Inference on the change point under a high dimensional sparse mean shift

Abstract

We study a plug in least squares estimator for the change point parameter where change is in the mean of a high dimensional random vector under subgaussian or subexponential distributions. We obtain sufficient conditions under which this estimator possesses sufficient adaptivity against plug in estimates of mean parameters in order to yield an optimal rate of convergence Op(ξ −2 ) in the integer scale. This rate is preserved while allowing high dimensionality as well as a potentially diminishing jump size) in the subgaussian and subexponential cases, respectively. Here s, p, T and l T represent a sparsity parameter, model dimension, sampling period and the separation of the change point from its parametric boundary, respectively. Moreover, since the rate of convergence is free of s, p and logarithmic terms of T , it allows the existence of limiting distributions under high dimensional asymptotics. These distributions are then derived as the argmax of a two sided negative drift Brownian motion or a two sided negative drift random walk under vanishing and non-vanishing jump size regimes, respectively, thereby allowing inference on the change point parameter. Feasible algorithms for implementation of the proposed methodology are provided. Theoretical results are supported with monte-carlo simulations.

Inference for Change Points in High Dimensional Mean Shift Models

“…In case of subgaussian errors, the sharper rate max 1≤j≤N |τ j − τ 0 j | ≤ c u σ 2 ξ −2 log T, w.p. 1 − o(1), would be obtained, due to the availability of sharper tail bounds on residual error terms (Kaul et al [2020]). Nevertheless, this rate is the sharpest available in the literature under high dimensionality.…”

“…Development of a near optimal estimator that is able to yield (4.5) under the weaker distributional assumptions made in this article remains a further question left to future work. The article Kaul et al [2020] provides an estimation method under these weaker conditions, but is limited to a single change point.…”

“…We also implement Step 1 with a second method and some heuristics. The ℓ 0 regularized near optimal estimator of Kaul et al [2020] (Remark 4.2 in that paper, referred to as KFJS below) designed for a single change point is extended to multiple change points via binary segmentation, i.e., recursive application of the method, repeated until no further change points are detected. This is described as Algorithm 3 in Appendix D. Tuning parameters are chosen via a BIC-type criterion at each segmenting recursion, as recommended in that paper.…”

“…Finally we also recall from Remark 1 that both models (1.1) and (2.1) have the same jump parameters thus no notational distinctions are made between the two for these parameters. Broadly speaking, the basic structure of the arguments to follow was first developed in Kaul et al [2020], this has been further generalized in and Kaul [2021] is other dynamic contexts. All mentioned articles in a single change point setting.…”

“…Fryzlewicz [2014], Frick et al [2014] and Wang and Samworth [2018] amongst several others, here the former two are under a fixed p framework and the latter under a high dimensional framework. Further, when a single change point has been assumed, the asymptotic distribution of the change point estimator has been established for various statistical models, see, e.g., [Bai, 1994[Bai, , 1997, Csorgo and Horváth [1997], under fixed p setting, and [Bhattacharjee et al, 2017[Bhattacharjee et al, , 2019, [Kaul et al, 2020, under diverging dimensionality, where the last two articles allow potential high dimensionality.…”