta}(n,t)) - (n_p - t_p) \log \left(1 + \frac{t_p}{n_p - t_p} \frac{b_{\alpha,\beta}(n,t)}{1 + b_{\alpha,\beta}(n,t)}\right) \nonumber \\ &\ge t_p \left(\log (1 + b_{\alpha,\beta}(n,t)) - \frac{b_{\alpha,\beta}(n,t)}{1 + b_{\alpha,\beta}(n,t)}\right), \tag{5.5} \end{align} $$

using $\log(1+u) \le u$ for $u > -1$. Using also $\log(1+u) \ge u - \frac{1}{2}u^2$ for $u \ge 0$, we obtain from (5.5),

$$ \begin{align} f(x_0) &\ge t_p \left( b_{\alpha,\beta}(n,t) - \frac{1}{2} b_{\alpha,\beta}^2(n,t) - \frac{b_{\alpha,\beta}(n,t)}{1 + b_{\alpha,\beta}(n,t)} \right) \nonumber \\ &\ge \frac{1}{2} t_p \frac{(1 - b_{\alpha,\beta}(n,t)) b_{\alpha,\beta}^2(n,t)}{(1 + b_{\alpha,\beta}(n,t))} \nonumber \\ &\ge \frac{1}{6} t_p b_{\alpha,\beta}^2(n,t). \tag{5.6} \end{align} $$
Using (5.3), $\beta > 1$ and $t_p \in \{[(3N^\alpha)^{1/\beta}], [(3N^\alpha)^{1/\beta}] + 1, \dots, [n_p/2] - 1]\}$, we deduce for $N$ large enough,

$$b_{\alpha,\beta}(n,t) \geq \frac{1}{2} \beta \frac{N^\alpha}{n_p^\beta t_p} (n_p - t_p) \geq \frac{\beta}{2\tau^\beta} N^{\alpha-\beta}.$$

Therefore, from (5.6) and always with $N$ large enough:

$$\begin{aligned} f(x_0) &\geq \frac{1}{6} (3N^\alpha)^{1/\beta} \left(\frac{\beta}{2\tau^\beta} N^{\alpha-\beta}\right)^2 \\ &\geq \frac{3^{1/\beta} \beta^2}{24\tau^{2\beta}} N^{\alpha/\beta - 2(\beta-\alpha)}. \end{aligned}$$

Then if $t_p \in \{[(3N^\alpha)^{1/\beta}], [(3N^\alpha)^{1/\beta}] + 1, \dots, [n_p/2] - 1]\}$ and for $N$ large enough,

$$\mathbb{P}(\widehat{Cr}(t) > \widehat{Cr}(n)) \leq \exp\left(-\frac{3^{1/\beta} \beta^2}{48 \tau^{2\beta}} N^{\alpha/\beta - 2(\beta-\alpha)}\right), \quad (5.7)$$

which explains the condition $\frac{\alpha}{2\beta} > \beta - \alpha$.

iii. Let $t_p \in \{1, 2, \dots, [(3N^\alpha)^{1/\beta}] - 1\}$. Simple computations imply that $b_{\alpha,\beta}(n,t) > \frac{1}{4}$ when $N$ is large enough. But it can be easily proved that $\log(1+u) - u(1+u)^{-1} \geq \frac{1}{10}\log(1+u)$ for all $u > 1/4$. Therefore, from (5.5),

$$f(x_0) \geq \frac{1}{10} t_p \log (1 + b_{\alpha,\beta}(n,t)).$$

But for $t_p \in \{1, 2, \dots, [(3N^\alpha)^{1/\beta}] - 1\}$ and when $N$ is large enough, $b_{\alpha,\beta}(n,t) \geq \frac{1}{2} \frac{N^\alpha}{t_p^\beta}$ and thus

$$f(x_0) \geq \frac{1}{10} t_p \log \left(1 + \frac{1}{2} \frac{N^\alpha}{t_p^\beta}\right).$$

As a consequence, $t_p \in \{1, 2, \dots, [(3N^\alpha)^{1/\beta}] - 1\}$ and when $N$ is large enough,

$$\mathbb{P}(\widehat{Cr}(t) > \widehat{Cr}(n)) \leq \left(1 + \frac{1}{2} \frac{N^\alpha}{t_p^\beta}\right)^{-t_p/20}. \quad (5.8)$$

2. Let $t > n$ such that $|t - n| > N^{1-\gamma}$, i.e. $[\tau N] + N^{1-\gamma} \leq t \leq N$. Then:

$$\begin{aligned} \mathbb{P}(\widehat{Cr}(t) < \widehat{Cr}(n)) &= \mathbb{P}\left(\frac{a_{\alpha,\beta}(t)}{t_p} \left\|(I(t) - H(t))(\varepsilon(t) + D(t))\right\|^2 < \frac{a_{\alpha,\beta}(n)}{n_p} \left\|(I(n) - H(n))\varepsilon(n)\right\|^2\right) \\ &= \mathbb{P}\left(\frac{a_{\alpha,\beta}(t)}{t_p} \left\|(V_n + W_{t-n}) + P_{[Z]_t^\perp} D(t)\right\|^2 < \frac{a_{\alpha,\beta}(n)}{n_p} \left\|V_n\right\|^2\right) \\ &= \mathbb{P}\left(\frac{a_{\alpha,\beta}(t)}{t_p} \left(\left\|V_n + P_{[Z]_t^\perp} D(t)\right\|^2 + \left\|W_{t-n}\right\|^2\right) < \frac{a_{\alpha,\beta}(n)}{n_p} \left\|V_n\right\|^2\right) \\ &= \mathbb{P}\left(\left(\left\|V_n - B P_{[Z]_t^\perp} D(t)\right\|^2 - B(B+1) \left\|P_{[Z]_t^\perp} D(t)\right\|^2 - B \left\|W_{t-n}\right\|^2\right) > 0\right), \end{aligned}$$

where $B = n_p a_{\alpha,\beta}(t) (t_p a_{\alpha,\beta}(n) - n_p a_{\alpha,\beta}(t))^{-1}$, and $V_n$ and $W_{t-n}$ defined as in the first part of the proof and $D(t)$ defined in (3.5). The conditional probability given $D(t)$ can be bounded using again (5.2). Indeed, given $D(t)$, $\sigma^{-2}\|V_n - B P_{[Z]_t^\perp} D(t)\|^2$ follows a noncentral chi-squared distribution with $n_p$ degrees of freedom, independent to $\sigma^{-2}\|W_{t-n}\|^2$ which follows a chi-squared distribution
with $(t_p - n_p)$ degrees of freedom. Moreover, given $D(t)$, the moment generating function of $\sigma^{-2}\|V_n - B P_{[Z]_t^\perp} D(t)\|^2$ is given by $(1-2x)^{n_p/2} \exp(\lambda x (1-2x)^{-1})$ where $\lambda = \sigma^{-2}B^2\|P_{[Z]_t^\perp} D(t)\|^2$.
Thus,

$$ \mathbb{P}(\widehat{Cr}(t) < \widehat{Cr}(n) | D(t)) \le \exp \left( \inf_{x \ge 0} \left\{ \log \left[ \frac{\exp\left(\frac{B^2 \|P_{[Z]_t^\perp} D(t)\|^2 x}{\sigma^2(1-2x)}\right)}{(1-2x)^{n_p/2}} \times \frac{\exp\left(-x \frac{B(B+1) \|P_{[Z]_t^\perp} D(t)\|^2}{\sigma^2}\right)}{(1+2Bx)^{(t_p-n_p)/2}} \right] \right\} \right) \\ \le \exp\left(-\frac{1}{2} \sup_{0 \le x \le 1/2} \{g(x)\}\right) $$

$$ \text{with } g(x) := n_p \log(1-2x) + (t_p - n_p) \log(1+2Bx) + \frac{2B \|P_{[Z]_t^\perp} D(t)\|^2 x}{\sigma^2} \frac{(1-2(B+1)x)}{1-2x}. $$

Using $n_p \sim \tau N$ and $t_p > n_p$, we obtain the following expansion:

$$ B = \frac{n_p}{t_p - n_p} \left( 1 - \beta \frac{N^\alpha}{n_p^\beta} (1 + o(1)) \right). $$

As a consequence we have $B \sim \frac{n_p}{t_p - n_p}$ and also $B \ge \frac{\tau}{2(1-\tau)}$ for $N$ large enough. Moreover, for any $u \in [-1/2, 1]$, $\log(1+u) \ge u - u^2$. Therefore, for $x$ such that $x \in [0, \max(1/4; 1/2B)]$ and $N$ large enough,

$$ \begin{aligned} g(x) &\ge n_p (-2x - 4x^2) + (t_p - n_p)(2Bx - 4B^2x^2) + \frac{2B \|P_{[Z]_t^\perp} D(t)\|^2 x}{\sigma^2} (1 - 2(B+1)x) \\ &\ge -4\beta \frac{N^\alpha}{n_p^\beta} (t_p - n_p)x - 4 \frac{t_p n_p}{t_p - n_p} x^2 + \frac{C}{\sigma^2} (t_p - n_p) B x (1 - 2(B+1)x), \end{aligned} $$

using Assumption C and Lemma 1, implying $\|P_{[Z]_t^\perp} D(t)\|^2 \ge \frac{1}{2} C (t_p - n_p)$ for $N$ large enough. For a given $\varepsilon > 0$ such that $\varepsilon \le \max(\frac{\tau}{8(1-\tau)}; \frac{1}{2})$, for $N$ large enough,

$$ \begin{aligned} g(\varepsilon/B) &\ge -4\beta \frac{N^\alpha}{n_p^\beta} \frac{(t_p - n_p)^2}{n_p} \varepsilon - 4 \frac{t_p}{n_p} (t_p - n_p) \varepsilon^2 + \frac{C}{\sigma^2} \left(1 - 2\varepsilon - 4 \frac{1-\tau}{\tau} \varepsilon\right) (t_p - n_p) \varepsilon \\ &\ge -4\beta \frac{1-\tau}{\tau} \frac{N^\alpha}{n_p^\beta} (t_p - n_p) \varepsilon - \frac{4}{\tau} (t_p - n_p) \varepsilon^2 + \frac{C}{\sigma^2} \left(1 - \frac{4}{\tau} \varepsilon\right) (t_p - n_p) \varepsilon \\ &\ge (t_p - n_p) \varepsilon \left(-\frac{5}{\tau} \varepsilon + \frac{C}{\sigma^2} \left(1 - \frac{4}{\tau} \varepsilon\right)\right). \end{aligned} $$

Therefore, for $\varepsilon < C \tau (4C + 5\sigma^2)^{-1}$, for instance $\varepsilon = \frac{C}{2} \tau (4C + 5\sigma^2)^{-1}$, we obtain:

$$ \sup_{0 \le x \le 1/2} \{g(x)\} \ge g(\varepsilon/B) \ge \frac{C^2 \tau}{4\sigma^2 (4C + 5\sigma^2)} (t_p - n_p). $$

Then, for $[\tau N] + N^{1-\gamma} \le t \le N$ and $N$ large enough,

$$ \begin{aligned} \mathbb{P}(\widehat{Cr}(t) < \widehat{Cr}(n) | D(t)) &\le \exp\left(-\frac{C^2 \tau}{8\sigma^2 (4C + 5\sigma^2)} (t_p - n_p)\right) \\ \Rightarrow \quad \mathbb{P}(\widehat{Cr}(t) < \widehat{Cr}(n)) &\le \exp\left(-\frac{C^2 \tau}{8\sigma^2 (4C + 5\sigma^2)} N^{1-\gamma}\right). \end{aligned} \quad (5.9) $$
Finally, using (5.1), (5.4), (5.7), (5.8) and (5.9) we deduce

$$
\begin{align}
\mathbb{P}(|\hat{\tau} - \tau| \ge N^{-\gamma}) &\le \sum_{t_p=1}^{[(3N^\alpha)^{1/\beta}]-1} \left(1 + \frac{1}{2} \frac{N^\alpha}{t_p^\beta}\right)^{-t_p/20} + \sum_{t_p=[(3N^\alpha)^{1/\beta}]}^{[n_p/2]-1} \exp\left(-\frac{3^{1/\beta} \beta^2}{48 \tau^{2\beta}} N^{\alpha/\beta - 2(\beta-\alpha)}\right) \nonumber \\
&\quad + \sum_{t_p=[n_p/2]}^{n_p - [N^{1-\gamma}]} \exp\left(-\frac{\beta^2}{8\tau^{2\beta}} N^{1-\gamma - 2(\beta-\alpha)}\right) + \sum_{t_p=n_p + [N^{1-\gamma}]}^{N-p} \exp\left(-\frac{C^2 \tau}{8\sigma^2 (4C + 5\sigma^2)} N^{1-\gamma}\right) \nonumber \\
&\le C N^{-\alpha/20} \underset{N\to\infty}{\longrightarrow} 0, \tag{5.10}
\end{align}
$$

with $C > 0$.

<figure><img src="image_8.png" /></figure>

*Proof of Corollary 1.* If $0 < \varepsilon < \alpha/\beta$, then using the previous proof we have for $N$ large enough,

$$
\begin{align*}
\mathbb{P}(|\tilde{n} - n| \ge N^{1-\gamma}) &\le \sum_{t_p=N^\varepsilon}^{[(3N^\alpha)^{1/\beta}]-1} \left(1 + \frac{1}{2} \frac{N^\alpha}{t_p^\beta}\right)^{-t_p/20} + \sum_{t_p=[(3N^\alpha)^{1/\beta}]}^{[n_p/2]-1} \exp\left(-\frac{3^{1/\beta} \beta^2}{48 \tau^{2\beta}} N^{\alpha/\beta - 2(\beta-\alpha)}\right) \\
&\quad + \sum_{t_p=[n_p/2]}^{n_p - [N^{1-\gamma}]} \exp\left(-\frac{\beta^2}{8\tau^{2\beta}} N^{1-\gamma - 2(\beta-\alpha)}\right) + \sum_{t_p=n_p + [N^{1-\gamma}]}^{N-p} \exp\left(-\frac{C^2 \tau}{8\sigma^2 (4C + 5\sigma^2)} N^{1-\gamma}\right) \\
&\le N \left[e^{-C_1 N^\varepsilon} + e^{-C_2 N^{\alpha/\beta - 2(\beta-\alpha)}} + e^{-C_3 N^{1-\gamma - 2(\beta-\alpha)}} + e^{-C_4 N^{1-\gamma}}\right] \\
&\le e^{-C N^\rho},
\end{align*}
$$

where $C_1, C_2, C_3, C_4$ and $C$ are positive real numbers and $0 < \rho < \min(\varepsilon, 1 - \gamma - 2(\beta - \alpha))$.

The case $\varepsilon \ge \alpha/\beta$ can be studied identically except that the first sum is omitted. $\square$

<figure><img src="image_9.png" /></figure>

*Proof of Corollary 2.* Let $x \in \mathbb{R}$ and denote $W_N = (Z'(\hat{n} - N^\delta) Z(\hat{n} - N^\delta))^{1/2} (\tilde{\theta}_N - \theta)$. Then:

$$
\mathbb{P}(W_N \le x) = \mathbb{P}(W_N \le x \mid \hat{n} - N^\delta \le n) \mathbb{P}(\hat{n} - N^\delta \le n) + \mathbb{P}(W_N \le x \cap \hat{n} - N^\delta \le n).
$$

But given $\hat{n} - N^{\delta} \le n$, since $(\varepsilon_t)_t$ is supposed to be a Gaussian white noise, $\tilde{\theta}_N$ is a Gaussian vector and therefore $W_N \stackrel{\mathcal{L}}{=} \mathcal{N}(0, \sigma^2)$. Moreover, from the proof of Theorem 3.1, $\mathbb{P}(\hat{n} \ge n + N^{1-\gamma}) \le N \exp(-C_4 N^{1-\gamma})$ with $C_4 > 0$ and therefore

$$
\mathbb{P}(\hat{n} - N^{\delta} \le n) = 1 - \mathbb{P}(\hat{n} \le n + N^{\delta}) \xrightarrow{N \to \infty} 1.
$$

Moreover $\mathbb{P}(W_N \le x \cap \hat{n} - N^\delta \le n) \le \mathbb{P}(\hat{n} - N^\delta \le n) \xrightarrow{N \to \infty} 0$. Thus, with $F_W$ the cumulative distribution function of $W \sim \mathcal{N}'0, \sigma^2)$, we have

$$
\mathbb{P}(W_N \le x) \xrightarrow{N \to \infty} F_W(x),
$$

and the asymptotic normality of $\tilde{\theta}_N$ can be deduced. $\square$

<figure><img src="image_10.png" /></figure>

*Proof of Theorem 3.2.* From Assumption B', we deduce that we can consider $N$ large enough, such that $n_0 \ge n_1 + N^{1-\gamma}$ and $n_0 + p \le n_2 - N^{1-\gamma}$. Now, using the notation $c(t_1, t_2) = \mathbb{P}(\tilde{C}r'(t_1, t_2) <$
$\widehat{Cr}'(n_1, n_2))$ and as it was already established in the proof of Theorem (3.1), we have

$$
\begin{align*}
& \mathbb{P}(|\hat{n}_1 - n_1| + |\hat{n}_2 - n_2| \ge N^{1-\gamma}) \\
\le & \mathbb{P} \left( \max \left\{ \widehat{Cr}'(t_1, t_2), t_1 \le n_0 \le t_2 - p, |t_1 - n_1| \ge N^{1-\gamma} \text{ and } |t_2 - n_2| \ge N^{1-\gamma} \right\} < \widehat{Cr}'(n_1, n_2) \right) \\
\le & \sum_{t_1=p}^{n_1-p-N^{1-\gamma}} \left\{ \sum_{t_2=n_0+p}^{n_2-N^{1-\gamma}} c(t_1, t_2) + \sum_{t_2=n_2+N^{1-\gamma}}^{N} c(t_1, t_2) \right\} \\
& + \sum_{t_1=n_1+N^{1-\gamma}}^{n_0} \left\{ \sum_{t_2=n_0+p}^{n_2-N^{1-\gamma}} c(t_1, t_2) + \sum_{t_2=n_2+N^{1-\gamma}}^{N} c(t_1, t_2) \right\} \\
\le & I_{11} + I_{12} + I_{21} + I_{22}.
\end{align*}
$$

From a time translation, the cases of $t_1$ and $t_2$ defined in $I_{11}$, $I_{12}$, $I_{21}$ and $I_{22}$ are similar to the one described in Theorem 3.1, i.e. at least $t_1$ or $t_2$ are in the "linear" zone. Therefore, these sums all satisfy $I_{..} \xrightarrow{N \to \infty} 0$.

We deduce that:

$$
\mathbb{P}(|\hat{n}_1 - n_1| + |\hat{n}_2 - n_2| \ge N^{1-\gamma}) \xrightarrow{N \to \infty} 0.
$$

<figure><img src="image_11.png" /></figure>

**Acknowledgments** The author is very grateful to the referee and the associated editor for many relevant suggestions, corrections and comments that helped to notably improve the paper.

References

[1] Arlot, A., Celisse, A. and Harchaoui, Z. (2019). A Kernel Multiple Change-point Algorithm via Model Selection. *Journal of Machine Learning Research*, 20, 1-56.

[2] Bai, J. (1998). Estimation of multiple-regime regressions with least absolutes deviation, *Journal of Statistical Planning and Inference*, **74**, 103–134.

[3] Bai J. and Perron P. (1998). Estimating and testing linear models with multiple structural changes. *Econometrica* 66, 47-78.

[4] Ben Hariz, S., Wylie J. and Zhang Q. (2007). Optimal rate of convergence for nonparametric change-point estimators for nonstationary sequences. *The Annals of Statistics* 35, 1802-1826.

[5] Carlstein, E. (1988). Nonparametric change-point estimation. *The Annals of Statistics* 16, 188-197.

[6] Chow, G. (1960). Tests of Equality Between Sets of Coefficients in Two Linear Regressions. *Econometrica*, 28, p. 591-605.

[7] Drygas, H. (1976). Weak and strong consistency of the least squares estimators in regression models. Z. Wahrscheinlichkeitstheorie Verw. Geb. 34, 119-127.
[8] Grassberger, P. and Proccacia, I. (1983). Measuring the strangeness of strange attractors. *Physica D* 9, 189-208.

[9] Lai T.L., Robbins, H. and Wei C.Z. (1978). Strong consistency of least squares estimates in multiple regression. *Proc. Natl. Acad. Sci. USA* 75, 3034-3036.

[10] Lavielle, M. (1999). Detection of multiple changes in a sequence of dependent variables. *Stoc. Proc. Appl.* 83, 79-102.

[11] Lavielle, M. and Moulines, E. (2000). Least-squares estimation of an unknown number of shifts in a time series. *J. of Time Series Anal.* 21, 33-59.

[12] Thommeret, N., Bailly, J.-S., Bardet, J.-M., Kaiser, B. and Puech, C. (2014). Dimensions fractales de réseaux vectoriels: méthodes d'estimation et robustesse des résultats. *Cybergeo: European Journal of Geography*