Regenerative block empirical likelihood for Markov chains

TL;DR

This paper introduces a regenerative block empirical likelihood method tailored for Markov chains, leveraging their regenerative structure to handle dependence and establish asymptotic validity for positive recurrent cases.

Contribution

It generalizes empirical likelihood to Markov chains using small blocks and regenerative structure, extending beyond mixing conditions.

Findings

Method is asymptotically valid for positive recurrent Markov chains.

Simulation results demonstrate practical effectiveness.

Approach handles dependence via regenerative blocks.

Abstract

Empirical likelihood is a powerful semi-parametric method increasingly investigated in the literature. However, most authors essentially focus on an i.i.d. setting. In the case of dependent data, the classical empirical likelihood method cannot be directly applied on the data but rather on blocks of consecutive data catching the dependence structure. Generalization of empirical likelihood based on the construction of blocks of increasing nonrandom length have been proposed for time series satisfying mixing conditions. Following some recent developments in the bootstrap literature, we propose a generalization for a large class of Markov chains, based on small blocks of various lengths. Our approach makes use of the regenerative structure of Markov chains, which allows us to construct blocks which are almost independent (independent in the atomic case). We obtain the asymptotic validity of the method for positive recurrent Markov chains and present some simulation results.