Efficient Estimation of Population-Level Summaries in General Semiparametric Regression Models

Abstract

This article considers a wide class of semiparametric regression models in which interest focuses on population-level quantities that combine both the parametric and the nonparametric parts of the model. Special cases in this approach include generalized partially linear models, generalized partially linear single-index models, structural measurement error models, and many others. For estimating the parametric part of the model efficiently, profile likelihood kernel estimation methods are well established in the literature. Here our focus is on estimating general population-level quantities that combine the parametric and nonparametric parts of the model (e.g., population mean, probabilities, etc.). We place this problem in a general context, provide a general kernel-based methodology, and derive the asymptotic distributions of estimates of these population-level quantities, showing that in many cases the estimates are semiparametric efficient. For estimating the population mean with no missing data, we show that the sample mean is semiparametric efficient for canonical exponential families, but not in general. We apply the methods to a problem in nutritional epidemiology, where estimating the distribution of usual intake is of primary interest and semiparametric methods are not available. Extensions to the case of missing response data are also discussed.

Estimation of generalized partially linear models with measurement error using sufficiency scores

“…The calculations given below basically follow those of Carroll et al (1998), with the uniformity of the expansion following as in Claeskens and Van Keilegom (2003), see also Maity et al (2007).…”

“…The regularity conditions given below are based on Claeskens and Carroll (2007) and Maity et al (2007):…”

Efficient Semiparametric Marginal Estimation for the Partially Linear Additive Model for Longitudinal/Clustered Data

“…Section 3.2 of Carroll et al (2009) discusses this issue. Also see Maity, et al (2007) for other comments on bandwidth estimation in semiparametric models.…”

Empirical likelihood confidence bands for distribution functions with missing responses

“…In semiparametric models, Wang et al (2004) developed empirical likelihood methods and proposed several estimators for the mean of the response variate by considering the partially linear models. Maity et al (2007) developed efficient estimation of population-level summaries in general semiparametric regression models. Titterington and Mill (1983) considered nonparametric estimation of the joint density of (X,Y) under missing completely at random (MCAR), utilizing random imputation to generate empirical versions of the joint density.…”