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\noindent Statistica Sinica {\bf 13}(2003), 351-365
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\centerline{\large\bf TESTS FOR HOMOGENEITY IN NORMAL MIXTURES IN}
\vspace{2pt}
\centerline{\large\bf THE PRESENCE OF A STRUCTURAL PARAMETER}
\vspace{.25cm}
\centerline{Hanfeng Chen and Jiahua Chen}
\vspace{.4cm}
\centerline{\it Bowling Green State University and University of Waterloo}
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\begin{quotation}
\noindent {\it Abstract:}
Often a question arises as to whether observed data are a sample from a
homogeneous population or from a heterogeneous population. If in particular,
one wants to test for a single normal distribution versus a mixture of two
normal distributions, classic asymptotic results do not apply since the model
does not satisfy regularity conditions. This paper investigates the large
sample behavior of the likelihood ratio statistic for testing homogeneity
in the normal mixture in location parameters with an unknown structural
parameter. It is proved that the asymptotic null distribution of the likelihood
ratio statistic is the maximum of a $\chi^2_2$-variable and the supremum of
the square of a truncated Gaussian process with mean 0 and variance 1. This
result exposes the unusual large sample behavior of the likelihood function
under the null distribution. The correlation structure of the process involved
in the limiting distribution is presented explicitly. From the large sample
study, it is also found that even though the structural parameter is not part
of the mixing distribution, the convergence rate of its maximum likelihood
estimate is $n^{-1/4}$ rather than $n^{-1/2}$, while the mixing distribution
has a convergence rate $n^{-1/8}$ rather than $n^{-1/4}$. This is in sharp
contrast to ordinary semi-parametric models and to mixture models without a
structural parameter. \par
\vspace{9pt}
\noindent {\it Key words and phrases:}
Asymptotic distribution, finite mixture, Gaussian process, genetic analysis,
likelihood ratio, non-regular model, semi-parametric model.\par
\end{quotation}
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