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

TL;DR

This paper introduces a new family of empirical phi-divergence test statistics for hypothesis testing, demonstrating their effectiveness and robustness compared to existing methods through theoretical derivations and simulation studies.

Contribution

It proposes a novel family of empirical phi-divergence test statistics for simple and composite hypotheses, with asymptotic properties and practical performance evaluation.

Findings

The new test statistic is competitive with the empirical likelihood ratio test.

It offers greater robustness in contaminated data scenarios.

Simulation results support its effectiveness for small samples.

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 are 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 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 empirical likelihood 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 empirical likelihood ratio test statistic, and is also more robust than the empirical likelihood 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.