Improved estimation method for high dimension semimartingale regression models based on discrete data

Evgeny Pchelintsev (), Serguei Pergamenshchikov and Maria Leshchinskaya
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Evgeny Pchelintsev: Tomsk State University
Serguei Pergamenshchikov: Université de Rouen Normandie
Maria Leshchinskaya: Tomsk State University

Statistical Inference for Stochastic Processes, 2022, vol. 25, issue 3, No 6, 537-576

Abstract: Abstract In this paper we study a high dimension (Big Data) regression model in continuous time observed in the discrete time moments with dependent noises defined by semimartingale processes. To this end an improved (shrinkage) estimation method is developed and the non-asymptotic comparison between shrinkage and least squares estimates is studied. The improvement effect for the shrinkage estimates showing the significant advantage with respect to the "small" dimension case is established. It turns out that obtained improvement effect holds true uniformly over observation frequency. Then, a model selection method based on these estimates is developed. Non-asymptotic sharp oracle inequalities for the constructed model selection procedure are obtained. Constructive sufficient conditions for the observation frequency providing the robust efficiency property in adaptive setting without using any sparsity assumption are found. A special stochastic calculus tool to guarantee these conditions for non-Gaussian Ornstein–Uhlenbeck processes is developed. Monte-Carlo simulations for the numeric confirmation of the obtained theoretical results are given.

Keywords: Non-parametric regression; Semimartingale noise; Big data; Incomplete observations; Improved non-asymptotic estimation; Least squares estimates; Robust quadratic risk; Ornstein–Uhlenbeck–Lévy process; Model selection; Sharp oracle inequality; Asymptotic efficiency; 62G08; 62G05 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s11203-021-09258-0

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