7 Sequential Estimation for Non-parametric Autoregressive Models

In this chapter, we present sequential estimation methods for non-parametric autoregression models. First, we consider the pointwise estimation problem and then we study the estimation problem for the quadratic risk. As to the pointwise estimation, we develop truncated sequential kernel estimation procedures for which we provide the minimax properties in an adaptive setting. It should be emphasized that for autoregressive models the adaptive minimax estimation is possible only in the sequential analysis framework. Moreover, using the sequential kernel estimators we develop new model selection estimation procedures for which we show non-asymptotic sharp oracle inequalities for the robust quadratic risk. This means that the constructed procedures are optimal in the sharp oracle non-asymptotic inequalities sense.

7.1. Introduction

The tradition of considering the problem of statistical estimation as that of estimating a finite number of parameters goes back to Fisher. Statistical models that explain the data more deeply are usually more complex: the unknowns of these models are, in general, some functions with certain properties of regularity. The problem of non-parametric estimation consists of estimating, from the observations, an unknown function belonging to a certain rather large functional class.

The theory of non-parametric estimation has been considerably developed in the last three decades, focusing on ...

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