Goodness-of-fit tests for semiparametric biased sampling models

Abstract

Consider an s-sample biased sampling model in which the distribution function for each of the first s−1 samples is related to the unknown distribution function G of the sth sample by a known parametric selection bias weight function. Gilbert et al. (Biometrika 86 (1999) 27) gave a procedure for semiparametric maximum likelihood estimation of the parameters in this model. In many applications, information are scarce for basing the choice of the parametric weight function(s), motivating the need for goodness-of-fit tests of the hypothesis that the weight functions are correctly specified. Cramér–von Mises-type, Anderson–Darling-type, and Kolmogorov–Smirnov-type test statistics are studied which compare discrepancies between the empirical distribution of G and the semiparametric maximum likelihood estimator of G. Finite-sample properties of the tests are evaluated with simulations and with a real example of HIV genetic sequence data.