Adaptive efficient analysis for big data ergodic diffusion models

Leonid I. Galtchouk () and Serge M. Pergamenshchikov ()
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Leonid I. Galtchouk: Tomsk State University
Serge M. Pergamenshchikov: Université de Rouen

Statistical Inference for Stochastic Processes, 2022, vol. 25, issue 1, No 8, 127-158

Abstract: Abstract We consider drift estimation problems for high dimension ergodic diffusion processes in nonparametric setting based on observations at discrete fixed time moments in the case when diffusion coefficients are unknown. To this end on the basis of sequential analysis methods we develop model selection procedures, for which we show non asymptotic sharp oracle inequalities. Through the obtained inequalities we show that the constructed model selection procedures are asymptotically efficient in adaptive setting, i.e. in the case when the model regularity is unknown. For the first time for such problem, we found in the explicit form the celebrated Pinsker constant which provides the sharp lower bound for the minimax squared accuracy normalized with the optimal convergence rate. Then we show that the asymptotic quadratic risk for the model selection procedure asymptotically coincides with the obtained lower bound, i.e this means that the constructed procedure is efficient. Finally, on the basis of the constructed model selection procedures in the framework of the big data models we provide the efficient estimation without using the parameter dimension or any sparse conditions.

Keywords: Adaptive nonparametric drift estimation; Asymptotic efficiency; Discrete time data; Nonasymptotic estimation; Model selection; Quadratic risk; Sharp oracle inequality; Primary 62G08; Secondary 62G05; 62G20 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s11203-021-09241-9

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