Semiparametric Likelihood Based Method for Goodness of Fit Tests and Estimation in Upgraded Mixture Models
Jing Qin · Scandinavian Journal of Statistics · 1998
We use Owen’s (1988, 1990) empirical likelihood method in upgraded mixture models. Two groups of independent observations are available. One is z1, ..., zn which is observed directly from a distribution F(z). The other one is x1, ..., xm which is observed indirectly from F(z), where the xis have density ∫p(x|z) dF(z) and p(x|z) is a conditional density function. We are interested in testing H0: p(x|z) = p(x|z; θ), for some specified smooth density function. A semiparametric likelihood ratio based statistic is proposed and it is shown that it converges to a chi‐squared distribution. This is a simple method for doing goodness of fit tests, especially when x is a discrete variable with finitely many values. In addition, we discuss estimation of θ and F(z) when H0 is true. The connection between upgraded mixture models and general estimating equations is pointed out.