Empirical phi-divergence test statistics for testing simple and composite null hypotheses

Abstract

The main purpose of this paper is to introduce first a new family of empirical test statistics for testing a simple null hypothesis when the vector of parameters of interest is defined through a specific set of unbiased estimating functions. This family of test statistics is based on a distance between two probability vectors, with the first probability vector obtained by maximizing the empirical likelihood (EL) on the vector of parameters, and the second vector defined from the fixed vector of parameters under the simple null hypothesis. The distance considered for this purpose is the phi-divergence measure. The asymptotic distribution is then derived for this family of test statistics. The proposed methodology is illustrated through the well-known data of Newcomb's measurements on the passage time for light. A simulation study is carried out to compare its performance with that of the EL ratio test when confidence intervals are constructed based on the respective statistics for small sample sizes. The results suggest that the ‘empirical modified likelihood ratio test statistic’ provides a competitive alternative to the EL ratio test statistic, and is also more robust than the EL ratio test statistic in the presence of contamination in the data. Finally, we propose empirical phi-divergence test statistics for testing a composite null hypothesis and present some asymptotic as well as simulation results for evaluating the performance of these test procedures.

Keywords:

• empirical likelihood
• empirical phi-divergence test statistics
• influence function
• phi-divergence measures
• power function
• empirical likelihood ratio
• empirical modified likelihood ratio

AMS 2001 Subject Classification:

• 62E20

We thank the referees for their helpful comments and suggestions. This research is partially supported by Grants MTM 2012-33740 and ECO-2011-25706 from Ministerio de Economia y Competitividad (Spain).