In applied problems it is common to specify a model for the conditional mean of a response given a set of regressors. A subset of the regressors may be missing for some study subjects either by design or happenstance. In this article we propose a new class of semiparametric estimators, based on inverse probability weighted estimating equations, that are consistent for parameter vector α0 of the conditional mean model when the data are missing at random in the sense of Rubin and the missingness probabilities are either known or can be parametrically modeled. We show that the asymptotic variance of the optimal estimator in our class attains the semiparametric variance bound for the model by first showing that our estimation problem is a special case of the general problem of parameter estimation in an arbitrary semiparametric model in which the data are missing at random and the probability of observing complete data is bounded away from 0, and then deriving a representation for the efficient score...

https://www.jstor.org/stable/pdfplus/2290910.pdf

Estimation of Regression Coefficients When Some Regressors are not Always Observed

On considere une variable aleatoire (X,Z) dans #7B-R P ×#7B-R q . On construit un estimateur generalisant l'estimateur des moindres carres ordinaires en inserant des estimateurs non parametriques de la regression dans la projection orthogonale non lineaire sur Z

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Root-n-consistent semiparametric regression

We propose a class of inverse probability of censoring weighted estimators for the parameters of models for the dependence of the mean of a vector of correlated response variables on a vector of explanatory variables in the presence of missing response data. The proposed estimators do not require full specification of the likelihood. They can be viewed as an extension of generalized estimating equations estimators that allow for the data to be missing at random but not missing completely at random. These estimators can be used to correct for dependent censoring and nonrandom noncompliance in randomized clinical trials studying the effect of a treatment on the evolution over time of the mean of a response variable. The likelihood-based parametric G-computation algorithm estimator may also be used to attempt to correct for dependent censoring and nonrandom noncompliance. But because of possible model misspecification, the parametric G-computation algorithm estimator, in contrast with the proposed w...

Analysis of semiparametric regression models for repeated outcomes in the presence of missing data

Introduction.- Asymptotic Inference for (Finite-Dimensional) Parametric Models.- Information Bounds for Euclidean Parameters in Infinite-Dimensional Models.- Euclidean Parameters: Further Examples.- Information Bounds for Infinite-Dimensional Parameters.- Infinite-Dimensional Parameters: Further Examples: Construction of Examples.

Efficient and Adaptive Estimation for Semiparametric Models

Let $(X, Y)$ be a pair of random variables such that $X = (X_1, \cdots, X_J)$ and let $f$ by a function that depends on the joint distribution of $(X, Y).$ A variety of parametric and nonparametric models for $f$ are discussed in relation to flexibility, dimensionality, and interpretability. It is then supposed that each $X_j \in \lbrack 0, 1\rbrack,$ that $Y$ is real valued with mean $\mu$ and finite variance, and that $f$ is the regression function of $Y$ on $X.$ Let $f^\ast,$ of the form $f^\ast(x_1, \cdots, x_J) = \mu + f^\ast_1(x_1) + \cdots + f^\ast_J(x_J),$ be chosen subject to the constraints $Ef^\ast_j = 0$ for $1 \leq j \leq J$ to minimize $E\lbrack(f(X) - f^\ast(X))^2\rbrack.$ Then $f^\ast$ is the closest additive approximation to $f,$ and $f^\ast = f$ if $f$ itself is additive. Spline estimates of $f^\ast_j$ and its derivatives are considered based on a random sample from the distribution of $(X, Y).$ Under a common smoothness assumption on $f^\ast_j, 1 \leq j \leq J,$ and some mild auxiliary assumptions, these estimates achieve the same (optimal) rate of convergence for general $J$ as they do for $J = 1.$

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Additive Regression and Other 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 $L^2$-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.

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Janet M. Begun

Papers

Information and Asymptotic Efficiency in Parametric-Nonparametric Models