UNIFORM CONSISTENCY FOR NONPARAMETRIC ESTIMATORS IN NULL RECURRENT TIME SERIES

Jiti Gao,Shin Kanaya,Degui Li,Dag Tjøstheim

doi:10.1017/s0266466614000577

Abstract

This paper establishes uniform consistency results for nonparametric kernel density and regression estimators when time series regressors concerned are nonstationary null recurrent Markov chains. Under suitable regularity conditions, we derive uniform convergence rates of the estimators. Our results can be viewed as a nonstationary extension of some well-known uniform consistency results for stationary time series.

Highlights

As shown in the literature, uniform consistency for nonparametric kernel density and regression estimators is important in estimation theory, and useful in deriving results in specification testing theory
This paper systematically studies the strong and weak uniform consistency results for a class of nonparametric kernel density and regression estimators for the case where the time series data involved are nonstationary null–recurrent Markov chains
The chain {Xt} is Harris recurrent if, given a neighborhood Nv of v (v ∈ E) with φ(Nv) > 0, {Xt} returns to Nv with probability one. It is well–known that for a Markov chain on a countable state space which has a point of recurrence, a sequence splitted by the regeneration times becomes independent and identically distributed (i.i.d.) by the Markov property

Summary

Introduction

As shown in the literature, uniform consistency for nonparametric kernel density and regression estimators is important in estimation theory, and useful in deriving results in specification testing theory. In the same period, Karlsen and Tjøstheim (1998, 2001) independently establish nonparametric kernel estimation in the nonstationary case where the time series regressors are nonstationary null–recurrent Markov chains. The supplementary material for the papers by Gao et al (2009a, 2009b) briefly discuss weak uniform consistency for a nonparametric kernel density estimator for the case where the time series involved follow a random walk process. This paper systematically studies the strong and weak uniform consistency results for a class of nonparametric kernel density and regression estimators for the case where the time series data involved are nonstationary null–recurrent Markov chains.

Some basic results for Markov chains

Main results

Applications in density and regression estimation

Conclusions

B: Proofs of the main results