A Progressive Block Empirical Likelihood Method for Time Series

Abstract

This article develops a new blockwise empirical likelihood (BEL) method for stationary, weakly dependent time processes, called the progressive block empirical likelihood (PBEL). In contrast to the standard version of BEL, which uses data blocks of constant length for a given sample size and whose performance can depend crucially on the block length selection, this new approach involves a data-blocking scheme where blocks increase in length by an arithmetic progression. Consequently, no block length selections are required for the PBEL method, which implies a certain type of robustness for this version of BEL. For inference of smooth functions of the process mean, theoretical results establish the chi-squared limit of the log-likelihood ratio based on PBEL, which can be used to calibrate confidence regions. Using the same progressive block scheme, distributional extensions are also provided for other nonparametric likelihoods with time series in the family of Cressie–Read discrepancies. Simulation evidence indicates that the PBEL method can perform comparably to the standard BEL in coverage accuracy (when the latter uses a “good” block choice) and can exhibit more stability, without the need to select a usual block length. Supplementary materials for this article are available online.

KEY WORDS:

• Arithmetic progression
• Blockwise empirical likelihood
• Cressie–Read family
• Stationarity
• Weak dependence

Acknowledgments

The authors are grateful to Professor He, an associate editor, and two referees for thoughtful comments that improved manuscript. This research is partially supported by NSF grants DMS-0906588, DMS-1007703, and NSA grant H98230-11-1-0130 (Lahiri).