Homogeneity Tests Against Central-Mixture Alternatives

Abstract

Abstract A test of homogeneity is derived for a general sampling model. The alternative hypothesis is a mixture over a parameter of the sampling model, centered at the null hypothesis. The test statistic is derived through a score test. Its functional form is independent of the particular mixing distribution. Suppose Yi (i = 1, …, n) are independent random variables with respective probability density (or mass) functions fi (yi ‖ λi). Under the null hypothesis, suppose all λ i homogeneous and equal to the common value λ0. Under the alternative hypothesis, suppose the λ i behave as random samples taken from a distribution with mean λ0 and finite third moment. The score statistic for testing these hypotheses rejects the null for large values of , where is the maximum likelihood estimate of λ0 under the null hypothesis and . When the sample size n is large, S is normally distributed under both the null hypothesis and a sequence of alternative hypotheses in which the variance of the mixing distribution tends to 0 as n −1/2. For example, reject the null hypothesis of Poisson observations yi in favor of a mixture of Poissons for large values of , where . In addition, suppose the Yi are independent normal random variables with respective means α + βxi and variance σ2. Reject the null hypothesis of a constant slope β in favor of a mixture of slopes for large values of , where , and are the usual maximum likelihood estimates of α, β, and σ, respectively.