Large sample theory of maximum likelihood estimates in semiparametric biased sampling models

Vardi [Ann. Statist. 13 178–203 (1985)] introduced an s-sample biased sampling model with known selection weight functions, gave a condition under which the common underlying probability distribution G is uniquely estimable and developed simple procedure for computing the nonparamet-ric maximum likelihood estimator (NPMLE) n of G. Gill, Vardi and Well-ner thoroughly described the large sample properties of Vardi’s NPMLE, giving results on uniform consistency, convergence of n n − G to a Gaussian process and asymptotic efficiency of n. Gilbert, Lele and Vardi considered the class of semiparametric s-sample biased sampling models formed by allowing the weight functions to depend on an unknown finite-dimensional parameter θ. They extended Vardi’s estimation approach by developing a simple two-step estimation procedure in which θ̂n is obtained by maximizing a profile partial likelihood and n ≡ nθ̂n is obtained by evaluating Vardi’s NPMLE at θ̂n. Here we examine the large sample behavior of the resulting joint MLE θ̂nn, characterizing conditions on the selection weight functions and data in order that θ̂nn is uniformly consistent, asymptotically Gaussian and efficient. Examples illustrated here include clinical trials (especially HIV vac-cine efficacy trials), choice-based sampling in econometrics and case-control studies in biostatistics