Asymptotic Distribution of the Likelihood Ratio Test That a Mixture of Two Binomials is a Single Binomial
Abstract
A problem of interest in genetics is that of testing whether a mixture of two binomial distributions B sub i (k, p) and B sub i (k, 1/2) is simply the pure distribution B sub i (k, 1/2). 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 < p < 0.5 where p is a recombination probability. Standard asymptotic theory breaks down for such problems which belong to a class of problems where a natural parameterization represents a single distribution, under the hypothesis to be tested, by infinitely many possible parameter points. That difficulty may be eliminated by a transformation of parameters. But in that case a second problem appears. The regularity conditions demanded by the applicability of the Fisher Information fails when k greater than 2. We present an approach where use is made off the Kullback Leibler information, of which the Fisher information is a limiting case. Several versions of the binomial mixture problem are studied.
Document Details
- Document Type
- Technical Report
- Publication Date
- May 01, 1991
- Accession Number
- ADA236714
Entities
People
- Eric Lander
- Herman Chernoff
Organizations
- Harvard University