Testing homogeneity in discrete mixtures

This paper introduces W-tests for assessing homogeneity in mixtures of discrete probability distributions. A W-test statistic depends on the data solely through parameter estimators and, if a penalized maximum likelihood estimation framework is used, has a tractable asymptotic distribution under the null hypothesis of homogeneity. The large-sample critical values are quantiles of a chi-square distribution multiplied by an estimable constant for which we provide an explicit formula. In particular, the estimation of large-sample critical values does not involve simulation experiments or random field theory. We demonstrate that W-tests are generally competitive with a benchmark test in terms of power to detect heterogeneity. Moreover, in many situations, the large-sample critical values can be used even with small to moderate sample sizes. The main implementation issue (selection of an underlying measure) is thoroughly addressed, and we explain why W-tests are well-suited to problems involving large and online data sets. Application of a W-test is illustrated with an epidemiological data set.