Improving Tests for Discrete Small Sample Data

This book provides a guide for improving test statistics that are based on phi-divergence for discrete models, which include various kinds of independence models of contingency tables as well as generalized linear models of binary data. The improvements are based on the theory of asymptotic expansion and lead to correct conclusions of a test even when sample sizes are not large. Without such an improvement, there is a risk that the results of a test will lead to the opposite conclusion, as a limiting distribution is used for an approximated distribution of test statistics.

Mainly, for the phi-divergence family of statistics that include Pearson’s chi-square statistic, the log-likelihood ratio statistic, and the power divergence family of statistics as a special case, the book derives the improvement of statistics as transformed statistics. This accomplishment is achieved by using the expression of approximation of the distribution of original phi-divergence statistics based on Edgeworth expansion. For an independence model of a contingency table, a complete independence model, an independence model among a group of factors, and a conditional independence model are considered. The test statistics of a contingency table for a log-linear model are also presented for consideration. Additionally, the selection of statistics for which the distribution is close to the limiting distribution is discussed using the evaluation of second-order correction of moments.

Nobuhiro Taneichi received his master degree from Hokkaido University in 1984 and the degree of Doctor of Philosophy in Engineering from Hokkaido University in 1995. His working career started as Assistant Professor at Obihiro University of Agriculture and Veterinary Medicine, 1988, Associate Professor at Obihiro University of Agriculture and Veterinary Medicine, 1996, Professor at Obihiro University of Agriculture, 2001, Professor at Department of Mathematics and Computer Science in Kagoshima University, 2006, Professor at Hokkaido University of Education, Sapporo Campus, 2017-2023. His research interest is primarily statistics, including categorical data analysis (generalized linear model, independence models of contingency table, loglinear models of contingency table), statistical inference based on divergence and asymptotic expansion for probability of discrete multivariate statistics. He has published works in several academic journals, including Journal of Multivariate Analysis, Annals of the Institute of Statistical Mathematics, Japanese Journal of Statistics and Data Science, Journal of Statistical Planning and Inference and Journal of the Japan Statistical Society.

Yuri Sekiya received her master degree from Hokkaido University in 1988 and the degree of Doctor of Philosophy in Engineering from Hokkaido University in 2004. Her working career started as Assistant at Hokkaido University of Education, 1989, Lecturer at Hokkaido University of Education, 1991, Assistant Professor at Hokkaido University of Education, 1994, Professor at Hokkaido University of Education, Kushiro Campus, 2005-2025. Her research interest is primarily statistics, including categorical data analysis (generalized linear model, independence models of contingency table, loglinear models of contingency table), statistical inference based on divergence and asymptotic expansion for probability of discrete multivariate statistics. She has published works in several academic journals, including Journal of Multivariate Analysis and Journal of the Japan Statistical Society.