Information and Asymptotic Efficiency in Parametric-Nonparametric Models

Asymptotic lower bounds for estimation of the parameters of models with both parametric and nonparametric components are given in the form of representation theorems (for regular estimates) and asymptotic minimax bounds. The methods used involve: (i) the notion of a "Hellinger-differentiable (root-) density", where part of the differentiation is with respect to the nonparametric part of the model, to obtain appropriate scores; and (ii) calculation of the "effective score" for the real or vector (finite-dimensional) parameter of interest as that component of the score function orthogonal to all nuisance parameter "scores" (perhaps infinite-dimensional). The resulting asymptotic information for estimation of the parametric component of the model is just (4 times) the squared L²-norm of the "effective score". A corollary of these results is a simple necessary condition for "adaptive estimation": adaptation is possible only if the scores for the parameter of interest are orthogonal to the scores for the nuisance function or nonparametric part of the model. Examples considered include the one-sample location model with and without symmetry, mixture models, the two-sample shift model, and Cox's proportional hazards model.