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Semiparametric Likelihood Based Method for Goodness of Fit Tests and Estimation in Upgraded Mixture Models
Author(s) -
Qin Jing
Publication year - 1998
Publication title -
scandinavian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.359
H-Index - 65
eISSN - 1467-9469
pISSN - 0303-6898
DOI - 10.1111/1467-9469.00129
Subject(s) - mathematics , goodness of fit , statistics , f distribution , empirical likelihood , density estimation , conditional probability distribution , likelihood ratio test , function (biology) , combinatorics , statistic , probability density function , empirical distribution function , probability distribution , estimator , evolutionary biology , biology
We use Owen’s (1988, 1990) empirical likelihood method in upgraded mixture models. Two groups of independent observations are available. One is z 1 , ..., z n which is observed directly from a distribution F ( z ). The other one is x 1 , ..., x m which is observed indirectly from F ( z ), where the x i s have density ∫ p ( x | z ) dF ( z ) and p ( x | z ) is a conditional density function. We are interested in testing H 0 : 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 H 0 is true. The connection between upgraded mixture models and general estimating equations is pointed out.