Statistica Sinica 14(2004), 1165-1177

LIKELIHOOD RATIO TEST FOR HOMOGENEITY

IN NORMAL MIXTURES IN THE PRESENCE

OF A STRUCTURAL PARAMETER

Yong Song Qin$^{1,2}$ and Bruce Smith$^{1}$

Dalhousie University and Guangxi Normal University

Abstract: This paper investigates the asymptotic properties of the likelihood ratio statistic for testing homogeneity in normal mixture models in the presence of a structural parameter. The asymptotic null distributions of the ordinary likelihood ratio statistic and the modified likelihood ratio statistic are the same, having the probability density function(pdf) $({1/2})g_1(x)+({1/2})g_2(x)$ where $g_1(x)$ and $g_2(x)$ are the probability density functions of $\chi^2_{(1)}$ and $\chi^2_{(2)}$, respectively. For the ordinary likelihood ratio statistic, we employ the assumption that $\min\{\alpha_1, \alpha_2\}\geq \epsilon$ for some $1/2>\epsilon>0$, where $\alpha_1$ and $\alpha_2$ are the coefficients of the mixtures.

Key words and phrases: Mixture model, likelihood ratio test, modified likelihood ratio test, asymptotic distribution.