2003, Volume 65, Pt. 3, 513--531

**On Likelihood Ratio Tests In
Gaussian Mixture Models**

By

C\'{E}LINE DELMAS, INRA, Jouy-en-Josas, France

SUMMARY. The aim of this paper is to calculate asymptotic threshold and power of likelihood ratio tests (LRT) in Gaussian mixture models. Since the LRT statistic is, asymptotically, the maximum of a Gaussian process on an interval, we use a Rice method to evaluate its tail probability and to obtain upper and lower bounds for asymptotic threshold and power. These likelihood ratio tests were then compared to different tests based on some moments of the observations. This comparison indicates that, asymptotically, the LRT is not always more powerful than more classical and simpler tests. An application to the detection of a major gene in an animal population is given.

*AMS (1991) subject classification. *62F05, 62F03, 62F30, 60G70, 60G15,
92D10.

*Key words and phrases. *Gaussian mixture models, likelihood ratio
tests, Rice method.