Adaptive efficient robust sequential analysis for autoregressive big data models

TL;DR

This paper introduces an adaptive, efficient, and robust sequential estimation method for high-dimensional autoregressive models, achieving minimax optimality without sparsity assumptions.

Contribution

It derives the first explicit Pinsker constant for such models and develops an adaptive estimation procedure that attains minimax risk bounds.

Findings

Sharp lower bound for robust risks using Van Trees inequality

Efficiency of the proposed estimator in the minimax sense

Achieves optimal risk bounds without sparsity assumptions

Abstract

In this paper we consider high dimension models based on dependent observations defined through autoregressive processes. For such models we develop an adaptive efficient estimation method via the robust sequential model selection procedures. To this end, firstly, using the Van Trees inequality, we obtain a sharp lower bound for robust risks in an explicit form given by the famous Pinsker constant. It should be noted, that for such models this constant is calculated for the first time. Then, using the weighted least square method and sharp non asymptotic oracle inequalities we provide the efficiency property in the minimax sense for the proposed estimation procedure, i.e. we establish, that the upper bound for its risk coincides with the obtained lower bound. It should be emphasized that this property is obtained without using sparse conditions and in the adaptive setting when the parameter dimension and model regularity are unknown.