Test for homogeneity in normal mixtures with unknown means and variances

This paper investigates the asymptotic properties of the likelihood ratio statistics for testing homogeneity in normal mixture models with unknown means and variances. The asymptotic null distribution of the ordinary likelihood ratio statistic is χ22 under the conditions that the means are bounded, the variances are in a compact space away from 0 and ∞, and the mixture coefficients are away from 0. The asymptotic null distribution of a modified likelihood ratio statistic is also χ22 under the conditions that the means are bounded, and the variances are in a compact space away from 0 and ∞.