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)where
and
are the probability density functions of
and
, respectively. For the ordinary likelihood ratio statistic, we employ the assumption that
for some
, where
and
are the coefficients of the mixtures.
Key words and phrases: Mixture model, likelihood ratio test, modified likelihood ratio test, asymptotic distribution.