TY - JOUR
T1 - Hellinger deviance tests
T2 - Efficiency, breakdown points, and examples
AU - Simpson, Douglas G.
N1 - Funding Information:
• Douglas G. Simpson is Assistant Professor, Department of Statistics and Institute for Environmental Studies, University of Illinois, Champaign, IL 61820. This research was supported by a National Science Foundation Mathematical Sciences Postdoctoral Research Fellowship and U.S. Air Force Office of Scientific Research Grant 87-0041 for computational equipment to the Department of Statistics at the University of Illinois. The author thanks the referees for suggestions leading to substantial improvements.
PY - 1989/3
Y1 - 1989/3
N2 - 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.
AB - 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.
KW - Deviance
KW - Hellinger distance
KW - Likelihood ratio test
KW - Nonparametric density estimate
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U2 - 10.1080/01621459.1989.10478744
DO - 10.1080/01621459.1989.10478744
M3 - Article
AN - SCOPUS:0001739061
SN - 0162-1459
VL - 84
SP - 107
EP - 113
JO - Journal of the American Statistical Association
JF - Journal of the American Statistical Association
IS - 405
ER -