On the distribution of the likelihood ratio test statistic for a mixture of two normal distributions

Abstract

A likelihood ratio statistic can be used to test whether a random sample is taken from a single normal distribution or from a mixture of two normal distributions with a common variance. The asymptotic distribution of the likelihood ratio test statistic G2 (minus twice the difference in the logc likelihoods) in this situation is often assumed to be χ2 with two degrees of freedom. In this study simulation is used to investigate the distribution of G2 for sample sizes up to 256,000. We determine several approximations to the empirical distribution for sample sizes between 50 and 500, including one that does not require the computation of a χ2 distribution with fractional degrees of freedom. We conclude that although the asymptotic distribution of G2 is not a χ2 with two degrees of freedom, this does appear to be a good approximation to the upper 15% of the distribution