Nonparametric regression with martingale increment errors

TL;DR

This paper develops adaptive kernel regression estimators for models with martingale increment noise, providing new theoretical bounds that do not rely on ergodicity, and links these to classical minimax rates under mixing conditions.

Contribution

It introduces a novel stability result for self-normalized martingales and establishes adaptive estimation bounds without ergodicity assumptions.

Findings

Provides adaptive upper bounds for kernel estimators with data-driven bandwidth.

Introduces a new stability result for self-normalized martingales.

Links the random rates to classical minimax rates under mixing conditions.

Abstract

We consider the problem of adaptive estimation of the regression function in a framework where we replace ergodicity assumptions (such as independence or mixing) by another structural assumption on the model. Namely, we propose adaptive upper bounds for kernel estimators with data-driven bandwidth (Lepski's selection rule) in a regression model where the noise is an increment of martingale. It includes, as very particular cases, the usual i.i.d. regression and auto-regressive models. The cornerstone tool for this study is a new result for self-normalized martingales, called ``stability'', which is of independent interest. In a first part, we only use the martingale increment structure of the noise. We give an adaptive upper bound using a random rate, that involves the occupation time near the estimation point. Thanks to this approach, the theoretical study of the statistical procedure is disconnected from usual ergodicity properties like mixing. Then, in a second part, we make a link with the usual minimax theory of deterministic rates. Under a beta-mixing assumption on the covariates process, we prove that the random rate considered in the first part is equivalent, with large probability, to a deterministic rate which is the usual minimax adaptive one.