Hellinger Deviance Tests: Efficiency, Breakdown Points, and Examples

Abstract

Hellinger distance analogs of likelihood ratio tests are proposed for parametric inference. The proposed tests are based on minimized Hellinger distances between nonparametric density estimates and densities corresponding to null and unconstrained parametric models. If the parametric model is correct, the Hellinger deviance test is asymptotically equivalent to the likelihood ratio test. A simulation study examines the relative performance of these tests in finite samples. Breakdown properties of the Hellinger deviance tests and likelihood ratio tests are compared. The Hellinger tests are generally high breakdown-point tests, whereas, in many instances, the likelihood ratio tests have breakdown points of 0. Two numerical examples illustrate the implementation and performance of the proposed tests. Likelihood-based methods are widely used in applications. They provide a routine method for generating efficient estimates and inference statements. In many instances, however, these methods are known to be sensitive to anomalous data points, and the careful data analyst will screen for outliers prior to a likelihood-based analysis. In large-scale data analysis, for instance, when processing the results of many similar experiments in a data base, the outlier screening procedure needs to be automated. This type of data processing is common in mutation research, where numerous chemicals are subjected to a battery of mutagenicity and carcinogenicity trials. An automated outlier screen is one method for obtaining robust inferences in this setting. The Hellinger test studied here is proposed as a more direct method for obtaining robust inferences. The test statistic simply measures how much farther the data are from the null model than from the unconstrained model. Regions of low predicted density receive very little weight, so the Hellinger deviance test is able to cope with anomalous data. The method adapts to a variety of parametric hypothesis testing situations.

Key Words:

• Deviance
• Hellinger distance
• Likelihood ratio test
• Nonparametric density estimate