  This paper develops a test for homogeneity in finite mixture models where the
mixing proportions are known a priori (taken to be 0.5) and a common nuisance
parameter is present. Statistical tests based on the notion of Projected
Likelihood Contrasts (PLC) are considered. The PLC is a slight modification of
the usual likelihood ratio statistic or the Wilk's $\Lambda$ and is similar in
spirit to the Rao's score test. Theoretical investigations have been carried
out to understand the large sample statistical properties of these tests.
Simulation studies have been carried out to understand the behavior of the null
distribution of the PLC statistic in the case of Gaussian mixtures with unknown
means (common variance as nuisance parameter) and unknown variances (common
mean as nuisance parameter). The results are in conformity with the theoretical
results obtained. Power functions of these tests have been evaluated based on
simulations from Gaussian mixtures.
