Asymptotically Distribution-Free Goodness-of-Fit Tests for Testing Independence in Contingency Tables of Large Dimensions

Abstract

AbstractWe discuss a possibility of using asymptotically distribution-free goodness-of-fit tests for testing independence of two discrete or categorical random variables in contingency tables. The tables considered are particularly of large dimension, in which the conventional chi-square test becomes less reliable when the table is relatively sparse. The main idea of the method is to apply the new Khmaladze transformation to transform the vector of the chi-square statistic components into another vector whose limit distribution is free of the parameters. The transformation is one-to-one and hence we can build up any statistic based on the transformed vector as an asymptotically distribution-free test statistic for the problem of interest where we recommend the analogue of the Kolmogorov-Smirnov test. Simulations are used to show that the new test not only converges relatively quickly but is also more powerful than the chi-square test in certain cases.KeywordsGoodness of fitContingency tablesLarge dimension