Marginal likelihood, conditional likelihood and empirical likelihood: Connections and applications
1 Biostatistics Research Branch, National Institute of Allergy and Infectious Diseases, 6700B Rockledge Drive, MSC 7609, Bethesda, Maryland 20892, U.S.A. jingqin{at}niaid.nih.gov, 2 Department of Mathematics, University of Toledo, Toledo, Ohio 43606, U.S.A. bzhang{at}utnet.utoledo.edu
Marginal likelihood and conditional likelihood are often used for eliminating nuisance parameters. For a parametric model, it is well known that the full likelihood can be decomposed into the product of a conditional likelihood and a marginal likelihood. This property is less transparent in a nonparametric or semiparametric likelihood setting. In this paper we show that this nice parametric likelihood property can be carried over to the empirical likelihood world. We discuss applications in case-control studies, genetical linkage analysis, genetical quantitative traits analysis, tuberculosis infection data and unordered-paired data, all of which can be treated as semiparametric finite mixture models. We consider the estimation problem in detail in the simplest case of unordered-paired data where we can only observe the minimum and maximum values of two random variables; the identities of the minimum and maximum values are lost. The profile empirical likelihood approach is used for maximum semiparametric likelihood estimation. We present some large-sample results along with a simulation study.
Key Words: Case-control study; Conditional likelihood; Empirical likelihood; Exponential-tilt model; Linkage analysis; Marginal likelihood; Semiparametric mixture model; Unordered-paired data
Received September 2003. Revised August 2004.