Estimating linear functionals in nonlinear regression with responses missing at random

TL;DR

This paper develops efficient estimators for linear functionals in nonlinear regression models with responses missing at random, leveraging empirical plug-in methods and residual-based weights, applicable to smooth transformations of expectations.

Contribution

It introduces a fully imputed estimator that achieves efficiency in nonlinear regression with missing responses, extending existing methods to smooth functionals of expectations.

Findings

Estimator is efficient under certain conditions.

Method applies to both linear and nonlinear models.

Provides new estimators for smooth transformations of expectations.

Abstract

We consider regression models with parametric (linear or nonlinear) regression function and allow responses to be ``missing at random.'' We assume that the errors have mean zero and are independent of the covariates. In order to estimate expectations of functions of covariate and response we use a fully imputed estimator, namely an empirical estimator based on estimators of conditional expectations given the covariate. We exploit the independence of covariates and errors by writing the conditional expectations as unconditional expectations, which can now be estimated by empirical plug-in estimators. The mean zero constraint on the error distribution is exploited by adding suitable residual-based weights. We prove that the estimator is efficient (in the sense of H\'{a}jek and Le Cam) if an efficient estimator of the parameter is used. Our results give rise to new efficient estimators of smooth transformations of expectations. Estimation of the mean response is discussed as a special (degenerate) case.