Abstract not found

Semiparametric maximum likelihood estimators have recently been proposed for a class of two-phase, outcome dependent sampling models. All of them were "restricted" maximum likelihood estimators, in the sense that the maximization is carried out only over distributions concentrated on the observed values of the covariate vectors. In this paper, the authors give conditions for consistency of these restricted maximum likelihood estimators. They also consider the corresponding unrestricted maximization problems, in which the "absolute" maximum likelihood estimators may then have support on additional points in the covariate space. Their main consistency result also covers these unrestricted maximum likelihood estimators, when they exist for all sample sizes.

A useful approach to asymptotic e ciency for estimators in semiparametric models is the study of lower bounds on asymptotic variances via convolution theorems. Such theorems are often applicable in models in which the classical assumptions of independence and identical distributions fail to hold, but to date, much of the research has focused on semiparametric models with independent and identically distributed (i.i.d.) data because tools are available in the i.i.d. setting for verifying pre-conditions of the convolution theorems. We develop tools for non-i.i.d. data that are similar in spirit to those for i.i.d. data and also analogous to the approaches used in parametric models with dependent data. This involves extending the notion of the tangent vector guring so prominently in the i.i.d. theory and providing conditions for smoothness, or differentiability, of the parameter of interest as a function of the underlying probability measures. As a corollary to the differentiability result we obtain sufficient conditions for equivalence, in terms of asymptotic variance bounds, of two models. Regularity and asymptotic linearity of estimators are also discussed.

In this paper we revisit the information bound calculations in Robins, Rotnitzky, and Zhao (1994) and Robins, Hsieh, and Newey (1995) for regression models with missing covariates. We present an approach to their calculations based on score operators which verifies the bounds established in sections 2-5 of RRZ and in section 3 of RHN (up to typos in the case of RHN). In the case of the Cox regression model for survival data treated in RRZ section 8, we obtain different results than those of RRZ. The integral equation we present for the least favorable direction simplifies to the known information bound for the Cox model with complete data. We give examples of our information calculations for case-cohort designs and errors in variables regression models for survival data.

Weighted likelihood, in which one solves Horvitz-Thompson or inverse probability weighted (IPW) versions of the likelihood equations, offers a simple and robust method for fitting models to two phase stratified samples. We consider semiparametric models for which solution of infinite dimensional estimating equations leads to √ N consistent and asymptotically Gaussian estimators of both Euclidean and nonparametric parameters. If the phase two sample is selected via Bernoulli (i.i.d.) sampling with known sampling probabilities, standard estimating equation theory shows that the influence function for the weighted likelihood estimator of the Euclidean parameter is the IPW version of the ordinary influence function. By proving weak convergence of the IPW empirical process, and borrowing results on weighted bootstrap empirical processes, we derive a parallel asymptotic expansion for finite population stratified sampling. Whereas the asymptotic variance for Bernoulli sampling involves the within strata second moments of the influence function, for finite population stratified sampling it involves only the within strata variances. The latter asymptotic variance also arises when the observed sampling fractions are used as estimates of those known a priori. A general procedure is proposed for fitting semiparametric models with estimated weights to two phase data. Several of our key results have already been derived for the special case of Cox regression with stratified case-cohort studies, other complex survey designs and missing data problems more generally. This paper is intended to help place this previous work in appropriate context and to pave the way for applications to other models. Key words: case-cohort, estimated weights, failure time, inverse probability weights, missing data 1 1

this paper we consider both the absolute maximizer ( b ; b G) of the likelihood and the restricted MLE, dened through maximizing the likelihood over all pairs (; G) such that

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We extend the Semiparametric Likelihood Ratio Theorem of Murphy and Van del ' Vaart for one-dimensional to Euclidean paramet(;rs of auy dimension. The as:VIrlptotic distribution of the likelihood ratio statistic for testing a k-dimensional Euclidean paramet'"r is shown to be the usual under the null hypothesis. This result is useful not only for testing purposes but also in forming likelihood ratio based confidence regions. We also obtain the behavior of the likelihood ratio statistic under local (contiguous) alternatives; this is non-central with a non-centrality parameter involving the direction of perturbation of the null hypothesis and the efficient information matrix for the Euclidean parameter under the null hypothesis in A rigoJwus first established in DER indexed an infinite dimensional considered and

In this paper we discuss the analysis of multi-phase, or multi-stage, case-control studies and present an efficient semiparametric maximum-likelihood approach that unifies and extends earlier work, including the seminal case-control paper by Prentice & Pyke (1979) as

Semi-parametric efficiency bounds for regression models under generalised case-control sampling: