Abstract
We consider nonlinear regression models that are solely defined by a parametric model for the regression function. The responses are assumed to be missing at random, with the missingness depending on multiple covariates. We propose estimators for expectations of a known function of response and covariates. Our estimator is a nonparametric estimator corrected for the regression function. We show that it is asymptotically efficient in the Hájek and Le Cam sense. Simulations and an example using real data confirm the optimality of our approach.
Highlights
IntroductionKnown as the conditional mean model. Here the regression function rθ is assumed to be known up to a parameter vector θ and X is a d-dimensional random vector
In this article we study efficient estimation of expectations in a nonlinear regression model that is defined solely by the conditional constraint
The nonlinear regression model is an important model for applications; see, for example, the books by Bates and Watts (1998 [1]) and Seber and Wild (1989 [18])
Summary
Known as the conditional mean model. Here the regression function rθ is assumed to be known up to a parameter vector θ and X is a d-dimensional random vector. In the literature usually only estimation of the mean response E(Y ) is considered; see, for example, Matloff (1981 [11]), Cheng (1994 [3]), Wang and Rao (2001, 2002 [22, 23]) and, for further references, Muller (2009 [12]) Other examples of such expectations are moments of Y or X, mixed moments, and probabilities involving X and Y such as P (X < Y ). We will show that the estimator proposed in this paper is efficient in the sense of Hajek and Le Cam. The efficiency results imply asymptotic normality, which is useful for constructing approximative confidence intervals for expectations E{h(X, Y )} of known square-integrable functions h(X, Y ).
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