SlowGuess/NeurIPS_2023_OCR huggingface dataset:
To develop refined estimators based on the preliminary estimators and study their limiting distributions, we would need to require a slightly stronger signal-to-noise ratio condition below.
Assumption 4 (Signal-to-noise ratio for inference). Assume that there exists an arbitrarily slow diverging sequence  $\gamma_T > 0$  such that
$$
\kappa^ {\frac {2 p}{r} + 3} \Delta > \gamma_ {T} \log (T) T ^ {\frac {p}{2 r + p}}.
$$
Assumption 4 is slightly stronger than Assumption 3. This is because our refined estimators are based on a sequence of random endpoints, i.e. the preliminary estimators. This brings theoretical challenges in deriving limiting distributions and estimating the long-run variances. It is worth noting that a similar phenomenon has been observed in the study on conducted by Xu, Wang, Zhao & Yu (2022).
Theorem 2. Suppose that Assumptions 1, 2 and 3 hold. Let  $\{\widetilde{\eta}_k\}_{k=1}^K$  be the refined change point estimators defined in Section 3, with the preliminary estimators  $\{\widehat{\eta}_k\}_{k=1}^K$  returned by Algorithm 1, the intervals  $\{(s_k, e_k)\}_{k=1}^K$  defined in (4), and  $\widehat{\kappa}_k$  defined as in (5). The following holds:
a. (Non-vanishing regime) Suppose the jump size at the change point  $\eta_{k}$  satisfies  $\lim_{T\to \infty}\kappa_k\rightarrow \varrho_k$  for some absolute constant  $\varrho_{k} > 0$ . Then, as  $T\to \infty$ , it holds that  $|\widetilde{\eta}_k - \eta_k| = O_p(1)$  and that
$$
(\widetilde {\eta} _ {k} - \eta_ {k}) \kappa_ {k} ^ {\frac {p}{r} + 2} \xrightarrow {\mathcal {D}} \underset {\widetilde {r} \in \mathbb {Z}} {\arg \min } P _ {k} (\widetilde {r})
$$
$$
\begin{array}{l} P _ {k} (\widetilde {r}) \\ = \left\{ \begin{array}{l l} \sum_ {t = \widetilde {r} + 1} ^ {0} 2 \big \langle F _ {\eta_ {k} + t, h _ {2}} - f _ {\eta_ {k} + t} * \mathcal {K} _ {h _ {2}}, (f _ {\eta_ {k}} - f _ {\eta_ {k + 1}}) * \mathcal {K} _ {h _ {2}} \big \rangle_ {L _ {2}} + \widetilde {r} \| (f _ {\eta_ {k + 1}} - f _ {\eta_ {k}}) * \mathcal {K} _ {h _ {2}} \| _ {L _ {2}} ^ {2}, & \widetilde {r} <   0; \\ 0, & \widetilde {r} = 0; \\ \sum_ {t = 1} ^ {\widetilde {r}} 2 \big \langle F _ {\eta_ {k} + t, h _ {2}} - f _ {\eta_ {k} + t} * \mathcal {K} _ {h _ {2}}, (f _ {\eta_ {k + 1}} - f _ {\eta_ {k}}) * \mathcal {K} _ {h _ {2}} \big \rangle_ {L _ {2}} + \widetilde {r} \| (f _ {\eta_ {k + 1}} - f _ {\eta_ {k}}) * \mathcal {K} _ {h _ {2}} \| _ {L _ {2}} ^ {2}, & \widetilde {r} > 0. \end{array} \right. \\ \end{array}
$$
Here  $*$  denotes convolution and  $h_2 = c_{\kappa_k}\kappa_k^{1 / r}$  for some absolute constant  $c_{\kappa_k} > 0$
b. (Vanishing regime) Suppose the jump size at the change point  $\eta_{k}$  satisfies  $\lim_{T\to \infty}\kappa_k = 0$ . Then, as  $T\to \infty$ , it holds that  $|\widetilde{\eta}_k - \eta_k| = O_p(\kappa_k^{-2 - p / r})$  and that
$$
\left(\widetilde {\eta} _ {k} - \eta_ {k}\right) \kappa_ {k} ^ {\frac {p}{r} + 2} \xrightarrow {\mathcal {D}} \underset {\widetilde {r} \in \mathbb {Z}} {\arg \min } \left\{\widetilde {\sigma} _ {\infty} (k) B (\widetilde {r}) + | \widetilde {r} | \right\}, \tag {7}
$$
where  $h_2 = c_{\kappa_k} \kappa_k^{1/r}$  for some absolute constant  $c_{\kappa_k} > 0$ . Here
$$
B (\widetilde {r}) = \left\{ \begin{array}{l l} B _ {1} (- \widetilde {r}), & \widetilde {r} <   0, \\ 0, & \widetilde {r} = 0, \\ B _ {2} (\widetilde {r}), & \widetilde {r} > 0, \end{array} \right.
$$
with  $B_{1}(r)$  and  $B_{2}(r)$  being two independent standard Brownian motions, and
$$
\widetilde {\sigma} _ {\infty} ^ {2} (k) = \lim  _ {T \rightarrow \infty} \frac {\kappa_ {k} ^ {\frac {p}{r} - 2}}{T} \operatorname {V a r} \left(\sum_ {t = 1} ^ {T} \left\langle F _ {t, h _ {2}} - f _ {t} * \mathcal {K} _ {h _ {2}}, \left(f _ {\eta_ {k}} - f _ {\eta_ {k + 1}}\right) * \mathcal {K} _ {h _ {2}} \right\rangle_ {L _ {2}}\right). \tag {8}
$$
Theorem 2 considers vanishing and non-vanishing regimes of the jump sizes. The upper bounds on the localization error in both regimes can be written as
$$
\max _ {1 \leq k \leq K} | \widetilde {\eta} _ {k} - \eta_ {k} | \kappa_ {k} ^ {\frac {p}{r} + 2} = O _ {p} (1).
$$
Therefore, when the Hölder smoothness parameter  $r = 1$ , our final estimator  $\{\widetilde{\eta}_k\}$  attains the minimax optimal convergence rate developed in Lemma 3 by Padilla et al. (2021). Furthermore, when  $r = 1$ , our resulting rate is sharper than that in Theorem 1 in Padilla et al. (2021), as we are able to remove the logarithmic factors from the upper bound. Additionally, our method can achieve optimal rates with choices of tuning parameters that do not depend on the unknown jump sizes  $\kappa_k$ .
Theorem 2 summarizes the limiting distributions of the refined estimators  $\{\widetilde{\eta}_k\}_{k=1}^K$ . In the nonvanishing case, the resulting limiting distribution can be approximated by a two-sided random walk and the change points can be accurately estimated within a constant error rate. In contrast, in the vanishing regime, a central limit theorem under mixing conditions leads to a two-sided Brownian motion distribution in the limit, which quantifies the asymptotic uncertainty of  $\{\widetilde{\eta}_k\}_{k=1}^K$ , enabling inference on change point locations, and allowing for the construction of confidence intervals.
4.3 Consistent long-run variance estimation
To obtain valid confidence intervals for change points using the limiting distributions in Theorem 2b., it is crucial to access robust estimators for the long-run (asymptotic) variances  $\{\widetilde{\sigma}_{\infty}^{2}(k)\}_{k=1}^{\widehat{K}}$  defined in (8). We propose a block-type long-run variance estimator in Algorithm 2 to fulfill this task and demonstrate its consistency in the following theorem.
Theorem 3. Suppose Assumptions 1, 2 and 3 hold. Let  $\{\widetilde{\sigma}_{\infty}^{2}(k)\}_{k = 1}^{\widetilde{K}}$  be the population long-run variance defined in (8) and  $\widehat{\sigma}_{\infty}^{2}(k)$  the output of Algorithm 2 with  $R = O(T^{(p + r) / (2r + p)} / \kappa_k^{p / (2r) + 3 / 2})$ . Then it holds that
$$
\max _ {k = 1} ^ {K} \left| \widehat {\sigma} _ {\infty} ^ {2} (k) - \widetilde {\sigma} _ {\infty} ^ {2} (k) \right| \stackrel {{P}} {{\longrightarrow}} 0 \quad a s \quad T \to \infty .
$$
4.4 Discussions on MNSBS
Tuning parameters. Our procedure comprises three steps: (1) preliminary estimation, (2) local refinement, and (3) confidence interval construction with three key tuning parameters. For step (1), we use a kernel density estimator with bandwidth  $h \asymp T^{-1/(2r+p)}$ , which follows from the classical non-parametric literature (e.g. Yu 1993, Tsybakov 2009). The threshold tuning parameter  $\tau$  is set to a high-probability upper bound on the CUSUM statistics when there is no change point, which reflects the requirement on the signal-to-noise ratio detailed in Assumption 3. For refined estimation in step (2) and long-run variance estimation in step (3), we set the bandwidth parameter  $h_1 \asymp \widehat{\kappa}_k^{1/r}$ . This choice of bandwidth is inspired by the minimax rate-optimal bandwidth used in Padilla et al. (2021).
Comparison with Padilla et al. (2021). Our main contribution is deriving the limiting distribution of multivariate non-parametric change point estimators. This problem has not been formally studied in the existing literature. Additionally, our Hölder assumption is more general than the Lipschitz assumption used in Padilla et al. (2021). Our Assumption 1d specifies changes through the  $L_{2}$ -norm of probability density functions, which is a weaker assumption than the  $L_{\infty}$ -norm used in Padilla
Algorithm 2 Long-run variance estimators
INPUT:  $\{X_{t}\}_{t = 1}^{T},\{\widehat{\eta}_{k}\}_{k = 1}^{\widehat{K}},\{\widehat{\kappa}_{k}\}_{k = 1}^{\widehat{K}},\{(s_{k},e_{k})\}_{k = 1}^{\widehat{K}}$  and tuning parameter  $R\in \mathbb{N}$    
for  $k = 1,\ldots ,\widehat{K}$  do   
Let  $h_1 = c_{\widehat{\kappa}}\widehat{\kappa}_k^{\frac{1}{r}}$    
for  $t\in \{s_k,\dots ,e_k - 1\}$  do   
 $Y_{t} = \widehat{\kappa}_{k}^{\frac{p}{2r} -1}\bigl {\langle}F_{t,h_{1}} - f_{t}*K_{h_{1}},(f_{\widehat{\eta}_{k}} - f_{\widehat{\eta}_{k + 1}})*K_{h_{1}}\bigr{\rangle}_{L_{2}}$    
end for   
 $S = \lfloor \frac{e_k - s_k}{R}\rfloor$    
for  $r\in \{1,\dots ,R\}$  do   
 $\mathcal{S}_r = \{s_k + (r - 1)S,\dots ,s_k + rS - 1\}$    
end for   
 $\widehat{\sigma}_{\infty}^{2}(k) = \frac{1}{R}\sum_{r = 1}^{R}\left(\frac{1}{\sqrt{S}}\sum_{i\in \mathcal{S}_r}Y_i\right)^2$    
end for   
OUTPUT:  $\{\widehat{\sigma}_{\infty}^{2}(k)\}_{k = 1}^{\widehat{K}}$
et al. (2021). Furthermore, our assumptions allow for temporal dependence captured by  $\alpha$ -mixing coefficients, whereas Padilla et al. (2021) assumed independent observations.
Comparison with existing literature in nonparametric, online, and inference change point.
In the nonparametric change point literature, different kernel-based methods are adopted for change point localisation and testing. In the offline setting, the penalized kernel least squares estimator, originally introduced by Harchaoui & Cappé (2007), was explored by Arlot et al. (2012) for multivariate change point problems, and an oracle inequality was derived. An upper bound on the localization rate provided by this method was established by Garreau & Arlot (2018) and was computationally enhanced further in Celisse et al. (2018). With a focus on a so-called running maximum partition strategy, Harchaoui et al. (2008) formulated a kernel-based test statistic to ascertain the existence of a change-point. In a similar vein, Zou et al. (2014) investigated a problem where  $s$  out of  $n$  sequences are anomalous and devised a test statistic using the kernel maximum mean discrepancy.
In the online setting, Kifer et al. (2004) introduces a meta-algorithm comparing data from a "reference window" to current data using empirical measures. Desobry et al. (2005) detects shifts by comparing two descriptor sets from the signal's immediate past and future, using a dissimilarity metric resembling the Fisher ratio in Gaussian cases via a soft margin single-class SVM. Meanwhile, Liu et al. (2013) adopts density ratio estimation with a non-parametric Gaussian kernel model for change-point detection, updating its parameters online through stochastic gradient descent.
The core methodology is largely shared but with different goals and performance measurements regarding online and offline change point literature comparisons. How to conduct inference in the online change point context is also unclear.
Compared to the existing work, in this paper, we follow the suit of using kernel-based CUSUM statistics but incorporate temporal dependence, which is rarely seen in the literature. Most importantly, we are unaware of existing work on nonparametric change point inference, which is the main selling point of our paper.
Most change point inference work focuses on fixed-dimensional parameters as well as lacks tracking of many model parameters. Xu, Wang, Zhao & Yu (2022), in terms of style, is indeed the most closely related. But tackles high-dimensional linear regression, fundamentally distinct from our nonparametric density estimation.
We refer to MNSBS as our final estimator, which is used for both localization and inference tasks. To evaluate its localization performance, we compare our proposed method against four competitors – MNP (Padilla et al. 2021), EMNCP (Matteson & James 2014), SBS (Cho & Fryzlewicz 2015) and DCBS (Cho 2016) – across a wide range of simulation settings, using corresponding R functions in changepoints (Xu, Padilla, Wang & Li 2022), ecp (James et al. 2019) and hdbinseg (Cho
& Fryzlewicz 2018) packages. However, to the best of our knowledge, no competitor is currently available for the inference task.
Tuning parameters. For MNSBS implementation, we use the Gaussian kernel and the false discovery rate control-based procedure of Padilla et al. (2021) for  $\tau$  selection. Preliminary estimators are set as  $h = 2 \times (1 / T)^{1 / (2r + p)}$ , while the second stage estimator has bandwidths respectively set as  $\widetilde{h} = 0.05$  and  $h_1 = 2 \times \widehat{\kappa}_k^{1 / r}$ . Selection of  $R = \left\lfloor \left(\max_{k=1}^{\widehat{K}} \{e_k - s_k\}\right)^{3/5} \right\rfloor$  with  $\{(s_k, e_k)\}_{k=1}^{\widehat{K}}$  is guided by Theorem 3 using  $\{(s_k, e_k)\}_{k=1}^{\widehat{K}}$  from (4). For the confidence interval construction, we use  $\{\widehat{\kappa}_k\}_{k=1}^{\widehat{K}}$  and  $\{\widehat{\sigma}_\infty^2(k)\}_{k=1}^{\widehat{K}}$  to estimate the required unknown quantities. We evaluate  $L_2$  based statistics in Change point estimation and Long-run variance estimation using the Subregion-Adaptive Vegas Algorithm<sup>1</sup> with a maximum of  $10^5$  function evaluations.
Evaluation measurements For a given set of true change points  $\mathcal{C} = \{\eta_k\}_{k=0}^{K+1}$ , to assess the accuracy of the estimator  $\widehat{\mathcal{C}} = \{\widehat{\eta}_k\}_{k=0}^{\widehat{K}+1}$  with  $\widehat{\eta}_0 = 1$  and  $\widehat{\eta}_{T+1} = T + 1f$ , we report (1) Misevaluation rate: the proportion of misestimating  $K$  and (2) Scaled Hausdorff distance:  $d_{\mathrm{H}}(\widehat{\mathcal{C}}, \mathcal{C})$ , defined by  $d_{\mathrm{H}}(\widehat{\mathcal{C}}, \mathcal{C}) = \frac{1}{T} \max \left\{ \max_{x \in \widehat{\mathcal{C}}} \min_{y \in \mathcal{C}} \{|x - y|\}, \max_{y \in \widehat{\mathcal{C}}} \min_{x \in \mathcal{C}} \{|x - y|\} \right\}$ .
The performance of our change point inference is measured by the coverage of  $\eta_{k}$ , defined as  $cover_{k}(1 - \alpha)$  for significance level  $\alpha \in (0,1)$ . For,  $k = 1,\dots ,K$
$$
c o v e r _ {k} (1 - \alpha) = \mathbb {1} \Bigg \{\eta_ {k} \in \left[ \widetilde {\eta} _ {k} + \frac {\widehat {q} _ {u} (\alpha / 2)}{\widehat {\kappa} _ {k} ^ {p / r + 2}}, \widetilde {\eta} _ {k} + \frac {\widehat {q} _ {u} (1 - \alpha / 2)}{\widehat {\kappa} _ {k} ^ {p / r + 2}} \right] \Bigg \},
$$
with  $\widehat{q}_u(\alpha /2)$  and  $\widehat{q}_u(1 - \alpha /2)$  are the  $\alpha /2$  and  $1 - \alpha /2$  empirical quantiles of the simulated limiting distribution given in (7),  $\widehat{\kappa}_k$  is defined in (5), and  $k = 1,\ldots ,K$ .
5.1 Localization
We consider three different scenarios with two equally spaced change points. For each scenario, we set  $r = 2$ , and vary  $T \in \{150,300\}$  and  $p \in \{3,5\}$ . Moreover, we consider  $\{Y_{t} = \mathbb{1}\{\lfloor T / 3\rfloor < t \leq \lfloor 2T / 3\rfloor\} Z_{t} + X_{t}\}_{t=1}^{T} \subset \mathbb{R}^{p}$  with  $X_{t} = 0.3X_{t-1} + \epsilon_{t}$ .
- Scenario 1 (S1) Let  $Z_{t} = \mu \in \mathbb{R}^{p}$ , where  $\mu_{j} = 0$  for  $j \in \{1, \dots, \lceil p / 2 \rceil\}$  and  $\mu_{j} = 2$  otherwise. Let  $\{\epsilon_t\}$  be i.i.d.  $\mathcal{N}(0_p, I_p)$ .
- Scenario 2 (S2) Let  $Z_{t}|\{u_{t} = 1\} = 1.5 \times 1_{p}, Z_{t}|\{u_{t} = 0\} = -1.5 \times 1_{p}$ , where  $\{u_{t}\}$  are i.i.d. Bernoulli(0.5) random variables. Let  $\{\epsilon_t\}$  be i.i.d.  $\mathcal{N}(0_p, I_p)$ .
- Scenario 3 (S3) Let  $Z_{t} = 0.3Z_{t - 1} + 0.5 \times 1_{p} + \epsilon_{t}^{*}$ , where  $\{\epsilon_t^*\} \subset \mathbb{R}^p$  and  $\{\epsilon_t\} \subset \mathbb{R}^p$  are mutually independent. They are i.i.d. with entries independently follow Unif  $(-\sqrt{3}, \sqrt{3})$  and the standardized Pareto(3, 1), respectively.
S1-S3 encompass a variety of simulation settings, including the same type of distributions, changed mean and constant covariance in S1; a mixture of distributions in S2; and change between light-tailed and heavy-tailed distributions in S3. We conduct 200 repetitions of each experiment and present the results for localization in Figure 1. Our proposed method, MNSBS, generally outperforms all other methods in all scenarios, except for S2, where ECP performs better. However, we observe that MNSBS achieves comparable performance to ECP for large  $T$  in S2.
5.2 Inference
In this section, we focus solely on analyzing the limiting distribution obtained in Theorem 2.b., which pertains to the vanishing regime. We explain this from two different perspectives. Firstly, in the non-vanishing regime (Theorem 2.a.), the localization error is at the order of  $O(1)$ . As a result, the construction of confidence intervals, which is a direct application of the limiting distribution, is of little demand with such estimation results. Secondly, since the localization error is only at the order of  $O(1)$ , the universality cannot come into play to produce a useful limiting distribution.
We consider the process  $\{Y_{t} = \mathbb{1}\{\lfloor T / 2\rfloor < t\leq T\} \mu +X_{t}\}_{t = 1}^{T}$ , with  $X_{t} = 0.3X_{t - 1} + \epsilon_{t}$ . Here,  $\mu = 1_p$  and  $\{\epsilon_t\}$  are i.i.d.  $\mathcal{N}(0_p,I_p)$ . We vary  $T\in \{100,200,300\}$  and  $p\in \{2,3\}$ , and observe
<sup>1</sup>The Subregion-Adaptive Vegas Algorithm is available in R package cubature (Narasimhan et al. 2022)
Figure 1: From top to bottom: Misestimation rate of the number of change point  $K$ ; Scaled Hausdorff distances. From left to right: Scenarios S1-S3. Different colors represent different methods, ordered as MNSBS, NMP, ECP, SBS, DCBS.
Table 1: Results for change point inference.
n	α = 0.01	α = 0.05
cover(1 - α)	width(1 - α)	cover(1 - α)	width(1 - α)
p = 2
100	0.864	17.613 (6.712)	0.812	14.005 (5.639)
200	0.904	22.940 (7.740)	0.838	18.407 (6.541)
300	0.993	26.144 (9.027)	0.961	20.902 (5.936)
p = 3
100	0.903	15.439 (5.792)	0.847	11.153 (4.361)
200	0.966	20.108 (7.009)	0.949	13.920 (5.293)
300	0.981	22.395 (6.904)	0.955	15.376 (4.763)
that our localization results are robust to the bandwidth parameters, yet sensitive to the smoothness parameter  $r$ . We thus set  $r = 1000$  in our simulations, as the density function of a multivariate normal distribution belongs to the Hölder function class with  $r = \infty$ . Table 1 shows that our proposed inference procedure produces good coverage in the considered setting.
We tackle the problem of change point detection for short range dependent multivariate non-parametric data, which has not been studied in the literature. Our two-stage algorithm MNSBS can consistently estimate the change points in stage one, a novelty in the literature. Then, we derived limiting distributions of change point estimators for inference in stage two, a first in the literature.
Our theoretical analysis reveals multiple challenging and interesting directions for future exploration. Relaxing the assumption  $\Delta \asymp T$  may be of interest. In addition, in Theorem 2.a, we can see the limiting distribution is a function of the data-generating mechanisms, lacking universality, therefore deriving a practical method to derive the limiting distributions in the non-vanishing regime may be interesting.
HX was supported by Swiss National Science Foundation Postdoc Mobility fellowship. OHMM is partially funded by NSF DMS-2015489. YY's research is partially funded by EPSRC EP/V013432/1.
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