Asymptotic distribution of the likelihood ratio test that a mixture of two binomials is a single binomial

An approach is presented where use is made of the Kullback Leibler information, of which the Fisher information is a limiting case, and it is shown that as n → ∞, the asymptotic distribution of twice the logarithm of the likelihood ratio corresponds to the square of the supremum of a Gaussian stochastic process with mean 0, variance 1 and a well behaved covariance function.

Abstract

A problem of interest in genetics is that of testing whether a mixture of two binomial distributions Bi(k, p) and Bi(k, 12) is simply the pure distribution Bi(k, 12). This problem arises in determining whether we have a genetic marker for a gene responsible for a heterogeneous trait, that is a trait which is caused by any one of several genes. In that event we would have a nontrivial mixture involving 0 2. We present an approach where use is made of the Kullback Leibler information, of which the Fisher information is a limiting case. Several versions of the binomial mixture problem are studied. The asymptotic analysis is supplemented by the results of simulations. It is shown that as n → ∞, the asymptotic distribution of twice the logarithm of the likelihood ratio corresponds to the square of the supremum of a Gaussian stochastic process with mean 0, variance 1 and a well behaved covariance function. As k → ∞ this limiting distribution grows stochastically as log k.