Nonparametric deconvolution problem for dependent sequences

TL;DR

This paper investigates how dependence in data affects nonparametric density estimation with noisy observations, revealing that dependence influences bandwidth choice and convergence rates differently depending on the smoothness of the density.

Contribution

It provides a detailed analysis of the impact of dependence on bandwidth selection and convergence in nonparametric deconvolution, highlighting differences between ordinary and supersmooth cases.

Findings

Dependence affects bandwidth choice in ordinary smooth case.

Moderate dependence does not alter convergence rates compared to i.i.d. case.

Strong dependence influences both bandwidth and central limit theorem outcomes.

Abstract

We consider the nonparametric estimation of the density function of weakly and strongly dependent processes with noisy observations. We show that in the ordinary smooth case the optimal bandwidth choice can be influenced by long range dependence, as opposite to the standard case, when no noise is present. In particular, if the dependence is moderate the bandwidth, the rates of mean-square convergence and, additionally, central limit theorem are the same as in the i.i.d. case. If the dependence is strong enough, then the bandwidth choice is influenced by the strength of dependence, which is different when compared to the non-noisy case. Also, central limit theorem are influenced by the strength of dependence. On the other hand, if the density is supersmooth, then long range dependence has no effect at all on the optimal bandwidth choice.