View article

Vardi (1985a) introduced an s-sample model for biased sampling, gave conditions which guarantee the existence and uniqueness of the nonparametric maximum likelihood estimator G_n of the common underlying distribution G and discussed numerical methods for calculating the estimator. Here we examine the large sample behavior of the NPMLE G_n, including results on uniform consistency of G_n, convergence of to a Gaussian process and asymptotic efficiency of G_n as an estimator of G. The proofs are based upon recent results for empirical processes indexed by sets and functions and convexity arguments. We also give a careful proof of identifiability of the underlying distribution G under connectedness of a certain graph G. Examples and applications include length-biased sampling, stratified sampling, "enriched" stratified sampling, "choice-based" sampling in econometrics and "case-control" …