A New Nuisance-Parameter Elimination Method With Application to the Unordered Homologous Chromosome Pairs Problem
Pengfei Li, Jing Qin · Journal of the American Statistical Association · 2011
Motivated by applications of the case–control model or exponential tilting model in the unordered homologous chromosome pairs problem in genetic studies and in the interim analysis in double-blinded clinical trials, we develop a new nuisance-parameter elimination method based on the empirical Shannon’s mutual information. The asymptotic behaviors of the maximum empirical Shannon’s mutual information estimation and the empirical Shannon’s mutual information test are similar to those of the maximum likelihood estimation and the likelihood ratio test, respectively. Interestingly, we have found a connection between the empirical Shannon’s mutual information and the profile empirical likelihood (Owen 1988) under some constraints. In the test of the null hypothesis that the unordered pairs come from the same distribution, the maximum Shannon’s mutual information estimation has a degenerate information matrix. As a result, we have to expand the empirical Shannon’s mutual information test statistic up to the fourth order to find the limiting distribution of the mixture of a distribution with point mass at zero and a chi-squared distribution with one degree of freedom. A real genetic dataset is employed for illustration. We also outline another application of Shannon’s mutual information in general genetic mixture models.