POWER PROPERTIES OF EMPIRICAL LIKELIHOOD FOR STATIONARY PROCESSES

Abstract

In this paper we discuss the second order power properties of empirical likelihood for stationary processes. The asymptotic distribution of empirical likelihood ratio statistics under a sequence of local alternatives is given. 1. Introduction. Since its introduction by Owen (1988, 1990), empirical likelihood has become a useful tool for nonparametric inference. It is well known that the empirical likelihood ratio statistic inherits a number of properties of the parametric likelihood ratio statistic. Owen has shown that the empirical likelihood ratio statistic has limiting chi- squared distribution. Qin and Lawless (1994) connected the theories of empirical likelihood and general estimating equations. Hence using empirical likelihood ratio statistics it is possible to obtain tests and confidence regions for a wide range of problems, including linear models (Owen, 1991). Another property of empirical likelihood which also resembles that of a parametric likelihood is Bartlett correction; see for example Hall and La Scala (1990) for the case of the mean parameters, DiCiccio, et al (1991) for the case of smooth functions of means, Chen (1993, 1994) for linear regression model, Chen and Cui (2006) in the presence of nuisance parameters. For dependent data, Monti (1997) applied the empirical likelihood approach to the derivative of the Whittle likelihood. Kitamura (1997) considered blockwise empirical likeli- hood ratios based on data blocks rather than individual observations. Recently, Nordman and Lahiri (2006) introduced a version of empirical likelihood based on the periodogram and spectral estimating equations. They elucidated the asymptotic properties of frequency domain empirical likelihood for linear processes exhibiting both short- and long-range de- pendence. In this paper we consider the second order power properties of empirical likelihood for stationary processes. Section 2 provides a survey of frequency domain empirical likelihood which is due to Nordman and Lahiri (2006). Section 3 gives the asymptotic distribution of empirical likelihood ratio statistics under a sequence of local alternatives. The proofs of results are relegated to Section 4.