Functional Extreme-PLS

Stéphane Girard, Cambyse Pakzad

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Stéphane Girard, Cambyse Pakzad. Functional Extreme-PLS. 2025. hal-04488561v2

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Functional Extreme-PLS

Stéphane Girard<sup>a</sup>, Cambyse Pakzad<sup>b</sup>

<sup>a</sup>Université Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK, Grenoble, 38000, France

<sup>b</sup>Université Paris Nanterre, Modal'X UMR CNRS 9023, Nanterre, 92001, France

**Abstract**

We propose an extreme dimension reduction method extending the Extreme-PLS approach to the discretized functional framework, where the covariate lies in the infinite-dimensional Hilbert space $L^2([0, 1])$ but is partially observed on a dense time grid. The ideas are partly borrowed from both Partial Least-Squares (PLS) and Sliced Inverse Regression (SIR) techniques. As such, the method relies on the projection of the covariate onto a subspace and maximizes the covariance between its projection and the response conditionally to an extreme event driven by a random threshold to capture the tail-information. The covariate and the heavy-tailed response are supposed to be linked through a non-linear inverse single-index model and our goal is to infer the index in this regression framework. We propose a new family of estimators and show its asymptotic consistency with convergence rates under the model. Assuming mild conditions on the noise, most of the assumptions are stated in terms of regular variation unlike the standard literature on SIR and single-index regression. Finally, our results are illustrated on a finite-sample study with synthetic functional data as well as high-frequency financial data, highlighting the effectiveness of the dimension reduction for capturing tail dependence and for extreme risk management.

**Keywords:**

2020 MSC: 60G70 Extreme value theory, extremal stochastic processes,

2020 MSC: 62G08 Nonparametric regression and quantile regression,

2020 MSC: 62G32 Statistics of extreme values and tail-inference;

2020 MSC: 62R10 Functional data analysis.

**1. Introduction**

Among the popular methods for dimension reduction, the Partial Least Squares (PLS) method, initiated by Wold (1975) and adapted to $L^2([0, 1])$ by Preda and Saporta (2005), has proven to be successful in both theoretical and practical sides; especially in chemometrics (Martens and Naes, 1992) where it takes root. It combines characteristics of Principal Component Analysis (PCA) and regression. Its core idea is to find a projection of the covariate/predictor $X$ having high variance and correlation with the response $Y$. Another standard line of research is Sliced Inverse Regression (SIR), originally introduced by Li (1991) and then by Dauxois et al. (2001) in the functional framework. It hinges on the inverse regression ‘$X$ against $Y$’ that happens to be simpler than the forward regression, usually at the price of a certain condition of linearity and constant variance. This somewhat contradicts the philosophy of Fisher, as noted and encouraged by Cook (2007), where the inference is rather conditioned on the covariate $X$.
In any case, the abovementioned techniques require the full sample and are not tailored for extreme-value analysis. Hence, one may wonder what happens in the extreme regime, i.e., when the focus is on the conditional tail of Y given a high-dimensional covariate X. This question belongs to the vivid research branch of the association of dimension reduction and extreme-value statistics. Here, a major challenge is the sparsity which is doubled in the sense that both the curse of dimensionality and the rarity of tail observations operate. Such hindrance may apply in the estimation of certain statistics or risk measures in the functional extreme regime (see e.g., Gardes and Girard (2012); Girard et al. (2022)). To circumvent this issue, one would build up the inference on the existence of a projection that captures all the information on the extreme values of Y. Next, instead of using standard estimation methods, one would then expect that replacing X by its projection as the covariate, and hence opting for a lower-dimensional method that would yield comparable or even better performances. The pioneer work may be identified as Gardes (2018), later followed up by Gardes (2020), which proposes a new notion of tail conditional independence. Applying this to the extremal quantile regression, the author then studies the inference under a single and multi-index model when the dimension is large but finite and when the dimension reduction subspace is known. Still, some theoretical properties are lacking and the computational cost is too high. In this direction, Xu et al. (2020) develop a three-step estimation procedure for such large conditional quantiles and show the $\sqrt{n}$-consistency under a new tail single-index model when the indices are unknown; yielding better computational performances. Alternatively, focusing on the prediction of tail events, Aghbalou et al. (2024) propose a new framework which introduces a revised definition of tail conditional independence and provides theoretical guarantees for estimating a low-dimensional projection of covariates that is sufficient for predicting extremes, using inverse regression under standard SIR assumptions.

Recently, a novel method based on the PLS in the extreme regime was introduced by Bousebata et al. (2023). Herein, the authors introduce an inverse single-index model. This means that the covariate is written in terms of the response and some unknown link function (instead of the converse) which results in a reduction of dimension. In particular, under some conditions and within the regular variation framework, this inverse point of view may also be interpreted as a forward single-index model. Next, they propose an estimator of the index which is shown to be consistent with explicit rate and under a deterministic threshold level. The nature of their assumptions is quite different from the standard literature of dimension reduction as no conditions on the inverse conditional mean $E(X|Y)$, such as linearity, nor noise independence or centering are required. We also refer to Arbel et al. (2024) for an adaptation to the Bayesian framework allowing the introduction of prior information on the index.

In this work, we take up the methodology introduced in Bousebata et al. (2023) for the supervised branch of extreme dimension reduction and convert it to the functional covariate setting. Specifically, we project the predictor/covariate onto a one-dimensional subspace while capturing most of the information explaining the extreme values of the response variable. As such, we consider covariates that lie in the infinite-dimensional Hilbert space $H = L^2([0, 1])$ and the projection is expressed in terms of the associated inner product. In addition, we assume the more realistic case where the functional covariate is discretized as in Cardot et al. (2003); Crambes et al. (2009), i.e., partially observed on a deterministic time grid. Our methodology would equally apply when the functional covariate is ideally/fully observed (in this case, $H$ could be any Hilbert space) and also when $H = \mathbb{R}^d$ with $d$ large but finite, thereby positioning Bousebata et al. (2023) as a particular sub-case. Following the philosophy of PLS, we maximize the covariance between the projected predictor/covariate and the response conditionally to the response being larger than an increasing threshold. This corresponds to the extreme framework where the response is as-
sumed to be heavy-tailed. As for the covariate, we echo to SIR by modelling its relationship to the response through a non-linear inverse regression model which relies on a single index, i.e., a deterministic vector in $H$. One may again recover a forward single-index model interpretation. Thereby, the method at hand induces an effective reduction of dimension as we provide an estimation of the aforementioned index based on the observations. Using a simple geometrical interpretation, we in fact propose a new family of estimators that generalizes the one in Bousebata et al. (2023) and hence allows more freedom to the statistician on which part of the extreme regime to take account. The consistency (with rate of convergence) of these estimators is established under similar conditions as in Arbel et al. (2024); Bousebata et al. (2023) while extending these two works to the more realistic case of a random threshold. The assumptions are expressed in the language of regular variation and require minimal conditions about the noise, unlike the standard literature on SIR and forward single-index model. In practice, the finite-sample properties of the method show good results on numerical simulations with synthetic functional data and on high-frequency financial data.

The remaining of the paper is organized as follows. First, we introduce in Section 2 the mathematical framework for the Functional Extreme Partial Least-Squares (FEPLS in short) and, more specifically, the inverse single-index model. The estimators are presented in Section 3 with the associated consistency result. Next, the FEPLS finite-sample properties are illustrated in Section 4 on simulated data. Beyond simulations, we illustrate in Section 5 the method’s practical value in a financial risk assessment context, using high-frequency asset data to estimate conditional extreme quantiles and compare tail dependence capture against standard dimension-reduction techniques. The respective code in Python is publicly available on the website of the authors<fnref n="1" />. The mathematical details needed for the derivation of our main result are gathered in Appendix A.

## 2. Theoretical framework

Let $H = L^2([0, 1], \mathbb{R})$ endowed with its usual scalar product $\langle \cdot, \cdot \rangle$ and Borel $\sigma$-field $\mathcal{B}(H)$, and let $(\Omega, \mathcal{A}, \mathcal{P})$ be a complete probability space. Consider a random pair $(Y, X)$, where $Y: \Omega \to \mathbb{R}$ and $X: \Omega \to H$ are random variables, i.e., $Y$ is Borel measurable and $X$ is strongly $\mathbb{P}$-measurable in the sense of Bochner, see Hytonen et al. (2016, Definition 1.1.3).

Let us first focus on the direction defined as

$$ w(y) := \arg \max_{\|w\|=1} \mathrm{Cov}(\langle w, X \rangle, Y | Y \geq y), \quad y \in \mathbb{R}, \quad (1) $$

where $\|\cdot\|$ is the norm induced by the inner product $\langle \cdot, \cdot \rangle$ and $y \in \mathbb{R}^+$ is a large threshold. The expectation underlying the covariance in (1) is understood in the sense of Lebesgue. The interpretation of (1) follows the core idea of PLS but in the extreme regime. Using an optimization framework, we seek the element in $H$ which maximizes the tail-information, by considering the covariance conditionally to the tail event of $Y$, shared between $Y$ and the projection of $X$ on a single direction.

In the sequel, the cumulative distribution function (cdf) of $Y$ is denoted by $F$ and the survival function by $\bar{F} = 1 - F$. In addition, let us define the tail-moment of a random object $W$ by $m_W(y) := \mathbb{E}(W1_{\{Y \geq y\}})$ whenever it exists. In the case where $W \in H$, we interpret the expectation

<footnote n="1">¹https://cpakzad.github.io/download/FEPLS.zip</footnote>
at hand as the Bochner expectation denoted by $E_G$. Note that, since we consider a separable space, we could equivalently use the Pettis integral instead. We refer to Hytonen et al. (2016) for more details about the notion.

A key fact is that the solution of the optimization problem (1) is explicit:

**Proposition 1.** Suppose that $Y$ is integrable and $X$ is Bochner-integrable. Then, the unique solution of the optimization problem (1) is given by, for any $y \in \mathbb{R}$,

$$w(y) = v(y)/\|v(y)\| \text{ where } v(y) = \bar{F}(y)m_{XY}(y) - m_X(y)m_Y(y). \quad (2)$$

It readily appears that $w(y)$ is a linear combination of $m_{XY}(y) \in H$ and $m_X(y) \in H$. Since both of them point in the same direction in $H$ which is given by $X$, we rather investigate the use of more general tail moments by considering

$$w_{\varphi}(y) = v_{\varphi}(y)/\|v_{\varphi}(y)\| \text{ with } v_{\varphi}(y) := m_{X\varphi(Y)}(y), \quad (3)$$

where $\varphi$ is some test function such that

$$y \in \mathbb{R} \mapsto \varphi(y) \in \mathcal{RV}_{\tau(+\infty)}, \quad \tau \in \mathbb{R}. \quad (4)$$

Let us recall that $\varphi \in \mathcal{RV}_{\tau(+\infty)}$ is a standard notation meaning that $\varphi$ is a positive measurable function defined on some neighbourhood $[x_0, +\infty)$ of infinity and such that, for any $x > x_0 \ge 0$,

$$\lim_{t \to +\infty} \frac{\varphi(tx)}{\varphi(t)} = x^{\tau}.$$

In such a case, $\varphi$ is said to be regularly-varying at infinity with index $\tau$, see Bingham et al. (1989) for further details. In the sequel, $w_{\varphi}(y)$ is referred to as the FEPLS direction.

Before undertaking an inference procedure based on the empirical tail-moments in (3), let us introduce the model characteristics. An extreme-value framework is considered where the response variable $Y$ is heavy-tailed to the second order. As such, denote the generalized inverse function or quantile function of the cdf $F$ by $y \mapsto F^{-}(y) := \inf\{x \in \mathbb{R}, F(x) \ge y\}$. The tail quantile function is then $U(t) := F^{-}(1 - 1/t)$ and is assumed to belong to the class of second-order regularly-varying functions, i.e.,

* $2\mathcal{RV}_{\gamma,\rho}(+\infty)$: There exist $\gamma \in (0,1)$, $\rho \le 0$ and an auxiliary function $A$ ultimately of constant sign with $A(t) \to 0$ as $t \to +\infty$ such that:

$$\lim_{t \to +\infty} \frac{1}{A(t)} \left( \frac{U(ty)}{U(t)} - y^{\gamma} \right) = y^{\gamma} H_{\rho}(y) := y^{\gamma} \int_{1}^{y} u^{\rho-1} du, \quad y > 0. \quad (5)$$

Note that for $y > 0$, $H_{\rho}(y) = \log(y)1_{\{\rho=0\}} + \frac{y^{\rho-1}}{\rho}1_{\{\rho<0\}}$. By de Haan and Ferreira (2006, Theorem 2.3.9), the following correspondence in terms of regular variation indices holds:

$$U \in 2\mathcal{RV}_{\gamma,\rho}(+\infty) \iff \bar{F} \in 2\mathcal{RV}_{-1/\gamma,\rho/\gamma}(+\infty).$$

A direct consequence of (5) is that $U \in \mathcal{RV}_{\gamma(+\infty)}$ while $\bar{F} \in \mathcal{RV}_{-1/\gamma(+\infty)}$. On the auxiliary functions level, de Haan and Ferreira (2006, Theorem 2.3.3) provides $|A| \in \mathcal{RV}_{\rho(+\infty)}$ and $|A| \circ (1/\bar{F}) \in \mathcal{RV}_{\rho/\gamma(+\infty)}$.
Next, in order to provide theoretical guarantees on the inference method described in the next section and based on (3), we consider the following (non-linear) inverse single-index functional model,

$$X = g(Y)\beta + \varepsilon, \quad \beta \in H, \quad \|\beta\| = 1, \qquad (6)$$

with $\varepsilon: \Omega \to H$ a random variable being the noise and an unknown deterministic link function g such that

$$y \in \mathbb{R} \mapsto g(y) \in \mathcal{RV}_{\kappa}(+\infty), \quad \kappa > 0. \qquad (7)$$

The appeal of the model (6) stems from the fact that, for instance when $\varepsilon$ is independent of $Y$ and centered, the FEPLS direction (3) coincides with the true index: $w_{\varphi}(y) = \beta$ for any test function $\varphi \in \mathcal{RV}_{\tau}$ and $y \ge 0$. Heuristically, one expects, assuming more generally that $Y$ and $\varepsilon$ are dependent but $\varepsilon$ has a small contribution in the tail regime of $Y$, that $w_{\varphi}(y) \to \beta$ as $y \to +\infty$.

To deal with the conditional expectation involving the tail event $\{Y > y\}$ for large $y$, we need some regularity on the distribution of $Y$. So let us assume that the density of $Y$ exists and denote it by $f := F'$ which is also regularly-varying with index $1/\gamma - 1$ when (5) holds.

### 3. Inference and consistency

Let $(X_i, Y_i)_{1 \le i \le n}$ be independent copies of $(X, Y) \in H \times \mathbb{R}$. Let $k := k_n \to +\infty$ be an intermediate sequence, i.e. integer deterministic such that $k/n \to 0$ as $n \to +\infty$, and consider a deterministic sequence $y_{n,k}$ such that $y_{n,k} \sim U(n/k)$ as $n \to +\infty$. Denoting the order statistics of the sample $(Y_i)_{1 \le i \le n}$ by $Y_{1,n} \le \dots \le Y_{n,n}$, one may see that the empirical version of $U(n/k)$ is $Y_{n-k+1,n}$. In the sequel and for $\alpha > 0$, we denote by $C_{\text{loc}}^{0,\alpha}([0, 1], \mathbb{R})$ the space of locally $\alpha$-Hölder continuous functions, i.e., the set of $f: [0, 1] \to \mathbb{R}$ such that for any compact subset $K \subset [0, 1]$, there exists a finite constant $c > 0$ satisfying

$$|f(x) - f(y)| \le C_K |x - y|^\alpha \quad \text{for all } x, y \in K.$$

Concerning the noise, we require the existence of $q > 2$ and $\alpha_e > 0$ independent of $n$ such that

$$\limsup_{n \to +\infty} \sup_{y \ge 0} \mathbb{E} \left( \|\varepsilon\|_{0, \alpha_e}^q \mid Y_{n-k+1, n} = y \right) < +\infty. \qquad (8)$$

For any random variable $W$, the empirical counterpart of the tail-moment $m_W(y)$ (whenever it exists) is defined by

$$\hat{m}_W(y) := \frac{1}{n} \sum_{i=1}^{n} W_i 1_{\{Y_i \ge y\}},$$

basing on the $n$-sample $(W_i, Y_i)_{1 \le i \le n}$. Two situations are investigated. When the full functions $X_1, \dots, X_n$ are observed, it is then possible to estimate $w_{\varphi}(y)$ by replacing the tail moment in (3) by its empirical counterpart defined as

$$\hat{\beta}_{\varphi}(y) := \frac{\hat{v}_{\varphi}(y)}{\|\hat{v}_{\varphi}(y)\|} \quad \text{with} \quad \hat{v}_{\varphi}(y) = \hat{m}_{X_{\varphi}(Y)}(y) = \frac{1}{n} \sum_{i=1}^{n} X_i \varphi(Y_i) 1_{\{Y_i \ge y\}}.$$
The FEPLS estimator $\hat{\beta}_{\varphi}(y)$ is thus a linear combination of the $X_i$'s for which the associated $Y_i$'s are located in the tail of $\bar{F}$. The function $\varphi$ tunes the weight applied to each selected tail observation.

Let us now consider the more realistic situation where each function $X_i$, $i \in \{1, \dots, n\}$ is observed on a finite deterministic grid $t_1, \dots, t_N$. Introducing $(e_j)_{j \ge 1}$ an orthonormal basis of $H$, the (unobserved) function $X_i$ can be expanded as $X_i = \sum_{j \ge 1} \langle X_i, e_j \rangle e_j$ so that

$$X_i(t) = \sum_{j \ge 1} \int_0^1 X_i(u)e_j(u)e_j(t)du = \int_0^1 \psi(t, u)X_i(u)du, \quad (9)$$

where $\psi(t, u) = \sum_{j \ge 1} e_j(t)e_j(u)$ for any $t, u \in [0, 1]$. In view of (9), $\psi$ can be considered as a reproducing kernel associated with $H$. Let us introduce a partition $\{I_1, \dots, I_N\}$ of $[0, 1]$ such that $t_j \in I_j$ for any $j \in \{1, \dots, N\}$. It follows that

$$X_i(t) = \sum_{j=1}^{N} \int_{I_j} \psi(t, u) X_i(u) du, \quad t \in [0, 1],$$

and one can then consider the reconstructed version of $X_i$ defined as

$$\tilde{X}_i = \sum_{j=1}^{N} \int_{I_j} \psi(\cdot, u) du X_i(t_j) =: \sum_{j=1}^{N} X_i(t_j) \bar{\psi}_j,$$

with $\bar{\psi}_j(t) := \int_{I_j} \psi(t, u) du, t \in [0, 1]$. In this context, the estimator of $\beta$ is given by

$$\tilde{\beta}_{\varphi}(y) := \frac{\tilde{\nu}_{\varphi}(y)}{\|\tilde{\nu}_{\varphi}(y)\|} \quad \text{with} \quad \tilde{\nu}_{\varphi}(y) := \hat{m}_{\tilde{X}_{\varphi}(Y)}(y) = \frac{1}{n} \sum_{i=1}^{n} \tilde{X}_i \varphi(Y_i) 1_{\{Y_i \ge y\}}.$$

Let us now state our consistency result under the inverse single-index model.

**Theorem 1.** Assume $\beta \in C_{\text{loc}}^{0, \alpha_b}([0, 1], \mathbb{R})$ for some $\alpha_b > 0$. Assume

$$0 < 2(\kappa + \tau)\gamma < 1, \quad (10)$$

$$q\kappa\gamma > 1, \quad (11)$$

along with (4), (5), (7), (8) and $\sqrt{k}A(n/k) = O(1)$. Assume that $\varphi$ and $g$ are continuously differentiable in a neighbourhood of infinity such that $t(\varphi g)'(t)/(\varphi g)(t) \to \tau + \kappa \neq 0$ and $t\varphi'(t)/\varphi(t) \to \tau \neq 0$ as $t \to +\infty$. Then, for any deterministic sequences $k := k_n \to +\infty$ with $k = o(n)$, any $y_{n,k} \sim U(n/k)$ and any $N := N_n \to +\infty$, it holds that:

$$\|\tilde{\beta}_{\varphi}(Y_{n-k+1,n}) - \beta\| \xrightarrow[n \to +\infty]{P} 0,$$

with rate $\delta_{n,k} \vee N^{-\alpha_b} \vee \{N^{-(\alpha_b \wedge \alpha_e)}\}_{e \in \mathcal{N}_{n,k}} \vee \{\delta_{n,k}^2 N^{-\alpha_e}\}$ where $\delta_{n,k} := (g(y_{n,k})(k/n)^{1/q})^{-1}$.

Let us highlight that the sub-rate $\delta_{n,k}$ is a regularly-varying function of $k/n$ with index $\gamma\kappa - 1/q > 0$. It only depends on the tail distribution of $Y$, the integrability order of the noise $\varepsilon$ and the asymptotic behaviour of the deterministic link function $g$.

We now state some comments about the set of assumptions and their implications.
**Remark 1 (Model hypotheses).** Assumption *U* ∈ ℝV<sub>γ</sub>(+∞) or F̄ ∈ ℝV<sub>-1/γ</sub>(+∞) is equivalent to assuming that the distribution of *Y* is in the Fréchet maximum domain of attraction with positive tail-index *γ*, see Bingham et al. (1987, Theorem 1.5.8) and de Haan and Ferreira (2006, Theorem 1.2.1). This domain of attraction consists of heavy-tailed distributions, such as Pareto, Burr and Student distributions, see Beirlant et al. (2004) for further examples. The larger *γ* is, the heavier the tail. Going beyond the first order, the control of some tail-moment as in Stupfler (2019) requires (5) which is expressed in terms of tail quantile functions. The restriction to *γ* < 1 ensures that the first-order moment E(Y1_{Y≥y}) exists for all y ≥ 0. Assumption (7) ensures that the link function *g* ultimately behaves like a power function. Combined with *U* ∈ ℝV<sub>γ</sub>(+∞), it implies that *g*(*Y*) is heavy-tailed with tail-index γ<sub>g(Y)</sub> := κγ. Similarly, (4) and *U* ∈ ℝV<sub>γ</sub>(+∞) yield that φ(*Y*) is heavy-tailed with tail-index γ<sub>φ(Y)</sub> := τγ. Next, (8) ensures the uniform control of the q-mean of ||ε||<sub>0,α<sub>c</sub></sub> given the random threshold, here modelled by the extreme order statistic Y<sub>n−k+1,n</sub>. A direct consequence is that E(||ε||<sup>q</sup>) < +∞. Since q > 2, one has E(||ε||) < +∞ and, by Hytonen et al. (2016, Proposition 1.2.2), ε is also Bochner-integrable since ε is strongly P-measurable by definition. More importantly, E(||ε||<sup>q</sup>) < +∞ may be interpreted as an assumption on the tail of ||ε||. It is satisfied, for instance, by distributions with exponential-like tails such as Gaussian, Gamma or Weibull distributions. Heuristically, it would imply that the tail-index associated with ||ε|| is such that γ<sub>||ε||</sub> < 1/q. Condition (11) thus imposes that γ<sub>g(Y)</sub> > γ<sub>||ε||</sub>, meaning that g(Y) has a heavier right tail than ||ε||. Under model (6), the tail behaviors of ⟨β, X⟩ and ||X|| are thus driven by g(Y), i.e., γ<sub>||X||</sub> = γ<sub>g(Y)</sub>, which is the desired property.

**Remark 2 (Inference hypotheses).** The regime $y_{n,k} \sim U(n/k)$ entails $n\bar{F}(y_{n,k}) \sim k \to \infty$ as $n \to \infty$ and it still holds when $y_{n,k}$ is replaced by its empirical counterpart, namely $Y_{n-k+1,n}$. Whence, the average number of tail observations, i.e. larger than the threshold being random or not, increases as the sample size grows. This roughly ensures that the threshold does not grow too fast so that there are enough data points in the inference scheme. The number of such extreme points is ruled by the usual condition $\sqrt{k}A(n/k) = O(1)$ in extreme-value theory, which is fulfilled when $k \sim c.n^{-2\rho/(1-2\rho)}$, $\rho < 0$, $c > 0$, see de Haan and Ferreira (2006, Equation (3.2.10)). The constraint (11) is the translation in the regular variation language of the fact that $\delta_{n,k} \to 0$ as $n \to \infty$ regardless of the underlying slowly-varying part of $g$ and $\bar{F}$. The rate $\delta_{n,k}$ is indeed of order $n^{(1/q-\gamma k)/(1-2\rho)}$. Condition (10) implies the existence of the second moment of $\varphi \cdot g(Y)1_{\{Y \ge y\}}$ for all $y \ge 0$. To this end, one should pick $\tau < 1/(2\gamma) - \kappa$, this upper bound being either positive or negative. The influence of $\tau$ on the FEPLS estimate is illustrated on simulations in the next Section 4.

**4. Illustration on simulated data**

The performance of our method is assessed by a Monte Carlo simulation experiment with 500 independent repetitions. The ambient space is $H = L^2([0, 1])$ endowed with its usual inner product $\langle f, g \rangle = \int_0^1 f(t)g(t)dt$. Functional observations are discretized on a regular grid of length $N \in \{101, 1001\}$ with nodes $t_j = (j-1)/(N-1)$ for $1 \le j \le N$. To implement the $L^2$-inner product in the discretized setting, we consider the symmetric positive definite matrix $\Psi \in \mathbb{R}^{N \times N}$ given by $\Psi_{j,j'} = \int_{I_{j'}} \bar{\psi}_j = \int_{I_j} \int_{I_{j'}} \psi(u, v)dudv$ for all $(j, j') \in \{1, \dots, N\}^2$ associated to the truncated Fourier basis functions (up to the first 50). For any discretized functions $u = (u(t_j))_{1 \le j \le N}, v = (v(t_j))_{1 \le j \le N} \in \mathbb{R}^N$, the quadratic form $\langle u, v \rangle_{\Psi} := u^T \Psi v$ induces an inner product and all subsequent computations use $\| \cdot \|_{\Psi} := \sqrt{\langle \cdot, \cdot \rangle_{\Psi}}$ as the induced norm.
### 4.1. Experimental design

We draw a data sample of $n = 500$ independent replications ($X_i, Y_i$) of $(X, Y)$ using the following scheme:

* $Y$ has a Burr distribution, i.e., $\bar{F}(y) = (1 + y^{-\rho/\gamma})^{1/\rho} \in 2\mathbb{R}^V_{-1/\gamma, \rho/\gamma}(+\infty)$ with $y \ge 0$, $\gamma \in (0, 1)$ being the tail-index and $\rho < 0$ being the second-order parameter. Here, $\gamma \in \{1/3, 1/2, 9/10\}$ while $\rho \in \{-2\gamma, -\gamma/2\}$.

* $\varepsilon | Y = y$ has the same distribution as $\sigma(y)B_y^H + \mu(y)$ where $B_y^H$ is the fractional Brownian motion (fBm) on $[0, 1]$ with conditional Hurst parameter $H(y) \in (0, 1)$, $\sigma(y)$ is the conditional noise deviation and $\mu(y)$ the conditional mean.

* $\beta : t \in [0, 1] \mapsto \sqrt{2} \sin(2\pi t)$ and $g : y \in \mathbb{R}^+ \mapsto y^\kappa$ with $\kappa \in \{1, 3/2, 2\}$.

Hence, conditionally on $Y = y$, the noise process based on $B_y^H$ is centered Gaussian with conditional covariance function

$$ (s, t) \mapsto \frac{\sigma^2(y)}{2} (t^{2H(y)} + s^{2H(y)} - |t - s|^{2H(y)}) $$

Assumption (8) thus holds for all $q > 0$. Note that $H(y) = 1/2$ yields the Brownian motion $B_y$ with mean $\mu(y)$ and variance $\sigma^2(y)$.

Once both the response variable $Y$ and the noise $\varepsilon$ are simulated, and the deterministic functions $\beta$ and $g$ are chosen, the covariate sampling $\{\tilde{X}_1, \dots, \tilde{X}_n\}$ is readily derived from (6). The choice of $\sigma$ is guided by the following reasonable requirement: One should ensure that the contribution in (6) of $\varepsilon$ does not overwhelm $g(Y)$, but also conversely, in order to avoid a trivial case where the noise is not impacting. To this end, we propose to pick $\sigma(y) = g(y)/10$. Moreover, we fix $(H, \mu) \equiv (1/3, 200)$ regardless of $Y = y$.

The FEPLS estimator is implemented with $\phi(y) = y^\tau$ as test function. The selected values of $\tau$ depend on the pair $(\gamma, \kappa)$ in order to fulfil condition (10). They are given in Table 1.

<table><caption>Table 1: Selected values of $\tau$ in the nine simulated situations.</caption><thead><tr><th></th><th>γ = 1/3</th><th>γ = 1/2</th><th>γ = 1/9</th></tr></thead><tbody><tr><td>κ = 1</td><td>{0, -1, -2}</td><td>{-1, -2, -3}</td><td>{-1, -2, -3}</td></tr><tr><td>κ = 3/2</td><td>{-1, -2, -3}</td><td>{-1, -2, -3}</td><td>{-1, -2, -3}</td></tr><tr><td>κ = 2</td><td>{-1, -2, -3}</td><td>{-2, -3, -4}</td><td>{-2, -3, -4}</td></tr></tbody></table>

### 4.2. Selection of the threshold

We consider the following data-driven selection rule of $k \in \{1, \dots, n\}$ based on the maximization of the sample covariance between $Y$ and $\langle \tilde{\beta}_\phi(Y_{n-k+1,n}), \tilde{X}_{(n-i+1,n)} \rangle$ when $Y \ge Y_{n-k+1,n}$:

$$ \tilde{k} := \arg\max_{5 \le k \le n/5} r(k), $$

with $r(k)$ being

$$ \frac{1}{k} \sum_{i=1}^{k} Y_{n-i+1,n} \langle \tilde{\beta}_{\phi}(Y_{n-k+1,n}), \tilde{X}_{(n-i+1,n)} \rangle - \frac{1}{k} \sum_{i=1}^{k} Y_{n-i+1,n} \frac{1}{k} \sum_{i=1}^{k} \langle \tilde{\beta}_{\phi}(Y_{n-k+1,n}), \tilde{X}_{(n-i+1,n)} \rangle, $$
and where $\tilde{X}_{(n-i+1,n)}$ denotes the concomitant of $Y_{n-i+1,n}$, i.e., the random variable $\tilde{X}_s$ with $s \in \{1, \dots, n\}$ being the unique index such that $Y_s = Y_{n-i+1,n}$. Let us highlight that the above selection rule imposes $\tilde{k} \ge 5$ in order to prevent instabilities of estimates built on too few data points.

**Remark 3.** Let us denote $X_i = (X_i(t_1), \dots, X_i(t_N))^\top$. For any $i, \ell \in \mathbb{R}^n$, one has $\langle \tilde{X}_i, \tilde{X}_\ell \rangle = \langle X_i, X_\ell \rangle_\psi$ so that

$$ \begin{aligned} \|\tilde{v}_\varphi(y)\|^2 &= \frac{1}{n^2} \sum_{i, \ell=1}^n \langle \tilde{X}_i, \tilde{X}_\ell \rangle \varphi(Y_i) \varphi(Y_\ell) 1_{\{Y_i \ge y\}} 1_{\{Y_\ell \ge y\}} \\ &= \frac{1}{n^2} \sum_{i, \ell=1}^n \langle X_i, X_\ell \rangle_\psi \varphi(Y_i) \varphi(Y_\ell) 1_{\{Y_i \ge y\}} 1_{\{Y_\ell \ge y\}} = \|V_\varphi(y)\|_\psi^2 \end{aligned} $$

where $V_\varphi(y) := \frac{1}{n} \sum_{i=1}^n X_i \varphi(Y_i) 1_{\{Y_i \ge y\}} \in \mathbb{R}^N$ can be interpreted as a sampled version of $\tilde{v}_\varphi(y)$. Now, to compute $\langle \tilde{\beta}_\varphi(y), \tilde{X}_s \rangle$, we write, for the numerator,

$$ \langle \tilde{v}_\varphi(y), \tilde{X}_s \rangle = \frac{1}{n} \sum_{i=1}^n \langle \tilde{X}_i, \tilde{X}_s \rangle \varphi(Y_i) 1_{\{Y_i \ge y\}} = \frac{1}{n} \sum_{i=1}^n \langle X_i, X_s \rangle_\psi \varphi(Y_i) 1_{\{Y_i \ge y\}}, $$

the denominator already given being by: $\|\tilde{v}_\varphi(y)\| = \|V_\varphi(y)\|_\psi$.

### 4.3. Results

Figure 1 displays the dot product curves $k \in \{1, \dots, 250\} \mapsto \langle \tilde{\beta}_\varphi(Y_{n-k+1,n}), \beta \rangle$ averaged on the 500 Monte Carlo replications of the nine considered simulated situations.

**Remark 4.** Computation of $\langle \tilde{\beta}_\varphi(y), \beta \rangle$. Let us focus on the numerator:

$$ \langle \tilde{v}_\varphi(y), \beta \rangle = \frac{1}{n} \sum_{i=1}^n \langle \tilde{X}_i, \beta \rangle \varphi(Y_i) 1_{\{Y_i \ge y\}} = \frac{1}{n} \sum_{i=1}^n \sum_{j=1}^N X_i(t_j) \langle \tilde{\psi}_j, \beta \rangle \varphi(Y_i) 1_{\{Y_i \ge y\}} $$

with

$$ \langle \tilde{\psi}_j, \beta \rangle = \int_0^1 \int_{I_j} \psi(t, u) \beta(t) du dt = \int_{I_j} \int_0^1 \psi(t, u) \beta(t) dt du = \int_{I_j} \beta(u) du =: \bar{\beta}_j, $$

and thus $\langle \tilde{v}_\varphi(y), \beta \rangle = V_\varphi(y)^\top \bar{\beta}$.

It appears that the estimated index $\tilde{\beta}_\varphi(Y_{n-k+1,n})$ is close to the true one only for small values of $k$, or equivalently for large values of the threshold $Y_{n-k+1,n}$. This phenomenon is coherent with the fact that $\langle \beta, X \rangle$ is designed to capture the tail-information of $Y$. Let us highlight that the range of accurate values for $k$ depends on the difficulty of the estimation problem. The higher $\gamma$ and $\kappa$, the heavier the tail of $g(Y)$, and the easier the estimation of $\beta$ is. In practice, the selection of $k$ is performed using the above described selection procedure since $\beta$ is unknown and therefore Figure 1 cannot be used in real data situations.

Finally, the mean over the 500 replications of the curves $t \in [0, 1] \mapsto \tilde{\beta}_\varphi(Y_{n-\tilde{k}+1,n})(t)$ is compared to the true one $t \in [0, 1] \mapsto \beta(t)$ on Figure 2 in the case where $\tau = -2$. It appears that the selected values of $k$ yield very accurate estimations on all nine considered situations whatever the values of $\gamma$ or $\kappa$. Pushing further, Figure 3 is the analogous to Figure 2 for when
the dimension of the underlying grid is $N = 1001$. This aims to a better reflection of the high dimensional regime as the data size is $n = 500$. One may observe a slight degradation of the estimation in comparison to Figure 2 mostly when $\kappa = 1$. It appears again that, the higher $\gamma$ (heavier tail), the easier the inference becomes. Similarly, the larger $\kappa$, the more accurate the inference. Indeed, in view of the inverse regression model $X = g(Y)\beta + \varepsilon$ and the composition rule for regularly-varying functions, $\kappa$ also contributes to the leading tail-index.

## 5. Application on real data

We illustrate the FEPLS method on a financial application where covariates naturally take a functional form due to high-frequency sampling. Introduce the list of assets<fnref n="2" />

$$ \mathcal{A} := \{\text{S&P500, DAX30, Nikkei225, ETH, BTC}\} $$

where ETH and BTC respectively stands for Ethereum and Bitcoin, everything understood in their canonical currency. We consider the daily maximum log-return of the some asset $\mathcal{A}_1 \in \mathcal{A}$ as the response variable $Y$, and the intraday log-return curve of some asset $\mathcal{A}_2 \in \mathcal{A}$, possibly the same as $\mathcal{A}_1$, as the functional covariate $X$. Any asset at disposal is recorded at one-minute frequency over the period 2013–2017.

The principal motivation for applying FEPLS in this context is to facilitate the inference of conditional extreme risk measures. By replacing the functional covariate with a univariate projection $Z = \langle \tilde{X}, \tilde{\beta} \rangle_{\psi}$, the estimation of quantities such as the conditional Value-at-Risk (VaR) or the conditional tail index becomes computationally simpler and statistically more stable in the extreme regime. The core question is then whether the one-dimensional projection retains the covariate information that is relevant for predicting extreme responses. To answer it we compare the performance of FEPLS against natural competitors on the ability to capture tail dependence, based on the particular pair $(\mathcal{A}_1, \mathcal{A}_2) = (\text{DAX30, ETH})$ and then on all the possible asset pairs from $\mathcal{A}$. Ultimately and as a practical tool for extreme risk assessment, we include the estimation of extreme quantiles.

Concerning the tail tuning parameter of $\varphi(y) = y^{\tau}$, we arbitrarily set throughout this section $\tau = -1$. A study on the impact of $\tau$ on the analysis is out of the scope of the paper.

### 5.1. Sample Construction

The raw price series of both $\mathcal{A}_1, \mathcal{A}_2$ are transformed into log-returns $R_t = \log(P_t/P_{t-1})$ where $P_t$ denotes the price at time $t$. The two resulting series are aligned by date so that only contemporaneous observations are kept. This yields approximately the same number of one-minute observations for both indices over the common period Jan 1, 2013 - Dec 31, 2017. The discretized functional covariate sample $\{X_i\}_{1\le i \le n}$ is constructed by cutting the $\mathcal{A}_1$ log-return series into consecutive blocks of $N = 1440$ minutes (24 hours). Each block corresponds to one trading day, and the vector $X_i = (X_i(t_1), \dots, X_i(t_N))^{\top}$ contains the intraday log-returns of the $\mathcal{A}_1$ on day $2i - 1$. The response sample $\{Y_i\}_{1\le i \le n}$ is obtained from the $\mathcal{A}_2$ log-returns: for each even day $2i$, we take the maximum of the absolute log-returns over that day,

$$ Y_i = \max_{t \in \text{day } 2i} |R_t^{\mathcal{A}_2}|. $$

<footnote n="2">²Available at https://github.com/FutureSharks/financial-data/tree/master/pyfinancialdata/data/</footnote>
<figure><img src="image_2.png" /><figcaption>Figure 1: Finite sample behaviour of $\tilde{v}_{\varphi}(Y_{n-k+1,n})$ for different admissible values of $\tau$ (see Table 1) under the simulated inverse model with Burr response and conditional fBm noise. The x-axis represents the number $k$ of exceedances and the y-axis is the average value of $\langle \tilde{\beta}_{\varphi}(Y_{n-k+1,n}), \beta \rangle$ over $N = 500$ Monte Carlo replications. Here, $\rho = -1/2$ and $N = 101$.</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>(a) $\kappa = 1, \gamma = 1/3$.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>(b) $\kappa = 1, \gamma = 1/2$.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>(c) $\kappa = 1, \gamma = 9/10$.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>(d) $\kappa = 3/2, \gamma = 1/3$.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>(e) $\kappa = 3/2, \gamma = 1/2$.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>(f) $\kappa = 3/2, \gamma = 9/10$.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>(g) $\kappa = 2, \gamma = 1/3$.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>(h) $\kappa = 2, \gamma = 1/2$.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>(i) $\kappa = 2, \gamma = 9/10$.</figcaption></figure>

Figure 2: Simulation results on the inverse model with Burr response and conditional fBm noise. The orange curve is the graph of $t \in [0, 1] \mapsto \beta(t)$ while the blue one is the averaged value of $t \in [0, 1] \mapsto \tilde{\beta}_{\varphi}(Y_{n-\tilde{k}+1,n})(t)$ over $N = 500$ Monte Carlo replications, where $\tilde{k} = \arg \max_{5 \le k \le n} r(k)$. The light blue area corresponds to the confidence region comprising the top 5–95% values of the Monte Carlo replications. Here, $\rho = -1/2$, $\tau = -2$ and $N = 101$.
<figure><img src="image_12.png" /><figcaption>(a) $\kappa = 1, \gamma = 1/3$.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>(b) $\kappa = 1, \gamma = 1/2$.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>(c) $\kappa = 1, \gamma = 9/10$.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>(d) $\kappa = 3/2, \gamma = 1/3$.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>(e) $\kappa = 3/2, \gamma = 1/2$.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>(f) $\kappa = 3/2, \gamma = 9/10$.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>(g) $\kappa = 2, \gamma = 1/3$.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>(h) $\kappa = 2, \gamma = 1/2$.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>(i) $\kappa = 2, \gamma = 9/10$.</figcaption></figure>

<caption>Figure 3: Simulation results akin to Figure 2 for $N = 1001$.</caption>
No missing data were detected and the final sample size *n* is among {265, 365, 433, 442}. It may even be equal to 901 when $\mathcal{A}_1 = \mathcal{A}_2 = \text{BTC}$. As for our particular study case, the pair DAX30-ETH consists in *n* = 265 observations.

**Remark 5** (Temporal dependence). Although the theoretical guarantees of Section 2 assume i.i.d. observations, financial returns typically exhibit serial dependence and volatility clustering. A common remedy is to thin the data by keeping only observations spaced by several days, see Cai et al. (2015, Section 4). Here, it means that for some fixed day-gap *m* ≥ 3, the value $Y_i$ corresponds to the mi-th day and the discretized curve $\tilde{X}_i$ to the day mi − 1.

5.2. Response Tail Behaviour

Before applying FEPLS, we need to verify that the response variable *Y* exhibits a heavy tail. The evidence for the pair DAX30-ETH is given by the Hill plot in Figure 4, a standard diagnostic tool from extreme-value theory which represents the graph of the Hill estimator

$$
\hat{\gamma}(k) = \frac{1}{k} \sum_{i=1}^{k} (\log Y_{n-i+1,n} - \log Y_{n-k,n})
$$

as a function of $k \in \{15, \dots, 65\}$, see Hill (1975). As such, Figure 4 exhibits a stable behavior
over a range of intermediate values of $k$, with an estimated tail index around $\hat{\gamma} \approx 1/2$.

5.3. Threshold Selection

Applying the selection rule of Section 4.2, we plot the logarithm of the empirical absolute covariance between *Y* and ⟨*β*<sub>ϕ</sub>, *X̃*⟩ in Figure 5, namely *k* ↦→ log |*r*(*k*)| for *k* ∈ {15, . . . , 65}. Se- lecting the argument of |*r*(*k*)| that gives its maximum value hence leads to *k̃* = 6, which means that the extreme regime approximatively consists in the top 2.26% of the 265 response values. This is used as the effective sample size for all subsequent extreme-tail analyses about the pair DAX30-ETH. We should mention that different τ would lead to very distinct outcomes for *k̃*, e.g., for τ = 1, one has *k̃* = 17.

5.4. Tail Dependence Competition

To assess the effectiveness of our method, we evaluate its performance in a tail covariance
competition against several projection strategies on the present high-dimensional discretized
functional data, i.e., for the pair DAX30-ETH and then on all possible pairs. A variety of baseline
competitor strategies is considered: random projections, along with standard dimension reduc-
tion methods and their extreme variants (with e-prefix) computed only on the tail region of the
data sample. More precisely, fix a threshold $y > 0$ and define the exceedances index set as

$$
\mathcal{I}_y := \{i \in \{1, \dots, n\} : Y_i > y\}, \quad n_y := |\mathcal{I}_y|.
$$

Let $\tilde{X}_y \in \mathbb{R}^{n_y \times N}$ denote the submatrix with rows $\{\tilde{X}_i^\top : i \in \tilde{I}_y\}$ and $Y_y \in \mathbb{R}^{n_y}$ the corresponding tail responses. Let $\mathcal{M} := \{\text{FEPLS}, \text{eFPCA}, \text{eFSIR}, \text{FPCA}, \text{FSIR}\}$, then each method $m \in \mathcal{M}$ produces a projection score

$$
Z_{m,y} := \tilde{X}_y \Psi \beta_m(y) \in \mathbb{R}^{n_y},
$$

where the method-specific direction $\beta_m(y) = (\beta_m(t_1), \dots, \beta_m(t_N))^\top \in \mathbb{R}^N$ (or $\beta_m$ if the method
does not depend on $y$) is defined below, and $\Psi \in \mathbb{R}^{N \times N}$ as in Section 4 associated to the first 50
Fourier basis functions.
For each $m \in \mathcal{M}$, we compute the unbiased empirical covariance between the method's score and the tail responses, as the metric performance for our analysis,

$$ \rho_m(y) := \frac{1}{n_y - 1} \sum_{i \in \mathcal{I}_y} ((Z_{m,y})_i - \bar{Z}_{m,y}) (Y_i - \bar{Y}_y), $$

with $\bar{Z}_{m,y} = n_y^{-1} \sum_{i \in \mathcal{I}_y} (Z_{m,y})_i$ and $\bar{Y}_y = n_y^{-1} \sum_{i \in \mathcal{I}_y} Y_i$. The standard full-sample versions are recovered as special cases when $\mathcal{I}_y = \{1, \dots, n\}$. All competitors are implemented in their discretized form with the $\Psi$-inner product rather than the standard $L^2$ inner product.

(i) Extreme Functional PCA (eFPCA). Let $\bar{\tilde{X}}_y := n_y^{-1} \sum_{i \in \mathcal{I}_y} \tilde{X}_i$ be the empirical mean over the tail. Define the empirical covariance matrix of the tail covariates:

$$ \hat{\Sigma}_y := \frac{1}{n_y} \sum_{i \in \mathcal{I}_y} (\tilde{X}_i - \bar{\tilde{X}}_y)(\tilde{X}_i - \bar{\tilde{X}}_y)^{\top} \in \mathbb{R}^{N \times N}. $$

The eFPCA direction $\beta_{eFPCA}(y)$ is the eigenvector corresponding to the largest eigenvalue of $\hat{\Sigma}_y$:

$$ \beta_{eFPCA}(y) := \arg \max_{\beta \in \mathbb{R}^N, \|\beta\|_{\Psi}=1} \beta^{\top} \hat{\Sigma}_y \beta, $$

where $\|\beta\|_{\Psi}^2 := \beta^{\top} \Psi \beta$.

(ii) Extreme Functional SIR (eFSIR). Partition $\mathcal{I}_y$ into 5 equal-frequency slices $S_1, \dots, S_5$ according to the ordered values of $\{Y_i : i \in \mathcal{I}_y\}$. For each slice $1 \le h \le 5$, define the slice mean $m_h := \frac{1}{|S_h|} \sum_{i \in S_h} \tilde{X}_i$. The between-slice covariance matrix is

$$ \hat{\Sigma}_{B,y} := \sum_{h=1}^{5} \frac{|S_h|}{n_y} (m_h - \bar{\tilde{X}}_y)(m_h - \bar{\tilde{X}}_y)^{\top}. $$

The eFSIR direction $\beta_{eFSIR}(y)$ is the solution of the generalized eigenvalue problem

$$ \hat{\Sigma}_{B,y} \beta = \lambda \hat{\Sigma}_y \beta, $$

taken for the eigenvalue $\lambda$ with largest absolute value. In practice we solve the regularized problem $(\hat{\Sigma}_y + \eta I_N)\beta = \lambda^{-1}\hat{\Sigma}_{B,y}\beta$ with $\eta = 10^{-6} \text{tr}(\hat{\Sigma}_y)/N$ and $I_N$ being the identity matrix of size $N$. The full-sample FSIR direction $\beta_{FSIR}$, obtained by replacing $\mathcal{I}_y$ with $\{1, \dots, n\}$, is using 10 slices.

(iii) Random projections. As a naive baseline, we generate 5000 independent random directions $(\beta^{(r)} / \|\beta^{(r)}\|_{\Psi})_{1 \le r \le 5000}$ by drawing $\beta^{(r)} \sim N(0, I_N)$.

### 5.5. Competition results

In Figure 6, we plot the graph of $k \mapsto |\rho_m(Y_{n-k+1,n})|$ for all $m \in \mathcal{M}$ and the fixed pair $(\mathcal{A}_1, \mathcal{A}_2) = (\text{DAX30, ETH})$. One may see that FEPLS dominates the competitors across almost all thresholds $k$. The gray shaded region represents the inter-quartile range (IQR) at level 5-95% of covariances obtained from 5000 random directions. It highlights the fact that the observed performance of FEPLS is far above what would be expected by chance. Lastly, one
may also notice that the extreme versions of the competitor strategies consistently outperform their full-sample counterpart.

Extending Figure 6 across dataset-threshold combinations, Figure 7 reports the evolution, as a function of the threshold level $k$, of the mean rank of the different dimension-reduction methods in terms of absolute tail covariance with the response, based upon the 25 possible pairs and all threshold levels $k$. For each value of $k$, methods are ranked from best (rank 1) to worst, and the displayed curves correspond to ranks averaged over the 25 asset pairs; hence, lower values indicate better overall performance. One may observe that FEPLS consistently attains the lowest mean rank across the entire range of thresholds. Moreover, classical full-sample methods systematically exhibit higher mean ranks than their extreme counterparts, especially for smaller values of $k$.

Finally and as a complement to the previous plot, Figure 8 displays the proportion of win rate in which FEPLS outperforms all other competing methods across different threshold levels $k$. To formally support the claim that the superior performance of FEPLS is statistically significant and not attributable to random variations, we employ a one-sided binomial test under the null hypothesis of equal performance across all five methods. Under the null hypothesis, each method has equal 20% of winning probability. With 25 (supposedly independent) datasets, the number of wins $W$ for FEPLS follows $W \sim \text{Binomial}(25, 0.20)$. The critical value for 5%-significance corresponds to 9 wins over those 25 trials, meaning a win rate of 36%, since $P(W \ge 9) = 0.0468$, as visualized by the horizontal green line. The plot shows that FEPLS achieves a win rate consistently above 36% across all dataset-threshold combinations, providing evidence against the null hypothesis.

## 5.6. Conditional extreme quantile inference

To illustrate how our method can be used for risk measurement in the extreme regime, we go back to the particular case ($A_1, A_2$) = (DAX30, ETH) and study its related conditional quantile inference. For a generic random variable $V \in H$, let us denote by $F_{Y|V=v}$ the conditional cumulative distribution function of $Y$ given $V = v$, where $v \in H$. The (functional) Nadaraya-Watson estimator of $F_{Y|V=v}$ is

$$ \hat{F}_{Y|V=v}(y) := \sum_{i=1}^{n} K\left(\frac{\|V_i - v\|}{h_n}\right) \mathbf{1}_{\{Y_i \le y\}} \left| \sum_{s=1}^{n} K\left(\frac{\|V_s - v\|}{h_n}\right) \right|, \quad y \in \mathbb{R}. \quad (12) $$

Here, the kernel $K$ is Gaussian and the parameter $h_n \downarrow 0$ is the window bandwidth. The conditional quantile function of $Y|V = v$ is defined thanks to the generalized inverse $q_{Y|V=v} := \hat{F}_{Y|V=v}^{-1}$ and the associated estimator writes as

$$ \hat{q}_{Y|V=v}(\alpha) := \inf\{y > 0, \hat{F}_{Y|V=v}(y) \ge \alpha\}, \quad \alpha \in (0, 1), $$

see Ferraty and Vieu (2006, Section 6.4) for details.

Figure 9 shows the scatterplot for the pair DAX30-ETH of the projected data on the FEPLS direction $((\tilde{X}_i, \tilde{\beta}_\psi), Y_i)$ for $1 \le i \le n$ endowed with the estimated conditional Value-at-Risk curves $t \mapsto \hat{q}_{Y|((\tilde{X}_i, \tilde{\beta}_\psi)=t)}(\alpha)$ associated to risk levels $\alpha \in \{0.95, 0.98, 0.995\}$. Herein, the kernel inference relies on inverting the univariate version of (12) with bandwidth $h_n$ arbitrarily chosen to be $5.10^{-5}$. The red dots are associated with the $\tilde{k}$ pairs with highest response values. Such a graphic may be used as a visualization tool to detect the most risky situations. Indeed, given a new pair $(x_0, y_0)$, one may determine whether $y_0$ given $\tilde{X} = x_0$ exceeds the conditional Value-at-Risk $\hat{q}_{Y|((\tilde{X}_i, \tilde{\beta}_\psi)=x_0, y_0)}(\alpha)$
simply by comparing its projected position $((x_0, \tilde{\beta}_{\epsilon}), y_0)$ to the quantile curve on Figure 9. Additional experiments (not reported there) show that the FEPLS outputs are not very sensitive to $\tau$ in this real data application as well as to $\alpha \le 0.98$ for the extreme quantile estimation.
<figure><img src="image_21.png" /><figcaption>Figure 4: Hill plot for the response variable in the pair DAX30-ETH showing $k \mapsto \hat{y}(k)$ where k is the number of extreme observations.</figcaption></figure>
<figure><img src="image_22.png" /><figcaption>Figure 5: Threshold selection for the DAX30-ETH pair. The plot displays $k \mapsto \log|r(k)|$. The optimal threshold is obtained at $\bar{k} = 6$ (red dot).</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>Figure 6: Comparison for the DAX30-ETH pair of FEPLS against four competing methods across threshold values *k* in the x-axis. Each curve shows $|\rho_m(Y_{n-k+1,n})|$ the absolute unbiased covariance between the method's projection and the response for the top *k* observations. The grey shaded region represents the 5–95% inter-quantile range of 5000 random projections.</figcaption></figure>
<figure><img src="image_24.png" /><figcaption>Figure 7: Evolution of mean rank performance across thresholds for the different methods. Results are based on the evaluation over 25 assets pairs.</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>Figure 8: Top 1 percentage of FEPLS across threshold values *k*. The green line shows the statistical significance threshold derived from a binomial test at level 5% with 25 datasets and 5 competing methods.</figcaption></figure>
<figure><img src="image_26.png" /><figcaption>Figure 9: Conditional extreme quantile curves estimated via FEPLS for the DAX30-ETH pair. The plot shows the FEPLS projection score $Z = \langle \hat{X}, \hat{\beta}_{\psi} \rangle_{\psi}$ versus the response $Y$, with estimated conditional Value-at-Risk curves at different levels. Red dots mark the $\bar{k} = 6$ most extreme observations; gray points show all projected data.</figcaption></figure>
# Appendix A. Technical results

This section gathers all the mathematical details behind Section 2 and Section 3. We start with Appendix A.1 providing useful technical tools which enable to manipulate inner products under conditional expectations, or the distribution of order statistics allowing an extension of Bousebata et al. (2023, Lemma 2) to the random threshold framework. This yields theoretical tail-moments asymptotics which will be extensively used throughout the paper. The section also contains the proof of Proposition 1. Next, the asymptotic behaviour of the relevant empirical tail-moments, involving the noise and its inner product with $\beta$, are investigated in Appendix A.2. Finally, our main Theorem 1 is established in Appendix A.3.

## Appendix A.1. Preliminaries

A standard result in the literature of Bochner spaces, e.g., Hytonen et al. (2016, Eq (1.2)), states that Bochner integrals commute with bounded linear operators such as orthogonal projections. Note that this is in fact the definition of Pettis integrals. In the Hilbert framework, projections are generally expressed in terms of the inner product. The following technical tool considers this commutation operation with the inner product when the two random variables are conditionally stochastically independent. The case of one variable being deterministic will also often occur in the proofs. In the following, we denote by $E_F$ the conditional expectation given a $\sigma$-field $\mathcal{F}$ and if moreover the integrand is Banach-valued, we denote by $E_{F,\mathcal{B}}$ the Bochner version of the conditional expectation.

**Lemma 1.** For any independent Bochner-integrable random variables $W_1, W_2$ having values in $(H, \langle \cdot, \cdot \rangle)$ a separable Hilbert space and being independent conditionally to $\mathcal{F} \in \mathcal{B}(\mathbb{R})$,

$$E_F((W_1, W_2)) = \langle E_{F,\mathcal{B}}(W_1), E_{F,\mathcal{B}}(W_2) \rangle.$$

*Proof.* Let $i \in \{1, 2\}$. We start by observing that $E(\|W_i\|) < +\infty$ in view of Hytonen et al. (2016, Proposition 1.2.2). Since $H$ is separable, it admits an orthonormal countable basis, i.e., there exists $(e_j)_{j \ge 1} \in H$ such that $\|e_j\| = 1$ for any $j \ge 1$ and $W_i = \sum_{j \ge 1} \langle W_i, e_j \rangle e_j$ as well as $\|W_i\| = \|(W_i, e_j)\|_{\ell^2(\mathbb{N})}$. Accordingly, $E(\|W_i\|) < +\infty$ entails $E(\sum_{j \ge 1} \langle W_i, e_j \rangle^2) < +\infty$ which is the same as $\|E((W_i, e_j)^2)\|_{\ell^1(\mathbb{N})} = \sum_{j \ge 1} E(\langle W_i, e_j \rangle^2) < +\infty$ by a straightforward application of the standard Fubini's theorem. Using norm injections, we conclude that $\|E\langle W_i, e_j \rangle\|_{\ell^1(\mathbb{N})} \le \sum_{j \ge 1} E|\langle W_i, e_j \rangle| < +\infty$. One may now invoke a conditional version of Fubini's theorem Hytonen et al. (2016, Proposition 1.2.7) for Bochner integrals to decompose

$$E_{F,\mathcal{B}}(W_i) = \sum_{j \ge 1} E_F((W_i, e_j))e_j.$$

Now, $w \mapsto \langle w, z \rangle$ is bounded linear for any fixed $z \in H$ and thus

$$E_F((W_i, e_j)) = \langle E_{F,\mathcal{B}}(W_i), e_j \rangle$$

which in turn gives

$$E_{F,\mathcal{B}}(W_i) = \sum_{j \ge 1} \langle E_{F,\mathcal{B}}(W_i), e_j \rangle e_j.$$

Overall,

$$\langle E_{F,\mathcal{B}}(W_1), E_{F,\mathcal{B}}(W_2) \rangle = \sum_{j \ge 1} \langle E_{F,\mathcal{B}}(W_1), e_j \rangle \langle E_{F,\mathcal{B}}(W_2), e_j \rangle$$
holds by continuity of the inner product on $H^2$. Commuting again the Bochner expectation with the inner product and using conditional independence, it follows that

$$ \langle \mathbb{E}_{\mathcal{F},\mathcal{B}}(W_1), \mathbb{E}_{\mathcal{F},\mathcal{B}}(W_2) \rangle = \sum_{j \ge 1} \mathbb{E}_{\mathcal{F}}(\langle W_1, e_j \rangle) \mathbb{E}_{\mathcal{F}}(\langle W_2, e_j \rangle) = \sum_{j \ge 1} \mathbb{E}_{\mathcal{F}}(\langle W_1, e_j \rangle \langle W_2, e_j \rangle). $$

The last step is to exchange the standard conditional expectation and the infinite sum in the RHS of the last equality. This would conclude the proof as it implies that

$$ \langle \mathbb{E}_{\mathcal{F},\mathcal{B}}(W_1), \mathbb{E}_{\mathcal{F},\mathcal{B}}(W_2) \rangle = \mathbb{E}_{\mathcal{F}}\left(\sum_{j \ge 1} \langle W_1, e_j \rangle \langle W_2, e_j \rangle\right) = \mathbb{E}_{\mathcal{F}}(\langle W_1, W_2 \rangle). $$

Whence, we seek to apply the standard conditional Fubini's theorem and we thus need to show that $\sum_{j \ge 1} \mathbb{E} |\langle W_1, e_j \rangle \langle W_2, e_j \rangle| < +\infty$. Using independence, Cauchy-Schwarz's inequality in $\ell^2(\mathbb{N})$, Jensen's inequality and the integrability of $W_1, W_2$, one may write

$$ \sum_{j \ge 1} \mathbb{E} |\langle W_1, e_j \rangle \langle W_2, e_j \rangle| = \sum_{j \ge 1} \mathbb{E} |\langle W_1, e_j \rangle| \mathbb{E} |\langle W_2, e_j \rangle| $$

and bound this quantity by

$$ \| \mathbb{E}(|\langle W_1, e_j \rangle|) \|_{\ell^2(\mathbb{N})} \| \mathbb{E}(|\langle W_2, e_j \rangle|) \|_{\ell^2(\mathbb{N})} \le \| \mathbb{E}(|\langle W_1, e_j \rangle|^2) \|_{\ell^1(\mathbb{N})}^{1/2} \| \mathbb{E}(|\langle W_2, e_j \rangle|^2) \|_{\ell^1(\mathbb{N})}^{1/2} < +\infty $$

which ends the proof. □

<figure><img src="image_27.png" /></figure>

We may now establish the explicit solution of the optimization problem (1) under the inverse single-index model (6). The following proof is a simpler version of Bousebata et al. (2023, Proposition 1) but it extends to the functional case.

**Proof of Proposition 1.** We start by rewriting the conditional covariance. For any $y \in \mathbb{R}$,

$$ \mathrm{Cov}(\langle w, X \rangle, Y | Y \ge y) = \frac{\mathbb{E}(\langle w, X \rangle Y 1_{\{Y \ge y\}})}{\bar{F}(y)} - \frac{\mathbb{E}(\langle w, X \rangle 1_{\{Y \ge y\}}) \mathbb{E}(Y 1_{\{Y \ge y\}})}{\bar{F}(y)^2}. $$

By Hytonen et al. (2016, Proposition 1.2.2), we have $\mathbb{E}(\|X\|) < +\infty$. Since $\|w\| = 1$, a straightforward application of Cauchy-Schwarz's inequality and monotony of expectations imply that the expectations in the last display are finite. Next, by an unconditional version of Lemma 1, one has

$$ \bar{F}^2(y) \mathrm{Cov}(\langle w, X \rangle, Y | Y \ge y) = \bar{F}(y) \langle w, m_{XY}(y) \rangle - \langle w, m_Y(y) m_X(y) \rangle = \langle w, v(y) \rangle. $$

The optimization problem now becomes arg max$_{\|w\|=1} \langle w, v(y) \rangle$ for which the solution is known to be $v(y)/\|v(y)\|$. Indeed, for any $w \in H$ such that $\|w\| = 1$, Cauchy-Schwarz's inequality implies that $\langle w, v(y) \rangle \le \|v(y)\|$, while taking $w = v(y)/\|v(y)\|$ reaches the latter upper-bound. □

<figure><img src="image_28.png" /></figure>

Toward the use of a random threshold, the following result states the joint density of $(Y_i, Y_{n-k+1,n})$ for any $1 \le i \le n$ and $2 \le k < n$.

**Lemma 2.** Let $\{Y_1, \dots, Y_n\}$ be i.i.d. random variables with common cdf $F$. Let $k$ an integer such that $2 \le k < n$ and let any $1 \le i \le n$. The joint density $(t, y) \mapsto f_{n,k}(t, y)$ of $(Y_i, Y_{n-k+1,n})$ is given for all $y \le t$ by:

$$ f_{n,k}(t, y) = f(t)f(y) \frac{(n-1)!}{(n-k)!(k-2)!} F^{n-k}(y) \bar{F}^{k-2}(y). $$
Proof. Let us start by computing the joint cdf $F_{n,k}$ of $(Y_i, Y_{n-k+1,n})$ so that next the joint density will follow from $f_{n,k} = \partial_{12}F_{n,k}$. For any $t, y \ge 0$ and denoting $[n] := \{1, \dots, n\}$,

$$ 
\begin{aligned}
F_{n,k}(t, y) &= \sum_{\ell=n-k+1}^{n} \mathbb{P}(Y_i \le t, \text{exactly } \ell \text{ of the } Y_1, \dots, Y_n \text{ are less than } y) \\
&= \sum_{\ell=n-k+1}^{n} \mathbb{P}(Y_i \le t, \exists! \mathcal{I} \subset [n], |\mathcal{I}| = \ell, \forall j \in \mathcal{I}, Y_j \le y),
\end{aligned}
$$

which may be split as $\sum_{\ell=n-k+1}^{n-1} F_{\ell}^{(1)}(t, y) + \sum_{\ell=n-k+1}^{n} F_{\ell}^{(2)}(t, y)$ where

$$ 
\begin{aligned}
F_{\ell}^{(1)}(t, y) &:= \mathbb{P}(Y_i \le t, \exists! \mathcal{I} \subset [n] \setminus \{i\}, |\mathcal{I}| = s, \forall j \in \mathcal{I}, Y_j \le y) \\
F_{\ell}^{(2)}(t, y) &:= \mathbb{P}(Y_i \le t, \exists! \mathcal{I} \subset [n], i \in \mathcal{I}, |\mathcal{I}| = \ell, \forall j \in \mathcal{I}, Y_j \le y) \\
&= \mathbb{P}(Y \le t \land y) \mathbb{P}(\exists! \mathcal{I} \subset [n-1], |\mathcal{I}| = \ell-1, \forall j \in \mathcal{I}, Y_j \le y).
\end{aligned}
$$

Note that $F_{\ell}^{(1)}(t, y)$ exists when $\ell \le n-1$ while $F_{\ell}^{(2)}(t, y)$ exists when $\ell \le n$. First, one has

$$ F_{\ell}^{(2)}(t, y) = F(t \land y) \binom{n-1}{\ell-1} F^{\ell-1}(y) \bar{F}^{n-\ell}(y). $$

When $y \le t$, the partial derivative of $F_{\ell}^{(2)}(t, y)$ with respect to $t$ is zero so that this term has no impact on the density $f_{n,k}$. As such, we only focus on $F^{(1)}(t, y)$ when $y \le t$ and, by independence,

$$ 
\begin{aligned}
\sum_{\ell=n-k+1}^{n-1} F_{\ell}^{(1)}(t, y) &= \sum_{\ell=n-k+1}^{n-1} F(t) \binom{n-1}{\ell} F^{\ell}(y) \bar{F}^{n-\ell-1}(y) \\
&= \sum_{\ell=n-k+1}^{n-2} F(t) \binom{n-1}{\ell} F^{\ell}(y) \bar{F}^{n-\ell-1}(y) + F(t) F^{n-1}(y).
\end{aligned}
$$

Now, taking the derivative, it follows that

$$ \partial_{12} F_{n,k}(t, y) = f(t) \sum_{\ell=n-k+1}^{n-2} \binom{n-1}{\ell} (F^{\ell} \cdot \bar{F}^{n-\ell-1})'(y) + f(t) f(y) (n-1) F^{n-2}(y). $$

The product rule gives us that

$$ \sum_{\ell=n-k+1}^{n-2} (F^{\ell} \cdot \bar{F}^{n-\ell-1})'(y) = f(y) \sum_{\ell=n-k+1}^{n-2} (\ell F^{\ell-1} \cdot \bar{F}^{n-\ell-1}(y) - (n-\ell-1)F^{\ell} \cdot \bar{F}^{n-\ell-2}(y)). $$

Thus, one may write $\partial_{12}F_{n,k}(t, y) = f(t)f(y)(D_1(y) + D_2(y))$ with

$$ 
\begin{aligned}
D_1(y) &= \sum_{\ell=n-k+1}^{n-1} \frac{(n-1)!}{(\ell-1)!(n-\ell-1)!} F^{\ell-1} \cdot \bar{F}^{n-\ell-1}(y), \\
D_2(y) &= - \sum_{\ell=n-k+1}^{n-2} \frac{(n-1)!}{\ell!(n-\ell-2)!} F^{\ell} \cdot \bar{F}^{n-\ell-2}(y).
\end{aligned}
$$
The change of variable $\ell \mapsto \ell - 1$ yields

$$D_1(y) + D_2(y) = \frac{(n-1)!}{(n-k)!(k-2)!} F^{n-k}(y) \bar{F}^{k-2}(y)$$

and the conclusion follows.

<figure><img src="image_29.png" /></figure>

The next result is an extension of Bousebata et al. (2023, Lemma 2) to a conditional framework taking into account the random threshold.

**Lemma 3.** Let any $h \in \mathcal{RV}_{\rho}(+\infty)$ with $\rho \in \mathbb{R}$ and i.i.d. random variables $\{Y_1, \dots, Y_n\}$ with common density $f \in \mathcal{RV}_{-1/\gamma-1}(+\infty)$, $\gamma > 0$. Assume that $\rho\gamma < 1$. Let $2 \le k \le n$ be some integer. Then, for any $y \ge 0$ and any $1 \le i \le n$,

$$\mathbb{E}(h(Y_i)1_{\{Y_i \ge Y_{n-k+1,n}\}} | Y_{n-k+1,n} = y) = \frac{(k-1)}{n} \cdot \frac{1}{\bar{F}(y)} \int_{y}^{+\infty} h(t)f(t)dt.$$

Moreover, when $y \to +\infty$, one has

$$\mathbb{E}(h(Y_i)1_{\{Y_i \ge Y_{n-k+1,n}\}} | Y_{n-k+1,n} = y) \sim \frac{(k-1)}{n} \cdot \frac{h(y)}{1 - \rho\gamma}.$$

*Proof.* The order statistic $Y_{n-k+1,n}$ has the following density:

$$f_{Y_{n-k+1,n}}(y) = f(y) \frac{n!}{(n-k)!(k-1)!} F^{n-k}(y) \bar{F}^{k-1}(y).$$

Altogether with Lemma 2 giving the joint density of $(Y_i, Y_{n-k+1,n})$, this implies that

$$\frac{f_{(Y_i, Y_{n-k+1,n})}(t, y)}{f_{Y_{n-k+1,n}}(y)} = (k-1)(n\bar{F}(y))^{-1} f(t).$$

Whence, using Bayes' formula, one may write the conditional expectation as

$$\mathbb{E}(h(Y)1_{\{Y \ge Y_{n-k+1}\}} | Y_{n-k+1,n} = y) = (k-1)(n\bar{F}(y))^{-1} \int_{y}^{+\infty} h(t)f(t)dt.$$

An application of Bousebata et al. (2023, Lemma 2) readily ends the proof.

<figure><img src="image_30.png" /></figure>

## Appendix A.2. Empirical tail-moments

The next lemma establishes a bound on the (random) tail moments of $\varphi(Y)\varepsilon$.

**Lemma 4.** Assume that $\bar{F} \in \mathcal{RV}_{-1/\gamma}(+\infty)$, (4), (7), (8), (10) and (11) hold. Let $k := k_n \to +\infty$ be an integer deterministic sequence such that $k/n \to 0$ and $y_{n,k} \sim U(n/k)$ as $n \to +\infty$. Let $\delta_{n,k} := (g(y_{n,k})(k/n)^{1/q})^{-1}$. Then,

$$\frac{\|\hat{m}_{\varphi(Y)\varepsilon}(Y_{n-k+1,n})\|}{m_{\varphi,g(Y)}(y_{n,k})} = O_{\mathbb{P}}(\delta_{n,k}) \xrightarrow{n \to +\infty} 0.$$
Proof. For clarity, denote $Z_1 := \varphi(Y_1)\varepsilon_1$ and $Y_{n,k} := Y_{n-k+1,n}$. Let $y > 0$ and highlighting the dependence on $y$, we denote the conditional expectation given $\{Y_{n,k} = y\}$ by $\mathbb{E}_{k,y}(\cdot) := \mathbb{E}(\cdot|Y_{n,k} = y)$ and samewise, let us denote the conditional tail-moments of any generic random variable $W_i$, $1 \le i \le n$, by

$$m_{k,y}(W_i) := \mathbb{E}_{k,y}(W_i 1_{\{Y_i \ge Y_{n,k}\}}).$$

We start by computing the expectation of $\|\hat{m}_Z(Y_{n,k})\|^2$ which is by construction,

$$\|\hat{m}_Z(Y_{n,k})\|^2 = \frac{1}{n^2} \sum_{i_1, i_2 = 1}^{n} \langle Z_{i_1}, Z_{i_2} \rangle 1_{\{Y_{i_1} \ge Y_{n,k}\}} 1_{\{Y_{i_2} \ge Y_{n,k}\}}.$$

Taking the conditional expectation and splitting the sum, it follows,

$$\mathbb{E}_{k,y}(\|\hat{m}_Z(Y_{n,k})\|^2) = \frac{1}{n^2} \sum_{i_1 \neq i_2}^{n} \mathbb{E}_{k,y}(\langle Z_{i_1} 1_{\{Y_{i_1} \ge Y_{n,k}\}}, Z_{i_2} 1_{\{Y_{i_2} \ge Y_{n,k}\}} \rangle) \\ + \frac{1}{n^2} \sum_{i=1}^{n} \mathbb{E}_{k,y}(\|Z_i\|^2 1_{\{Y_i \ge Y_{n,k}\}}).$$

Lemma 1 and equidistribution entail

$$\mathbb{E}_{k,y}(\|\hat{m}_Z(Y_{n,k})\|^2) = \|m_{k,y}(Z)\|^2 + \frac{1}{n}(m_{k,y}(\|Z\|^2) - \|m_{k,y}(Z)\|^2).$$

The main tool of the proof is the conditional Markov's inequality which gives, for any $\eta' > 0$,

$$\mathbb{P}_{k,y} \left( \frac{\|\hat{m}_Z(Y_{n,k})\|}{m_{\varphi,g(Y)}(Y_{n,k})} > \eta' \right) \le \frac{1}{\eta'^2} \mathbb{E}_{k,y} (\|\hat{m}_Z(y)\|^2) m_{\varphi,g(Y)}^{-2}(y_{n,k}). \quad (A.1)$$

According to Bousebata et al. (2023, Lemma 2) under (10), there exists some constant $c \in (0, +\infty)$ independent of $n$ such that, for $n$ large enough, Markov's inequality (A.1) becomes

$$\mathbb{P}_{k,y} \left( \frac{\|\hat{m}_Z(Y_{n,k})\|}{m_{\varphi,g(Y)}(Y_{n,k})} > \eta' \right) \le \frac{c}{\eta'^2} (\varphi \cdot g \cdot \bar{F})^{-2}(y_{n,k}) \left( \|m_{k,y}(Z_1)\|^2 \vee \frac{1}{n} m_{k,y}(\|Z_1\|^2) \right).$$

The next step is to bound the term involving the conditional tail-moments. To this end, observe that we may write by Lemma 1,

$$\|m_{k,y}(Z_1)\|^2 = \langle \mathbb{E}_{\mathcal{F},\mathcal{B}}(Z_1 1_{\{Y_1 \ge Y_{n,k}\}}), \mathbb{E}_{\mathcal{F},\mathcal{B}}(Z_2 1_{\{Y_2 \ge Y_{n,k}\}}) \rangle = \mathbb{E}_{k,y}(\langle Z_1, Z_2 \rangle 1_{\{Y_1, Y_2 \ge Y_{n,k}\}}).$$

By conditional independence and Cauchy-Scharwz's inequality in $H$ and then by conditional Cauchy-Schwarz's inequality in $L^2(\Omega, \mathcal{A}, \mathbb{P})$, we may bound

$$\|m_{k,y}(Z_1)\|^2 \le m_{k,y}^2 (\varphi(Y_1) \|\varepsilon_1\|) \le m_{k,y}(\|Z_1\|^2) m_{k,y}(1), \quad (A.2)$$

and therefore

$$\|m_{k,y}(Z_1)\|^2 \vee \frac{1}{n} m_{k,y}(\|Z_1\|^2) \le m_{k,y}(\|Z_1\|^2) \left( \frac{1}{n} \vee m_{k,y}(1) \right).$$
Besides, Lemma 2 yields

$$m_{k,y}(1) = \mathbb{P}(Y_1 > Y_{n,k} | Y_{n,k} = y) = \int_y^{+\infty} \frac{f_{(Y_1, Y_{n-k+1,n})}(t, y)}{f_{Y_{n,k}}(y)} dt = \frac{k-1}{n},$$

so that

$$\|m_{k,y}(Z_1)\|^2 \vee \frac{1}{n} m_{k,y}(\|Z_1\|^2) \le \frac{k-1}{n} m_{k,y}(\|Z_1\|^2). \quad (A.3)$$

The focus now turns to the tail-moment of $\|Z_1\|^2$ in the previous inequality. The $q$-moment of $\|\varepsilon_1\|$ is finite under (8) so that Hölder's inequality yields

$$m_{k,y}(\|Z_1\|^2) \le \mathbb{E}_{k,y} (\|\varepsilon_1\|^q)^{2/q} m_{k,y}^{1-2/q} (\varphi(Y_1)^{2q/(q-2)}). \quad (A.4)$$

Note that the latter tail moment is finite if $2\tau q/(q-2) < 1/\gamma$ which is satisfied for any $q > 2$ and $\gamma \in (0, 1)$ whenever $\tau \le 0$. This existence condition is also satisfied when $\tau > 0$ under (10)-(11). Indeed, the condition at hand is equivalent to $\tau < (1-2/q)/(2\gamma)$ while (10) is the same as $\tau < (1-2\kappa\gamma)/(2\gamma)$ and (11) writes as $2/q < 2\kappa\gamma$. Moreover, $\varphi^{2q/(q-2)} \in \mathcal{R}V_{2\tau q/(q-2)}(+\infty)$ so that Lemma 3 yields as $n \to +\infty$:

$$m_{k,y}^{1-2/q}(\varphi(Y_1)^{2q/(q-2)}) \sim \left(1 - \frac{2\tau q\gamma}{q-2}\right)^{(2/q)-1} \varphi^2(y) \left(\frac{k-1}{n}\right)^{1-2/q}. \quad (A.5)$$

It follows from (A.4) and (A.5) that there exists a constant $c \in (0, +\infty)$ independent of $n$ such that for $n$ large enough,

$$m_{k,y}(\|Z_1\|^2) \le c \mathbb{E}_{k,y} (\|\varepsilon_1\|^q)^{2/q} \varphi^2(y) \left(\frac{k-1}{n}\right)^{1-2/q}. \quad (A.6)$$

Combining (A.3) and (A.6), one has for some constant $c > 0$ independent of $n$ and for $n$ large enough,

$$\|m_{k,y}(Z_1)\|^2 \vee \frac{1}{n} m_{k,y}(\|Z_1\|^2) \le c \mathbb{E}_{k,y} (\|\varepsilon_1\|^q)^{2/q} \varphi^2(y) \left(\frac{k-1}{n}\right)^{2(1-1/q)}.$$

Going back to Markov's bound (A.1), for some constant $c \in (0, +\infty)$ independent of $n$,

$$\mathbb{P}_{k,y} \left( \frac{\|\hat{m}_{Z_1}(Y_{n,k})\|}{m_{\varphi,g(Y_1)(Y_{n,k})}} > \eta' \right) \le \frac{c}{\eta'^2} \mathbb{E}_{k,y} (\|\varepsilon_1\|^q)^{2/q} \Delta_{n,k}(y),$$

where

$$\Delta_{n,k}(y) := (\varphi \cdot g \cdot \bar{F})^{-2}(y_{n,k}) \left(\frac{k-1}{n}\right)^{2(1-1/q)} \varphi^2(y). \quad (A.7)$$

Define two sequences $y_{n,k}^+ := 3y_{n,k}/2$ and $y_{n,k}^- := y_{n,k}/2$. Clearly, when $y \in [y_{n,k}^-, y_{n,k}^+]$ and $n$ is large enough, $\varphi(y)$ is bounded by $\varphi(y_{n,k}^-) \vee \varphi(y_{n,k}^+)$. Moreover, $\varphi \in \mathcal{R}V_\tau(+\infty)$ and thus, $\varphi(y_{n,k}^+) = (1 \pm 1/2)^\tau \varphi(y_{n,k})(1+o(1))$. Hence, when $y \in [y_{n,k}^-, y_{n,k}^+]$ and $n$ is large enough, $\varphi^2(y) \le c\varphi^2(y_{n,k})$ for some constant $c > 0$ independent of $n$. At last, $y_{n,k} \sim U(n/k)$ implies $\bar{F}(y_{n,k}) \sim \bar{F}(U(n/k)) = k/n$
as $n \to +\infty$ since asymptotic equivalences are stable under regularly-varying composition, and therefore, for some other constant $c > 0$,

$$ \Delta_{n,k}(y) \leq c g^{-2}(y_{n,k}) \left(\frac{k}{n}\right)^{-2/q}, $$

Taking account of (8), it follows that there exists a constant $c > 0$ independent of $n$ such that for $n$ large enough,

$$ \sup_{y \in [y_{n,k}^{-}, y_{n,k}^{+}]} \{ \mathbb{E}_{k,y} (\|\varepsilon_1\|^q)^{2/q} \Delta_{n,k}(y) \} \leq c \delta_{n,k}^2. $$

Now, let any $\eta > 0$. Since $\eta'$ is arbitrary, one may pick $\eta'$ of the form $\delta_{n,k}(c/\eta)^{1/2}$ and combining with what precedes, we have shown that (A.1) becomes, when $y_{n,k}^{-} \le y \le y_{n,k}^{+}$,

$$ \mathbb{P}_{k,y} \left( \delta_{n,k}^{-1} \frac{\| \hat{m}_Z(Y_{n,k}) \|}{m_{\varphi \cdot g(Y)}(y_{n,k})} > \eta^{-1/2} \right) \leq \eta. \quad (A.8) $$

In order to derive the result, we need in fact to integrate the conditional probability (A.8) with respect to $y$ on the whole real positive line, in view of Bayes' formula. After splitting the integral domain, the proof is complete if the next quantity is negligible when $n$ is large,

$$ P_{n,k,\eta}(0, +\infty) = P_{n,k,\eta}(0, y_{n,k}^{-}) + P_{n,k,\eta}(y_{n,k}^{-}, y_{n,k}^{+}) + P_{n,k,\eta}(y_{n,k}^{+}, +\infty), $$

where

$$ P_{n,k,\eta}(a, b) := \int_a^b \mathbb{P}_{k,y} \left( \delta_{n,k}^{-1} \frac{\| \hat{m}_Z(Y_{n,k}) \|}{m_{\varphi \cdot g(Y)}(y_{n,k})} > \eta^{-1/2} \right) f_{Y_{n,k}}(y) dy, \quad -\infty \le a \le b \le +\infty. $$

First, note that (A.8) readily provides that $P_{n,k,\eta}(y_{n,k}^{-}, y_{n,k}^{+}) \le \eta/2$ for any $\eta > 0$. Concerning the remaining two terms, we bound the conditional probability by one, use the complementary event and invoke the fact that $Y_{n,k}/U(n/k) \xrightarrow{\mathbb{P}} 1$ as $n \to +\infty$ to get, for $n$ large enough:

$$ P_{n,k,\eta}(0, y_{n,k}^{-}) + P_{n,k,\eta}(y_{n,k}^{+}, +\infty) \le 1 - \mathbb{P}(y_{n,k}^{-} < Y_{n,k} < y_{n,k}^{+}) \le \eta/2. $$

By Bayes' formula, the conclusion follows as we have shown that for $n$ large enough,

$$ \mathbb{P} \left( \delta_{n,k}^{-1} \frac{\| \hat{m}_Z(Y_{n,k}) \|}{m_{\varphi \cdot g(Y)}(y_{n,k})} > \eta^{-1/2} \right) \leq \eta/2 + \eta/2 = \eta, \quad \forall \eta > 0. $$

□

Finally, Lemma 5 below is dedicated to the control of the (random) tail moments of $\langle \beta, \varphi(Y)\varepsilon \rangle$.

**Lemma 5.** Let any $\beta \in H$ with $\|\beta\| = 1$ and suppose $\bar{F} \in \mathcal{R}V_{-1/\gamma}(+\infty)$. Assume that (4), (7), (8), (10) and (11) hold. Let $k := k_n \to +\infty$ be an integer deterministic sequence such that $k/n \to 0$ and $y_{n,k} \sim U(n/k)$ as $n \to +\infty$. Let $\delta_{n,k} := (g(y_{n,k})(k/n)^{1/q})^{-1}$. Then,

$$ \frac{\hat{m}_{\langle \beta, \varphi(Y)\varepsilon \rangle}(Y_{n-k+1,n})}{m_{\varphi \cdot g(Y)}(y_{n,k})} = O_{\mathbb{P}}(\delta_{n,k}) \xrightarrow{n \to +\infty} 0. $$
*Proof.* Let $y > 0$. Again, we denote $Z_1 := \varphi(Y_1)\varepsilon_1$, $Y_{n,k} := Y_{n-k+1,n}$, the conditional expectation given $\{Y_{n,k} = y\}$ by $\mathbb{E}_{k,y}(\cdot) := \mathbb{E}(\cdot|Y_{n,k} = y)$ and $m_{k,y}(W_i) := \mathbb{E}_{k,y}(W_i 1_{\{Y_i \ge Y_{n,k}\}})$. The proof relies on the conditional Markov's inequality which states that

$$ \mathbb{P}_{k,y} \left( \left| \frac{\hat{m}_{(\beta,\varphi(Y)\varepsilon)}(Y_{n,k})}{m_{\varphi,g(Y)}(Y_{n,k})} \right| > \eta' \right) \le \frac{1}{\eta'^2} \frac{\mathbb{E}_{k,y} \left( \hat{m}_{(\beta,\varphi(Y)\varepsilon)}^2(Y_{n,k}) \right)}{m_{\varphi,g(Y)}^2(Y_{n,k})}, \quad \forall \eta' > 0. \quad (A.9) $$

The goal is to show that the RHS in (A.9) converges to zero for large $n$. Let us start with a preliminary computation of the expectation. One has

$$ \hat{m}_{(\beta,\varphi(Y)\varepsilon)}^2(Y_{n,k}) = \frac{1}{n^2} \sum_{i,j=1}^{n} \langle \beta, \varphi(Y_i)\varepsilon_i \rangle \langle \beta, \varphi(Y_j)\varepsilon_j \rangle 1_{\{Y_i \ge Y_{n,k}\}} 1_{\{Y_j \ge Y_{n,k}\}}. $$

Taking the conditional expectation, splitting the sum and using Lemma 1 yield

$$ \begin{aligned} \mathbb{E}_{k,y} (\hat{m}_{(\beta,\varphi(Y)\varepsilon)}^2(Y_{n,k})) = & \left(1 - \frac{1}{n}\right) \mathbb{E}_{k,y} (\varphi(Y_1)\langle \beta, \varepsilon_1 \rangle 1_{\{Y_1 \ge Y_{n,k}\}})^2 \\ & + \frac{1}{n} \mathbb{E}_{k,y} (\varphi^2(Y_1)\langle \beta, \varepsilon_1 \rangle^2 1_{\{Y_1 \ge Y_{n,k}\}}). \end{aligned} \quad (A.10) $$

On the first hand, Cauchy-Schwarz's inequality in $H$ with $\|\beta\| = 1$ as well as the bound (A.6) in the proof of Lemma 4, which is allowed since (8), (10) and (11) hold, entail that there exists a constant $c \in (0, +\infty)$ independent of $n$ such that for $n$ large enough,

$$ \begin{aligned} \mathbb{E}_{k,y} (\varphi^2(Y_1)\langle \beta, \varepsilon_1 \rangle^2 1_{\{Y_1 \ge Y_{n,k}\}}) & \le m_{k,y} (\varphi^2(Y) \|\varepsilon\|^2) \\ & \le c \mathbb{E}_{k,y} (\|\varepsilon\|^q)^{2/q} \varphi^2(y) \left(\frac{k-1}{n}\right)^{1-2/q}. \end{aligned} \quad (A.11) $$

On the other hand, recalling that $Z_1 := \varphi(Y_1)\varepsilon_1$, one may apply Cauchy-Schwarz's inequality in $H$ to obtain

$$ \mathbb{E}_{k,y} (\varphi(Y_1)\langle \beta, \varepsilon_1 \rangle 1_{\{Y_1 \ge Y_{n,k}\}})^2 \le \mathbb{E}_{k,y} (\varphi(Y_1) \|\varepsilon_1\| 1_{\{Y_1 \ge Y_{n,k}\}})^2 = m_{k,y}^2 (\|Z_1\|). \quad (A.12) $$

The focus now turns to the conditional tail-moment of $\|Z_1\|$ in the previous inequality. The $q$-moment of $\|\varepsilon\|$ is finite under (8) so that Hölder's inequality allows us to write,

$$ m_{k,y}^2 (\|Z_1\|) \le \mathbb{E}_{k,y} (\|\varepsilon_1\|^q)^{2/q} m_{k,y}^{2(1-q)} (\varphi^{q/(q-1)}(Y_1)). \quad (A.13) $$

Lemma 3 applies since $\tau q/(q-1) < 1/\gamma$ whenever $\tau \le 0$. This very condition is also satisfied when $\tau > 0$ under (10) and (11). Indeed, the condition at hand is equivalent to $\tau < (1 - 1/q)/\gamma$ while (10) gives $\tau < (1/2 - \kappa\gamma)/\gamma < (1 - \kappa\gamma)/\gamma$ and (11) writes as $1/q < \kappa\gamma$ which concludes. Finally, it yields, for some constant $c > 0$ independent of $n$ and for $n$ large enough,

$$ m_{k,y}^{2(1-1/q)} (\varphi^{q/(q-1)}(Y_1)) \le c \varphi^2(y) \left(\frac{k-1}{n}\right)^{2(1-1/q)}. \quad (A.14) $$

Collecting (A.12), (A.13) and (A.14) entails

$$ \mathbb{E}_{k,y} (\varphi(Y_1)\langle \beta, \varepsilon_1 \rangle 1_{\{Y_1 \ge Y_{n,k}\}})^2 \le c \mathbb{E}_{k,y} (\|\varepsilon_1\|^q)^{2/q} \varphi^2(y) \left(\frac{k-1}{n}\right)^{2(1-1/q)}, \quad (A.15) $$
for $n$ large enough, where $c$ is another constant independent of $n$. It readily follows from (A.10), (A.11) and (A.15) that the numerator of the RHS in (A.9) can be upper bounded as

$$ \mathbb{E}_{k,y} (\hat{m}_{\beta,\varphi(Y)\varepsilon}^2(Y_{n,k})) \leq c \mathbb{E}_{k,y} (\|\varepsilon_1\|^q)^{2/q} \varphi^2(y) \left(\frac{k-1}{n}\right)^{2(1-1/q)} . \quad (A.16) $$

Besides, the denominator of the RHS in (A.9) is controlled through Bousebata et al. (2023, Lemma 2), for some constant $c > 0$ independent of $n$ and for $n \to +\infty$,

$$ m_{\varphi \cdot g(Y)}^2(y_{n,k}) \sim c \varphi^2(y_{n,k}) g^2(y_{n,k}) \bar{F}^2(y_{n,k}). \quad (A.17) $$

Thus, (A.16) altogether with (A.17) give, for some constant $c > 0$ independent of $n$, for $n$ large enough,

$$ \frac{\mathbb{E}_{k,y} (\hat{m}_{\beta,\varphi(Y)\varepsilon}^2(Y_{n,k}))}{m_{\varphi \cdot g(Y)}^2(y_{n,k})} \leq c (\varphi \cdot g \cdot \bar{F})^{-2}(y_{n,k}) \mathbb{E}_{k,y} (\|\varepsilon\|^q)^{2/q} \varphi^2(y) \left(\frac{k-1}{n}\right)^{2(1-1/q)}. $$

Overall, the Markov's inequality (A.9) becomes, for some constant $c \in (0, +\infty)$ independent of $n$, for any $n$ large enough and any $\eta' > 0$,

$$ \mathbb{P}_{k,y} \left( \left| \frac{\hat{m}_{\beta,\varphi(Y)\varepsilon}(Y_{n,k})}{m_{\varphi \cdot g(Y)}(Y_{n,k})} \right| > \eta' \right) \leq \frac{c}{\eta'^2} \mathbb{E}_{k,y} (\|\varepsilon\|^q)^{2/q} \Delta_{n,k}(y), $$

where $\Delta_{n,k}(y)$ is defined by (A.7) in the proof of Lemma 4.

The remaining part of the proof is a mere repetition of the same steps as in the proof of Lemma 4. $\square$

## Appendix A.3. Consistency

The proof of Theorem 1 results in the application of the triangular inequality with the upcom-
ing Proposition 2, which is proved based on Lemma 4 and Lemma 5, combined with Proposition
3 which itself is based on Lemma 6.

**Proposition 2.** Assume (4)–(11) with $\sqrt{k}A(n/k) = O(1)$. Assume $\varphi$ and $g$ are continuously differentiable in a neighbourhood of infinity such that $t(\varphi g)'(t)/(\varphi g)(t) \to \tau + \kappa \neq 0$ as $t \to +\infty$. Let $\delta_{n,k} := (g(y_{n,k})(k/n)^{1/q})^{-1}$ where $y_{n,k} \sim U(n/k)$ as $n \to +\infty$. Then,

$$ \|\hat{\beta}_{\varphi}(Y_{n-k+1,n}) - \beta\| = O_{\mathbb{P}}(\delta_{n,k}) \xrightarrow{n \to +\infty} 0. $$

*Proof of Proposition 2.* Denote $Y_{n,k} := Y_{n-k+1,n}$. Under (6), the estimator $\hat{v}_{\varphi}$ may be expressed as

$$ \hat{v}_{\varphi}(Y_{n,k}) = \hat{m}_{\varphi \cdot g(Y_1)}(Y_{n,k})\beta + \hat{m}_{\varphi(Y_1)\varepsilon_1}(Y_{n,k}). $$

So that, since $\|\beta\| = 1$, one may write the inner product between $\beta$ and $\hat{v}_{\varphi}(Y_{n,k})$ as

$$ \langle \hat{v}_{\varphi}(Y_{n,k}), \beta \rangle = \hat{m}_{\varphi \cdot g(Y_1)}(Y_{n,k}) + \hat{m}_{\varphi(Y_1)\varepsilon_1}(Y_{n,k}), $$

and $\|\hat{v}_{\varphi}(Y_{n,k})\|^2 = \hat{m}_{\varphi \cdot g(Y_1)}^2(Y_{n,k}) + \|\hat{m}_{\varphi(Y_1)\varepsilon_1}(Y_{n,k})\|^2 + 2\hat{m}_{\varphi \cdot g(Y_1)}(Y_{n,k})\hat{m}_{\varphi(Y_1)\varepsilon_1}(Y_{n,k})$. It follows that

$$ 1 - \frac{\langle \hat{v}_{\varphi}(Y_{n,k}), \beta \rangle^2}{\|\hat{v}_{\varphi}(Y_{n,k})\|^2} = \frac{\|\hat{m}_{\varphi(Y_1)\varepsilon_1}(Y_{n,k})\|^2 - \hat{m}_{\varphi(Y_1)\varepsilon_1, \beta}^2(Y_{n,k})}{\hat{m}_{\varphi \cdot g(Y_1)}^2(Y_{n,k}) + 2\hat{m}_{\varphi \cdot g(Y_1)}(Y_{n,k})\hat{m}_{\varphi(Y_1)\varepsilon_1}(Y_{n,k}) + \|\hat{m}_{\varphi(Y_1)\varepsilon_1}(Y_{n,k})\|^2}, $$
which equals

$$ \frac{\left(\|\hat{m}_{\varphi(Y_1)\varepsilon_1}(Y_{n,k})\|/\hat{m}_{\varphi\cdot g(Y_1)}(Y_{n,k})\right)^2 - \left(\hat{m}_{\langle\varphi(Y_1)\varepsilon_1\beta\rangle}(Y_{n,k})/\hat{m}_{\varphi\cdot g(Y_1)}(Y_{n,k})\right)^2}{1 + 2\left(\hat{m}_{\langle\varphi(Y_1)\varepsilon_1\beta\rangle}(Y_{n,k})/\hat{m}_{\varphi\cdot g(Y_1)}(Y_{n,k})\right) + \left(\|\hat{m}_{\varphi(Y_1)\varepsilon_1}(Y_{n,k})\|/\hat{m}_{\varphi\cdot g(Y_1)}(Y_{n,k})\right)^2} $$

We have everything needed to apply Stupfler (2019, Theorem 2). Indeed, $(\varphi \cdot g)'$ is regularly-varying with index $\tau + \kappa - 1 \in [-1, -1/2]$ from the assumptions $t(\varphi g)'(t)/(\varphi g)(t) \to \tau + \kappa \neq 0$ as $t \to +\infty$, (4) and (7). The same goes for Lemma 4 and Lemma 5. Therefore, their combination together with Bousebata et al. (2023, Lemma 2) and the well-known fact that $\bar{F}(y_{n,k}) \sim \bar{F}(U(n/k)) = k/n$ since $y_{n,k} \sim U(n/k)$, allows us to chain asymptotic equivalences in probability:

$$ \|\hat{v}_{\varphi}(Y_{n,k})\|^2 = \hat{m}_{\varphi\cdot g(Y_1)}^2(Y_{n,k})(1 + o_{\mathbb{P}}(1)) = m_{\varphi\cdot g(Y_1)}^2(Y_{n,k})(1 + o_{\mathbb{P}}(1)). \quad (A.18) $$

Whence after a first order Taylor expansion, $1 - (\hat{v}_{\varphi}(Y_{n,k}), \beta)^2 / \|\hat{v}_{\varphi}(Y_{n,k})\|^2$ is asymptotically equivalent, with respect to the convergence in probability, to

$$ \frac{\left(\|\hat{m}_{\varphi(Y_1)\varepsilon_1}(Y_{n,k})\|/m_{\varphi\cdot g(Y_1)}(Y_{n,k})\right)^2 - \left(\hat{m}_{\langle\varphi(Y_1)\varepsilon_1\beta\rangle}(Y_{n,k})/m_{\varphi\cdot g(Y_1)}(Y_{n,k})\right)^2}{1 + 2\left(\hat{m}_{\langle\varphi(Y_1)\varepsilon_1\beta\rangle}(Y_{n,k})/m_{\varphi\cdot g(Y_1)}(Y_{n,k})\right) + \left(\|\hat{m}_{\varphi(Y_1)\varepsilon_1}(Y_{n,k})\|/m_{\varphi\cdot g(Y_1)}(Y_{n,k})\right)^2} $$

Again, Lemma 4 and Lemma 5, together with an Taylor expansion, yield

$$ 1 - \frac{\langle \hat{v}_{\varphi}(Y_{n,k}), \beta \rangle^2}{\|\hat{v}_{\varphi}(Y_{n,k})\|^2} = O_{\mathbb{P}}(\delta_{n,k}^2) \xrightarrow{n \to +\infty} 0. $$

Finally, note that $\|\hat{\beta}_{\varphi}(Y_{n,k}) - \beta\|^2 = 2(1 - \langle \hat{v}_{\varphi}(Y_{n,k}), \beta \rangle / \|\hat{v}_{\varphi}(Y_{n,k})\|)$ to conclude the proof with the classic identity $1 - \lambda^2 = (1 + \lambda)(1 - \lambda)$. $\square$

**Lemma 6.** Assume that the conditions of Lemma 4 or Lemma 5 are satisfied and (6) is met with $\beta \in C_{\text{loc}}^{0,\alpha_b}([0, 1], \mathbb{R})$ and $\varepsilon \in C_{\text{loc}}^{0,\alpha_e}([0, 1], \mathbb{R})$ for some $\alpha_b, \alpha_e > 0$. Assume that $\varphi$ and $g$ are continuously differentiable in a neighbourhood of infinity such that $t(\varphi g)'(t)/(\varphi g)(t) \to \tau + \kappa \neq 0$ and $t\varphi'(t)/\varphi(t) \to \tau \neq 0$ as $t \to +\infty$. Then, for any deterministic sequences $k = o(n)$, $k \to +\infty$, $y_{n,k} \sim U(n/k)$ and $N := N_n \to +\infty$, as $n \to +\infty$,

$$ \|\tilde{v}_{\varphi}(Y_{n-k+1,n})\| = \|\hat{v}_{\varphi}(Y_{n-k+1,n})\|(1 + o_{\mathbb{P}}(1)) = m_{\varphi\cdot g(Y_1)}(Y_{n,k})(1 + o_{\mathbb{P}}(1)). $$

*Proof of Lemma 6.* Recall the notation $Y_{n,k} = Y_{n-k+1,n}$ and denote

$$ \hat{\Delta}^{(2)}(Y_{n,k}) := \langle \hat{v}_{\varphi}(Y_{n,k}), \hat{v}_{\varphi}(Y_{n,k}) \rangle - \langle \tilde{v}_{\varphi}(Y_{n,k}), \tilde{v}_{\varphi}(Y_{n,k}) \rangle = \|\hat{v}_{\varphi}(Y_{n,k})\|^2 - \|\tilde{v}_{\varphi}(Y_{n,k})\|^2, $$

As a consequence of (A.18), it is enough for proving the result to show that:

$$ \frac{|\hat{\Delta}^{(2)}(Y_{n,k})|}{m_{\varphi\cdot g(Y_1)}^2(Y_{n,k})} = \frac{\|\hat{v}_{\varphi}(Y_{n,k})\|^2 - \|\tilde{v}_{\varphi}(Y_{n,k})\|^2}{m_{\varphi\cdot g(Y_1)}^2(Y_{n,k})} \xrightarrow{n \to +\infty} 0. $$

To this end, we are going to show that the LHS of the previous display may be bounded by some negligible quantity:

$$ \frac{|\hat{\Delta}^{(2)}(Y_{n,k})|}{m_{\varphi\cdot g(Y_1)}^2(Y_{n,k})} \lesssim \frac{\hat{\Lambda}_N^{(2)}(Y_{n,k})}{m_{\varphi\cdot g(Y_1)}^2(Y_{n,k})} := \hat{\Lambda}_{N,n,k}^{(2)} \xrightarrow{n \to +\infty} 0. \quad (A.19) $$
We start by investigating $\Delta^{(2)}(i, \ell)$ the building block of $\hat{\Delta}^{(2)}(y)$ and defined as, for $1 \le i, \ell \le n$,

$$ \Delta^{(2)}(i, \ell) := \langle X_i, X_\ell \rangle - \langle \tilde{X}_i, \tilde{X}_\ell \rangle. $$

First, one has:

$$ \langle \tilde{X}_i, \tilde{X}_\ell \rangle = \sum_{j, j'=1}^{N} \int_0^1 \tilde{\psi}_j(t) \tilde{\psi}_{j'}(t) dt X_i(t_j) X_\ell(t_{j'}) $$

By the reproducing kernel property,

$$ \int_0^1 \tilde{\psi}_j(t) \tilde{\psi}_{j'}(t) dt = \int_0^1 \int_{I_j} \psi_j(t, u) \psi_{j'}(t, v) du dv dt = \int_{I_j} \int_{I_{j'}} \psi_j(u, v) du dv := \tilde{\psi}_{jj'} $$

Hence, one may express $\langle \tilde{X}_i, \tilde{X}_\ell \rangle = \sum_{j, j'=1}^{N} \tilde{\psi}_{jj'} X_i(t_j) X_\ell(t_{j'})$ which is, under the model (6),

$$ \begin{aligned} \langle \tilde{X}_i, \tilde{X}_\ell \rangle = & g(Y_i)g(Y_\ell) \sum_{j, j'=1}^{N} \tilde{\psi}_{jj'} \beta(t_j) \beta(t_{j'}) + g(Y_i) \sum_{j, j'=1}^{N} \tilde{\psi}_{jj'} \beta(t_j) \varepsilon_\ell(t_{j'}) \\ & + g(Y_\ell) \sum_{j, j'=1}^{N} \tilde{\psi}_{jj'} \varepsilon_i(t_j) \beta(t_{j'}) + \sum_{j, j'=1}^{N} \tilde{\psi}_{jj'} \varepsilon_i(t_j) \varepsilon_\ell(t_{j'}) \end{aligned} $$

Using once more the model (6), we finally arrive at:

$$ |\Delta^{(2)}(i, \ell)| \le g(Y_i)g(Y_\ell)|\Delta^{(2,1)}| + g(Y_i)|\Delta^{(2,2)}(\ell)| + g(Y_\ell)|\Delta^{(2,2)}(i)| + |\Delta^{(2,3)}(i, \ell)| $$

where,

$$ \begin{aligned} \Delta^{(2,1)} &:= 1 - \sum_{j, j'=1}^{N} \tilde{\psi}_{jj'} \beta(t_j) \beta(t_{j'}), & \Delta^{(2,2)}(i) &:= \langle \beta, \varepsilon_i \rangle - \sum_{j, j'=1}^{N} \tilde{\psi}_{jj'} \varepsilon_i(t_j) \beta(t_{j'}), \\ \Delta^{(2,3)}(i, \ell) &:= \langle \varepsilon_i, \varepsilon_\ell \rangle - \sum_{j, j'=1}^{N} \tilde{\psi}_{jj'} \varepsilon_i(t_j) \varepsilon_\ell(t_{j'}) \end{aligned} $$

We study each $\Delta^{(2,s)}$, $s \in \{1, 2, 3\}$ and start by using the reproducing kernel property on $\Delta^{(2,1)}$,

$$ |\Delta^{(2,1)}| = |1 - \sum_{j, j'=1}^{N} \int_{I_j \times I_{j'}} \psi(u, v) du dv \beta(t_j) \beta(t_{j'})|. $$

Notice that $\int_0^1 \int_0^1 \psi(u, v) \beta(u) \beta(v) du dv = \int_0^1 \beta^2(u) du = 1$. Thus, using again the reproducing kernel property,

$$ \begin{aligned} |\Delta^{(2,1)}| & \le \sum_{j, j'=1}^{N} \int_{I_j \times I_{j'}} |\psi(u, v)\{\beta(u)\beta(v) - \beta(t_j)\beta(t_{j'})\}| du dv \\ & \le \sum_{j, j'=1}^{N} \int_{I_j \times I_{j'}} \{|\psi(u, v)\beta(u)\{\beta(v) - \beta(t_{j'})\}| + |\psi(u, v)\beta(t_{j'})\{\beta(u) - \beta(t_j)\}|\} du dv \\ & \le \sum_{j=1}^{N} \int_{I_j} |\beta(u)\{\beta(u) - \beta(t_j)\}| du + \sum_{j, j'=1}^{N} \int_{I_j} |\tilde{\psi}_{j'}(u)\beta(t_{j'})\{\beta(u) - \beta(t_j)\}| du \\ & := |\Delta^{(2,1,1)}| + |\Delta^{(2,1,2)}|. \end{aligned} $$
Repeating the argument with $\beta \in C_{\text{loc}}^{0, \alpha_b}([0, 1])$, we have that

$$|\Delta^{(2,1,1)}| \leq \|\beta\|_{0,\alpha_b} N^{-\alpha_b}. \qquad (A.20)$$

Next, since $\beta$ is continuous,

$$ \begin{aligned} |\Delta^{(2,1,2)}| & \leq \|\beta\|_{L^{\infty}([0,1])} \|\beta\|_{0,\alpha_b} N^{-\alpha_b} \sum_{j,j'=1}^{N} \int_{I_j} |\bar{\psi}_{j'}(u)| du \\ & \leq \|\beta\|_{L^{\infty}([0,1])} \|\beta\|_{0,\alpha_b} N^{-\alpha_b} \sum_{j=1}^{N} \|\bar{\psi}_j\|_{L^1([0,1])} \\ & \leq \|\beta\|_{L^{\infty}([0,1])} \|\beta\|_{0,\alpha_b} N^{-\alpha_b} \|\psi\|_{L^1([0,1]^2)}. \end{aligned} \qquad (A.21) $$

Combining (A.20) and (A.21), it gives

$$|\Delta^{(2,1)}| \leq \|\beta\|_{0,\alpha_b} N^{-\alpha_b} (1 + \|\beta\|_{L^{\infty}([0,1])} \|\psi\|_{L^1([0,1]^2)}). \qquad (A.22)$$

The term $\Delta^{(2,2)}(i)$ is treated as before:

$$ \begin{aligned} |\Delta^{(2,2)}(i)| &= \left| \sum_{j=1}^{N} \int_{I_j} \int_{0}^{1} \psi(u, t) \beta(t) dt \epsilon_i(u) du - \sum_{j,j'=1}^{N} \bar{\psi}_{jj'} \epsilon_i(t_j) \beta(t_j') \right| \\ &\leq \sum_{j,j'=1}^{N} \int_{I_j \times I_{j'}} |\psi(u, t)(\beta(t)\epsilon_i(u) - \beta(t_j)\epsilon_i(t_j))| dt du \\ &\leq \sum_{j,j'=1}^{N} \int_{I_j \times I_{j'}} \left\{ |\psi(u, t)\beta(t)(\epsilon_i(u) - \epsilon_i(t_j))| + |\psi(u, t)\epsilon_i(t_j)(\beta(t) - \beta(t_j))| \right\} dt du \end{aligned} $$

Using the previous arguments, one may conclude that:

$$|\Delta^{(2,2)}(i)| \leq \|\beta\|_{L^{\infty}([0,1])} \|\epsilon_i\|_{0,\alpha_c} N^{-\alpha_c} + \|\epsilon_i\|_{L^{\infty}([0,1])} \|\psi\|_{L^1([0,1]^2)} \|\beta\|_{0,\alpha_b} N^{-\alpha_b}. \qquad (A.23)$$

The term $\Delta^{(2,3)}(i, \ell) = \sum_{j,j'=1}^{N} \int_{I_j \times I_{j'}} \psi(t, z) \{\epsilon_i(t)\epsilon_\ell(z) - \epsilon_i(t_j)\epsilon_\ell(t_{j'})\} dt dz$ may be decomposed as $\Delta^{(2,3)}(i, \ell) = \Delta^{(2,3,1)}(i, \ell) + \Delta^{(2,3,2)}(i, \ell)$ where,

$$\Delta^{(2,3,1)}(i, \ell) := \sum_{j,j'=1}^{N} \int_{I_j \times I_{j'}} \psi(t, z) \epsilon_i(t) \{\epsilon_\ell(z) - \epsilon_\ell(t_{j'})\} dt dz,$$

$$\Delta^{(2,3,2)}(i, \ell) := \sum_{j,j'=1}^{N} \int_{I_j \times I_{j'}} \psi(t, z) \epsilon_\ell(t_{j'}) \{\epsilon_i(t) - \epsilon_i(t_j)\} dt dz.$$

In the same manner as before,

$$|\Delta^{(2,3,1)}(i, \ell)| \leq \sum_{j,j'=1}^{N} \int_{I_j \times I_{j'}} |\epsilon_\ell(z)\{\epsilon_\ell(z) - \epsilon_\ell(t_{j'})\}| dt dz \leq \|\epsilon_\ell\|_{L^1([0,1])} \|\epsilon_\ell\|_{0,\alpha_c} N^{-\alpha_c}, \qquad (A.24)$$

$$|\Delta^{(2,3,2)}(i, \ell)| \leq \|\epsilon_\ell\|_{L^{\infty}([0,1])} \|\psi\|_{L^1([0,1]^2)} \|\epsilon_i\|_{0,\alpha_c} N^{-\alpha_c}. \qquad (A.25)$$
Therefore combining (A.24)-(A.25),

$$|\Delta^{(2,3)}(i, \ell)| \le (\|\varepsilon_i\|_{L^1([0,1])} \|\varepsilon_\ell\|_{0,\alpha_e} + \|\varepsilon_\ell\|_{L^\infty([0,1])} \|\psi\|_{L^1([0,1]^2)} \|\varepsilon_i\|_{0,\alpha_e})N^{-\alpha_e}. \quad (A.26)$$

From (A.22), (A.23) and (A.26) together, it follows that,

$$\begin{aligned} |\Delta^{(2)}(i, \ell)| & \le g(Y_i)g(Y_\ell)\|\beta\|_{0,\alpha_b} N^{-\alpha_b} (1 + \|\beta\|_{L^\infty([0,1])} \|\psi\|_{L^1([0,1]^2)}) \\ & \quad + g(Y_i)(\|\beta\|_{L^\infty([0,1])} \|\varepsilon_\ell\|_{0,\alpha_e} N^{-\alpha_e} + \|\varepsilon_\ell\|_{L^\infty([0,1])} \|\psi\|_{L^1([0,1]^2)} \|\beta\|_{0,\alpha_b} N^{-\alpha_b}) \\ & \quad + g(Y_\ell)(\|\beta\|_{L^\infty([0,1])} \|\varepsilon_i\|_{0,\alpha_e} N^{-\alpha_e} + \|\varepsilon_i\|_{L^\infty([0,1])} \|\psi\|_{L^1([0,1]^2)} \|\beta\|_{0,\alpha_b} N^{-\alpha_b}) \\ & \quad + \|\varepsilon_i\|_{L^1([0,1])} \|\varepsilon_\ell\|_{0,\alpha_e} N^{-\alpha_e} + \|\varepsilon_\ell\|_{L^\infty([0,1])} \|\psi\|_{L^1([0,1]^2)} \|\varepsilon_i\|_{0,\alpha_e} N^{-\alpha_e}. \end{aligned}$$

As a consequence, (A.19) holds with, $\hat{\Lambda}_N^{(2)}(y)$, $y \ge 0$, being

$$\begin{aligned} & \hat{m}_{\varphi \cdot g(Y)}^2(y) \|\beta\|_{0,\alpha_b} N^{-\alpha_b} (1 + \|\beta\|_{L^\infty([0,1])} \|\psi\|_{L^1([0,1]^2)}) \\ & \quad + 2\hat{m}_{\varphi \cdot g(Y)}(y) \hat{m}_{\|\varepsilon\|_{0,\alpha_e} \varphi(Y)}(y) \|\beta\|_{L^\infty([0,1])} N^{-\alpha_e} \\ & \quad + 2\hat{m}_{\varphi \cdot g(Y)}(y) \hat{m}_{\|\varepsilon\|_{L^\infty([0,1])} \varphi(Y)}(y) \|\psi\|_{L^1([0,1]^2)} \|\beta\|_{0,\alpha_b} N^{-\alpha_b} \\ & \quad + \hat{m}_{\|\varepsilon\|_{L^1([0,1])} \varphi(Y)}(y) \hat{m}_{\|\varepsilon\|_{0,\alpha_e} \varphi(Y)}(y) N^{-\alpha_e} \\ & \quad + \hat{m}_{\|\varepsilon\|_{L^\infty([0,1])} \varphi(Y)}(y) \hat{m}_{\|\varepsilon\|_{0,\alpha_e} \varphi(Y)}(y) \|\psi\|_{L^1([0,1]^2)} N^{-\alpha_e}. \end{aligned}$$

The second step of the proof is to show that $\hat{\Lambda}_N^{(2)}(Y_{n,k}) = o_P(m_{\varphi \cdot g(Y)}^2(y_{n,k}))$. As such, we start by decomposing $\hat{\Lambda}_{N,n,k}^{(2)} = \sum_{s=1}^5 \hat{\Lambda}_{N,n,k}^{(2,s)}$ with, according to (A.18),

$$\begin{aligned} \hat{\Lambda}_{N,n,k}^{(2,1)} &= \frac{\hat{m}_{\varphi \cdot g(Y_1)}^2(Y_{n,k})}{m_{\varphi \cdot g(Y)}^2(Y_{n,k})} \|\beta\|_{0,\alpha_b} N^{-\alpha_b} (1 + \|\beta\|_{L^\infty([0,1])} \|\psi\|_{L^1([0,1]^2)}) \\ &= \|\beta\|_{0,\alpha_b} N^{-\alpha_b} (1 + \|\beta\|_{L^\infty([0,1])} \|\psi\|_{L^1([0,1]^2)}) (1 + o_P(1)), \end{aligned}$$

$$\begin{aligned} \hat{\Lambda}_{N,n,k}^{(2,2)} &= 2 \frac{\hat{m}_{\varphi \cdot g(Y_1)}(Y_{n,k}) \hat{m}_{\|\varepsilon_1\|_{0,\alpha_e} \varphi(Y_1)}(Y_{n,k})}{m_{\varphi \cdot g(Y)}^2(Y_{n,k})} \|\beta\|_{L^\infty([0,1])} N^{-\alpha_e} \\ &= 2 \frac{\hat{m}_{\|\varepsilon_1\|_{0,\alpha_e} \varphi(Y_1)}(Y_{n,k})}{m_{\varphi \cdot g(Y)}(Y_{n,k})} \|\beta\|_{L^\infty([0,1])} N^{-\alpha_e} (1 + o_P(1)), \end{aligned}$$

$$\begin{aligned} \hat{\Lambda}_{N,n,k}^{(2,3)} &= 2 \frac{\hat{m}_{\varphi \cdot g(Y_1)}(Y_{n,k}) \hat{m}_{\|\varepsilon_1\|_{L^\infty([0,1])} \varphi(Y_1)}(Y_{n,k})}{m_{\varphi \cdot g(Y)}^2(Y_{n,k})} \|\psi\|_{L^1([0,1]^2)} \|\beta\|_{0,\alpha_b} N^{-\alpha_b} \\ &= 2 \frac{\hat{m}_{\|\varepsilon_1\|_{L^\infty([0,1])} \varphi(Y_1)}(Y_{n,k})}{m_{\varphi \cdot g(Y)}(Y_{n,k})} \|\psi\|_{L^1([0,1]^2)} \|\beta\|_{0,\alpha_b} N^{-\alpha_b} (1 + o_P(1)), \end{aligned}$$

$$\hat{\Lambda}_{N,n,k}^{(2,4)} = \frac{1}{m_{\varphi \cdot g(Y)}^2(Y_{n,k})} \hat{m}_{\|\varepsilon_1\|_{L^1([0,1])} \varphi(Y_1)}(Y_{n,k}) \hat{m}_{\|\varepsilon_1\|_{0,\alpha_e} \varphi(Y_1)}(Y_{n,k}) N^{-\alpha_e}$$

$$\hat{\Lambda}_{N,n,k}^{(2,5)} = \frac{1}{m_{\varphi \cdot g(Y)}^2(Y_{n,k})} \hat{m}_{\|\varepsilon_1\|_{L^\infty([0,1])} \varphi(Y_1)}(Y_{n,k}) \hat{m}_{\|\varepsilon_1\|_{0,\alpha_e} \varphi(Y_1)}(Y_{n,k}) \|\psi\|_{L^1([0,1]^2)} N^{-\alpha_e}.$$
We seek to show that their conditional expectations converge to zero as for to show that $\hat{\Lambda}_{N,n,k}^{(2,s)}$ converges to zero in probability for any $1 \le s \le 5$ by means of the Markov's inequality. In virtue of (A.18), the first term $\mathbb{E}(\hat{\Lambda}_{N,n,k}^{(2,1)}) = O(N^{-\alpha_b})$ converges to zero for any $\alpha_b > 0$ whenever $N := N_n \to +\infty$. Next, in order to deal with $\hat{\Lambda}_{N,n,k}^{(2,s)}$, $s \ge 2$, we consider two generic (semi)-norms $\|\cdot\|_s$, $s \in \{1, 2\}$ and write the conditional expectation product $\mathbb{E}_{k,y}(\hat{m}_{\|\cdot\|_1,\varphi(Y)}(Y_{n,k})\hat{m}_{\|\cdot\|_2,\varphi(Y)}(Y_{n,k}))$ using independence and equidistribution, as:

$$ \frac{1}{n} m_{k,y}(\|\varepsilon\|_1 \|\varepsilon\|_2 \varphi^2(Y)) + (1 - \frac{1}{n}) m_{k,y}(\|\varepsilon\|_1 \varphi(Y)) m_{k,y}(\|\varepsilon\|_2 \varphi(Y)). $$

Each term of the previous display is bounded using the generalized Hölder's inequality:

$$ \begin{aligned} m_{k,y}(\|\varepsilon\|_1 \|\varepsilon\|_2 \varphi^2(Y)) &\le \mathbb{E}_{k,y}(\|\varepsilon\|_1^q)^{1/q} \mathbb{E}_{k,y}(\|\varepsilon\|_2^q)^{1/q} m_{k,y}^{1-2/q}(\varphi^{\frac{q}{q-2}}(Y)), \\ m_{k,y}(\|\varepsilon\|_s \varphi(Y)) &\le \mathbb{E}_{k,y}(\|\varepsilon\|_s^q)^{1/q} m_{k,y}^{1-1/q}(\varphi^{\frac{q}{q-1}}(Y)), \quad s \in \{1, 2\}. \end{aligned} $$

By (8) and Lemma 3 since $\tau q/(q-2) < \tau q/(q-1) < 1/\gamma$ is satisfied for any $q > 2$ and $\gamma \in (0, 1)$ whenever $\tau \le 0$ and for $\tau > 0$ under (10)-(11), we may write, for $y := y_n \to +\infty$ deterministic and some constant $c > 0$ independent of $(n, y)$ such that for $n$ large enough,

$$ m_{k,y}(\|\varepsilon\|_1 \|\varepsilon\|_2 \varphi^2(Y)) \le c\varphi(y)\left(\frac{k-1}{n}\right)^{1-2/q}, \quad m_{k,y}(\|\varepsilon\|_s \|\varphi(Y)) \le c\varphi(y)\left(\frac{k-1}{n}\right)^{1-1/q}. \quad (A.27) $$

Therefore, collecting the bounds, it yields, for some constant $c > 0$, that may change from line to line, independent of $(n, y)$ such that for $n$ large enough and any $y \in [y_{n,k}^-, y_{n,k}^+]$,

$$ \begin{aligned} \mathbb{E}_{k,y}(\hat{\Lambda}_{N,n,k}^{(2,2)}) + \mathbb{E}_{k,y}(\hat{\Lambda}_{N,n,k}^{(2,3)}) &\le c \frac{(N^{-\alpha_e} + N^{-\alpha_b})\varphi(y)\left(\frac{k-1}{n}\right)^{1-1/q}}{\varphi(y_{n,k})g(y_{n,k})\bar{F}(y_{n,k})} \\ &= c(N^{-\alpha_e} + N^{-\alpha_b}) \sqrt{\Delta_{n,k}(y)} \le c\delta_{n,k}(N^{-\alpha_e} + N^{-\alpha_b}), \end{aligned} $$

$$ \begin{aligned} \mathbb{E}_{k,y}(\hat{\Lambda}_{N,n,k}^{(2,4)}) \vee \mathbb{E}_{k,y}(\hat{\Lambda}_{N,n,k}^{(2,5)}) &\le c \frac{N^{-\alpha_e} \varphi(y) \left(\frac{k-1}{n}\right)^{1-2/q}}{n \varphi^2(y_{n,k}) g^2(y_{n,k}) \bar{F}^2(y_{n,k})} + \frac{N^{-\alpha_e} \varphi^2(y) \left(\frac{k-1}{n}\right)^{2-2/q}}{\varphi^2(y_{n,k}) g^2(y_{n,k}) \bar{F}^2(y_{n,k})} \\ &= N^{-\alpha_e} \Delta_{n,k}(y) \left(1 + \frac{1}{n\varphi(y)}\left(\frac{k-1}{n}\right)^{-1}\right) \\ &\le c \delta_{n,k}^2 N^{-\alpha_e} \left(1 + \frac{1}{k\varphi(y)}\right), \end{aligned} $$

where $\Delta_{n,k}(y)$ comes from (A.7) and $\delta_{n,k} := (g(y_{n,k})(k/n)^{1/q})^{-1} \to 0$. It is now clear that $\sup_{y \in [y_{n,k}^-, y_{n,k}^+]} \mathbb{E}_{k,y}(\hat{\Lambda}_{N,n,k}^{(2,s)})$, $s \le 3$, is very small for large $n$ and for any $\alpha_e, \alpha_b \in (0, 1)$. The same goes for $\sup_{y \in [y_{n,k}^-, y_{n,k}^+]} \mathbb{E}_{k,y}(\hat{\Lambda}_{N,n,k}^{(2,s)})$, $3 < s \le 5$, since if $\tau \ge 0$ then $k\varphi(y) \to +\infty$, and if otherwise $\tau < 0$, then $k \inf_{y \in [y_{n,k}^-, y_{n,k}^+]} \varphi(y) = k\varphi(y_{n,k}^+) \le ck\varphi(U(n/k)) \sim n^{\tau\gamma}k^{1-\tau\gamma} \to +\infty$ in virtue of (10). The rest of the proof follows the lines of Lemma 4 as we invoke the Markov's inequality. In conclusion, since $\delta_{n,k} = o(1)$, the proof is finished as we have shown that $\hat{\Lambda}_{N,n,k}^{(2)}$ converges to zero in probability with sub-rates $\hat{\Lambda}_{N,n,k}^{(2,1)} = O_P(N^{-\alpha_b})$ and,

$$ \hat{\Lambda}_{N,n,k}^{(2,2)} \vee \hat{\Lambda}_{N,n,k}^{(2,3)} = O_P(N^{-\alpha_b \wedge \alpha_e}) \delta_{n,k}, \quad \hat{\Lambda}_{N,n,k}^{(2,4)} \vee \hat{\Lambda}_{N,n,k}^{(2,5)} = O_P(N^{-\alpha_e} \delta_{n,k}^2). $$

□
**Proposition 3.** Assume that the conditions of Lemma 4 or Lemma 5 hold with $\beta \in C_{\text{loc}}^{0,\alpha_b}([0, 1], \mathbb{R})$ and $\varepsilon \in C_{\text{loc}}^{0,\alpha_e}([0, 1], \mathbb{R})$ for some $\alpha_b, \alpha_e > 0$. Assume that $\varphi$ and $g$ are continuously differentiable in a neighbourhood of infinity such that $t(\varphi g)'(t)/(\varphi g)(t) \to \tau + \kappa \neq 0$ and $t\varphi'(t)/\varphi(t) \to \tau \neq 0$ as $t \to +\infty$. Then, for any deterministic sequences $k := k_n \to +\infty$ with $k = o(n)$, any $y_{n,k} \sim U(n/k)$ and any $N := N_n \to +\infty$, it holds that:

$$ \|\tilde{\beta}_{\varphi}(Y_{n-k+1,n}) - \hat{\beta}_{\varphi}(Y_{n-k+1,n})\| \xrightarrow[n \to +\infty]{} 0, $$

with rate $O_P(N^{-\alpha_b} \vee \{N^{-(\alpha_b \wedge \alpha_e)} \delta_{n,k}\} \vee \{\delta_{n,k}^2 N^{-\alpha_e}\})$.

*Proof of Proposition 3.* Denote, for $1 \le i, \ell \le n$ and $y \ge 0$,

$$ \hat{\Delta}^{(3)}(y) := \langle \hat{v}_{\varphi}(y), \hat{v}_{\varphi}(y) - \tilde{v}_{\varphi}(y) \rangle = \|\hat{v}_{\varphi}(y)\|^2 - \langle \hat{v}_{\varphi}(y), \tilde{v}_{\varphi}(y) \rangle, $$

and its building block counterpart $\hat{\Delta}^{(3)}(i, \ell) := \langle X_i, X_\ell - \tilde{X}_\ell \rangle$. Note that

$$ \langle \tilde{v}_{\varphi}(y), \hat{v}_{\varphi}(y) \rangle = \frac{1}{n^2} \sum_{i, \ell=1}^{n} \varphi(Y_i) \varphi(Y_{\ell}) \langle X_i, \tilde{X}_{\ell} \rangle 1_{\{Y_i \ge y\}} 1_{\{Y_{\ell} \ge y\}}. $$

We begin the proof by using Lemma 6 to get:

$$ \begin{aligned} \frac{1}{2} \|\tilde{\beta}_{\varphi}(Y_{n,k}) - \hat{\beta}_{\varphi}(Y_{n,k})\|^2 &= \frac{\|\tilde{v}_{\varphi}(Y_{n,k})\|\|\hat{v}_{\varphi}(Y_{n,k})\| - \langle \tilde{v}_{\varphi}(Y_{n,k}), \hat{v}_{\varphi}(Y_{n,k}) \rangle}{\|\tilde{v}_{\varphi}(Y_{n,k})\|\|\hat{v}_{\varphi}(Y_{n,k})\|} \\ &\le \frac{\hat{\Delta}^{(3)}(Y_{n,k}) + o_P(\|\hat{v}_{\varphi}(Y_{n,k})\|^2)}{\|\hat{v}_{\varphi}(Y_{n,k})\|^2} (1 + o_P(1)). \end{aligned} $$

In view of (A.18), the proof follows if we show that, one may find $\hat{\Lambda}_N^{(3)}$, or equivalently $\hat{\Lambda}_{N,n,k}^{(3)}$, such that

$$ \frac{|\hat{\Delta}^{(3)}(Y_{n,k})|}{m_{\varphi,g(Y)}^2(Y_{n,k})} \le \frac{\hat{\Lambda}_N^{(3)}(Y_{n,k})}{m_{\varphi,g(Y)}^2(Y_{n,k})} (1 + o_P(1)) := \hat{\Lambda}_{N,n,k}^{(3)} (1 + o_P(1)) \xrightarrow[n \to +\infty]{} 0. $$

By the reproducing kernel property and under the model (6), with obvious extension of notations, one may write

$$ \begin{aligned} \langle X_i, \tilde{X}_\ell \rangle &= \sum_{j=1}^{N} X_\ell(t_j) \langle \tilde{\psi}_j, X_i \rangle = \sum_{j=1}^{N} X_\ell(t_j) \int_0^1 \int_{I_j} X_i(t) \psi(t, u) du dt = \sum_{j=1}^{N} X_\ell(t_j) \int_{I_j} X_i(t) dt \\ &= \sum_{j=1}^{N} X_\ell(t_j) \tilde{X}_{ij} = \sum_{j=1}^{N} X_\ell(t_j) \{g(Y_i)\tilde{\beta} + \tilde{\varepsilon}_{ij}\} \\ &= \sum_{j=1}^{N} \{g(Y_i)\tilde{\beta} + \tilde{\varepsilon}_{ij}\} \{g(Y_\ell)\beta(t_j) + \varepsilon_\ell(t_j)\} \\ &= \sum_{j=1}^{N} \{g(Y_i)g(Y_\ell)\tilde{\beta}_j\beta(t_j) + g(Y_\ell)\tilde{\varepsilon}_{ij}\beta(t_j) + g(Y_i)\tilde{\beta}_j\varepsilon_\ell(t_j) + \tilde{\varepsilon}_{ij}\varepsilon_\ell(t_j)\}. \end{aligned} $$
Since $\langle X_i, X_l \rangle = g(Y_i)g(Y_l) + g(Y_i)\langle \beta, \varepsilon_l \rangle + g(Y_l)\langle \beta, \varepsilon_i \rangle + \langle \varepsilon_i, \varepsilon_l \rangle$, it follows that,

$$|\Delta^{(3)}(i, \ell)| \leq g(Y_i)g(Y_l)|\Delta^{(3,1)}| + g(Y_i)|\Delta^{(3,2)}(i)| + g(Y_l)|\Delta^{(3,3)}(i)| + |\Delta^{(3,4)}(i, \ell)|,$$

where, for $1 \le i, \ell \le n$,

$$\Delta^{(3,1)} := 1 - \sum_{j=1}^{N} \bar{\beta}_j \beta(t_j), \quad \Delta^{(3,2)}(i, \ell) := \langle \beta, \varepsilon_l \rangle - \sum_{j=1}^{N} \bar{\beta}_j \varepsilon_i(t_j),$$

$$\Delta^{(3,3)}(i, \ell) := \langle \beta, \varepsilon_i \rangle - \sum_{j=1}^{N} \bar{\varepsilon}_{ij} \beta(t_j), \quad \Delta^{(3,4)} := \langle \varepsilon_i, \varepsilon_l \rangle - \sum_{j=1}^{N} \bar{\varepsilon}_{ij} \varepsilon_l(t_j).$$

Since $\beta \in C_{\text{loc}}^{0, \alpha_b}([0, 1])$ for some $\alpha_b > 0$,

$$\begin{aligned} |\Delta^{(3,1)}| &= \|\beta\|_{L^2([0,1])}^2 - \sum_{j=1}^{N} \int_{I_j} \beta(u) du \beta(t_j) du = |\sum_{j=1}^{N} \int_{I_j} \{\beta^2(u) - \beta(u)\beta(t_j)\} du| \\ &\leq \sum_{j=1}^{N} \int_{I_j} |\beta(u)\|\beta(u) - \beta(t_j)| du \leq \|\beta\|_{0, \alpha_b} N^{-\alpha_b}. \end{aligned} \quad (A.28)$$

Concerning $\Delta^{(3,2)}$, the fact that $\varepsilon_i \in C_{\text{loc}}^{0, \alpha_e}([0, 1])$ for some $\alpha_e > 0$, that $\text{diam}(I_j) = 1/N$ and $\|\beta\|_{L^2([0,1])} = 1$ allows us to write:

$$\begin{aligned} |\Delta^{(3,2)}(i)| &= |\sum_{j=1}^{N} \int_{I_j} \beta(u) \varepsilon_i(u) du - \bar{\beta}_j \varepsilon_i(t_j)| \le \sum_{j=1}^{N} |\int_{I_j} \beta(u) \varepsilon_i(u) du - \int_{I_j} \beta(u) \varepsilon_i(t_j) du| \\ &\le \sum_{j=1}^{N} \int_{I_j} |\beta(u)\|\varepsilon_i(u) - \varepsilon_i(t_j)| du \le \|\varepsilon_i\|_{0, \alpha_e} \|\beta\|_{L^1([0,1])} N^{-\alpha_e} \le \|\varepsilon_i\|_{0, \alpha_e} N^{-\alpha_e}. \end{aligned} \quad (A.29)$$

Observe that the same arguments lead to

$$|\Delta^{(3,3)}(i)| := |\langle \beta, \varepsilon_i \rangle - \sum_{j=1}^{N} \bar{\varepsilon}_{ij} \beta(t_j)| \leq \|\beta\|_{0, \alpha_b} \|\varepsilon_i\|_{L^1([0,1])} N^{-\alpha_b}, \quad (A.30)$$

$$|\Delta^{(3,4)}(i, \ell)| \leq \|\varepsilon_i\|_{L^1([0,1])} \|\varepsilon_\ell\|_{0, \alpha_e} N^{-\alpha_e}. \quad (A.31)$$

Combining (A.28), (A.29), (A.30) and (A.31) with the bound on $\Delta^{(3)}(i, \ell)$, the upper-bound $\Lambda_N^{(3)}(y)$ may be chosen as:

$$\begin{aligned} & \hat{m}_{\varphi,g(Y)}^2(y) \|\beta\|_{0, \alpha_b} N^{-\alpha_b} + \hat{m}_{\|\varepsilon\|_{0, \alpha_e}, \varphi(Y)}(y) \hat{m}_{\varphi(Y)}(y) N^{-\alpha_e} \\ & \quad + \hat{m}_{\|\varepsilon\|_{L^1([0,1])}, \varphi(Y)}(y) \hat{m}_{\varphi(Y)}(y) \|\beta\|_{0, \alpha_b} N^{-\alpha_b} + \hat{m}_{\|\varepsilon\|_{L^1([0,1])}, \varphi(Y)}(y) \hat{m}_{\|\varepsilon_\ell\|_{0, \alpha_e}, \varphi(Y)}(y) N^{-\alpha_e} \end{aligned}$$

So our goal is now to show that $\hat{\Lambda}_{N,n,k}^{(3)} := \Lambda_N^{(3)}(Y_{n,k})/m_{\varphi,g(Y)}^2(y_{n,k})$, converges to zero in probability as $n \to +\infty$. First, we decompose $\hat{\Lambda}_{N,n,k}^{(3)} = \hat{\Lambda}_N^{(3)}(Y_{n,k})/m_{\varphi,g(Y)}^2(y_{n,k})$ as $\hat{\Lambda}_{N,n,k}^{(3)} = \sum_{s=1}^4 \hat{\Lambda}_{N,n,k}^{(3,s)}$ where

$$\hat{\Lambda}_{N,n,k}^{(3,1)} = \frac{\hat{m}_{\varphi,g(Y_1)}^2(Y_{n,k})}{m_{\varphi,g(Y)}^2(Y_{n,k})} \|\beta\|_{0, \alpha_b} N^{-\alpha_b}, \quad \hat{\Lambda}_{N,n,k}^{(3,2)} = \frac{\hat{m}_{\|\varepsilon_1\|_{0, \alpha_e}, \varphi(Y_1)}(Y_{n,k}) \hat{m}_{\varphi(Y_1)}(Y_{n,k})}{m_{\varphi,g(Y)}^2(Y_{n,k})} N^{-\alpha_e},$$

$$\hat{\Lambda}_{N,n,k}^{(3,3)} = m_{\varphi,g(Y)}^{-2}(Y_{n,k}) \hat{m}_{\|\varepsilon_1\|_{L^1([0,1])}, \varphi(Y_1)}(Y_{n,k}) \hat{m}_{\varphi(Y_1)}(Y_{n,k}) \|\beta\|_{0, \alpha_b} N^{-\alpha_b},$$

$$\hat{\Lambda}_{N,n,k}^{(3,4)} = m_{\varphi,g(Y)}^{-2}(Y_{n,k}) \hat{m}_{\|\varepsilon_1\|_{L^1([0,1])}, \varphi(Y_1)}(Y_{n,k}) \hat{m}_{\|\varepsilon_1\|_{0, \alpha_e}, \varphi(Y_1)}(Y_{n,k}) N^{-\alpha_e}.$$
On one hand, Stupfler (2019, Theorem 2) gives that $\hat{m}_{\varphi(Y_1)}(Y_{n,k}) = m_{\varphi(Y)}(y_{n,k})(1 + o_{\mathbb{P}}(1))$ and
Bousebata et al. (2023, Lemma 2) implies $\hat{m}_{\varphi(Y_1)}(Y_{n,k}) = \varphi(y_{n,k})\bar{F}(y_{n,k})(1 + o_{\mathbb{P}}(1))$ since $\gamma\tau < 1$.
On the other hand, we mimick what we already did for the term $\hat{\Lambda}_{N,n,k}^{(2)}$ in the proof of Lemma 6.
Specifically, (A.27), implies that for some constant $c$ independent of $(n, y)$ such that for $n$ large
enough and any $y \in [y_{n,k}^{-}, y_{n,k}^{+}]$, we have $\mathbb{E}_{k,y}(\hat{\Lambda}_{N,n,k}^{(3,1)}) = o_{\mathbb{P}}(N^{-\alpha_b})$ and,

$$
\begin{align*}
\mathbb{E}_{k,y}(\hat{\Lambda}_{N,n,k}^{(3,2)}) + \mathbb{E}_{k,y}(\hat{\Lambda}_{N,n,k}^{(3,3)}) &\le c(N^{-\alpha_b} + N^{-\alpha_e}) \frac{\varphi(y)(\frac{k-1}{n})^{1-1/q}}{\varphi(y_{n,k})g(y_{n,k})\bar{F}(y_{n,k})} \\
&\le c(N^{-\alpha_b} + N^{-\alpha_e})\delta_{n,k},
\end{align*}
$$

$$
\mathbb{E}_{k,y}(\hat{\Lambda}_{N,n,k}^{(3,4)}) \le c \mathbb{E}_{k,y}(\hat{\Lambda}_{N,n,k}^{(2,5)}) \le c \delta_{n,k}^2 N^{-\alpha_e} \left(1 + \frac{1}{k\varphi(y)}\right).
$$

Again, we follow the same lines as in Lemma 4 and the Markov’s inequality argument gives that
$\hat{\Lambda}_{N,n,k}^{(3)}$ converges to zero in probability with rate

$$
\hat{\Lambda}_{N,n,k}^{(3)} = O_{\mathbb{P}}\left(N^{-\alpha_b} \vee \{N^{-(\alpha_b \wedge \alpha_e)} \delta_{n,k}\} \vee \{\delta_{n,k}^2 N^{-\alpha_e}\}\right)
$$

since we have shown that:

$$
\hat{\Lambda}_{N,n,k}^{(3,1)} = O_{\mathbb{P}}(N^{-\alpha_b}), \quad \hat{\Lambda}_{N,n,k}^{(3,2)} \vee \hat{\Lambda}_{N,n,k}^{(3,3)} = O_{\mathbb{P}}(N^{-(\alpha_b \wedge \alpha_e)} \delta_{n,k}), \quad \hat{\Lambda}_{N,n,k}^{(3,4)} = O_{\mathbb{P}}(\delta_{n,k}^2 N^{-\alpha_e}).
$$

<figure><img src="image_31.png" /></figure>

Acknowledgements

This work is partially supported by the French National Research Agency (ANR) under the grant ANR-23-CE40-0009. S. Girard acknowledges the support of the Chair “Stress Test, Risk Management and Financial Steering”, led by the French Ecole Polytechnique and its Foundation and sponsored by BNP Paribas.

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