A PROJECTED LIKELIHOOD FUNCTION FOR SEMIPARAMETRIC MODELS

In a sequence of papers, McLeish (1984), Hutton & Nelson (1986) and Godambe & Heyde (1987) have motivated the use of the quasi-score function as an estimating function for semiparametric models by showing that it is the projection of the true score into a class of unbiased estimating functions. Thus the quasi-score approximates the score within this class of functions. In this paper, we introduce a class of inference functions for independent observations that contain analogues of likelihood ratios under semiparametric assumptions. This class of functions, which contains the semiparametric estimating functions as a subspace, is built through the use of tensor products for models with independent observations. We consider the projection of the likelihood ratios into this class of inference functions. It is shown that this projected likelihood is first order equivalent to the quasi-likelihood.