Semiparametric Maximum Likelihood Estimation Research Articles

Parametric Models for Response-Biased Sampling

Summary Suppose that subjects in a population follow the model f(y*|x*;θ) where y* denotes a response, x* denotes a vector of covariates and θ is the parameter to be estimated. We consider response-biased sampling, in which a subject is observed with a probability which is a function of its response. Such response-biased sampling frequently occurs in econometrics, epidemiology and survey sampling. The semiparametric maximum likelihood estimate of θ is derived, along with its asymptotic normality, efficiency and variance estimates. The estimate proposed can be used as a maximum partial likelihood estimate in stratified response-selective sampling. Some computation algorithms are also provided.

Consistency of semiparametric maximum likelihood estimators for two‐phase sampling

AbstractSemiparametric 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.

Maximum likelihood for generalised case-control studies

Suppose that we have a population of units with values (x,y), where x is a vector of covariates and y is a response variable, generated according to the model f(y|x;θ)g(x). Both x and y can be multivariate and any elements of x and y can be either discrete or continuous. The first term, f(y|x;θ), is some regression model linking response-variable behaviour to that of the covariates and we are interested in fitting this model. We discuss semiparametric maximum-likelihood estimation of θ for a class of study designs and data structures in which some units are not fully observed and in which it is impossible or undesirable to model the covariate distribution, g(x), which is typically of little interest in its own right. The key feature of the designs is that the probability that a unit is fully observed can depend on the value of the response, y. Applications include a number of useful generalisations of the basic case-control design and a broad class of missing data and measurement-error problems.

Semiparametric maximum likelihood for measurement error model regression.

This paper presents an EM algorithm for semiparametric likelihood analysis of linear, generalized linear, and nonlinear regression models with measurement errors in explanatory variables. A structural model is used in which probability distributions are specified for (a) the response and (b) the measurement error. A distribution is also assumed for the true explanatory variable but is left unspecified and is estimated by nonparametric maximum likelihood. For various types of extra information about the measurement error distribution, the proposed algorithm makes use of available routines that would be appropriate for likelihood analysis of (a) and (b) if the true x were available. Simulations suggest that the semiparametric maximum likelihood estimator retains a high degree of efficiency relative to the structural maximum likelihood estimator based on correct distributional assumptions and can outperform maximum likelihood based on an incorrect distributional assumption. The approach is illustrated on three examples with a variety of structures and types of extra information about the measurement error distribution.

Efficient estimation of binary choice models under symmetry

This paper proposes a semiparametric maximum likelihood estimator for both the intercept and slope parameters in a binary choice model under symmetry and index restrictions. The estimator attains the semiparametric efficiency bound in Cosslett (1987) under the symmetry and independence restrictions. Compared with the estimator of Klein and Spady (1993) which attains the semiparametric efficiency bound in Chamberlain (1986) and Cosslett (1987) under the independence restriction, we show that there are possible efficiency gains in estimating the slope parameters by imposing the additional symmetry restriction. A small Monte Carlo study is carried out to illustrate the usefulness of our estimator.

Observed Information in Semi-Parametric Models

We discuss the estimation of the asymptotic covariance matrix of semi-parametric maximum likelihood estimators by the observed profile information. We show that a discretized version of the second derivative of the profile likelihood function yields consistent estimators of minus the efficient information matrix.

Maximum likelihood estimation in semiparametric selection bias models with application to AIDS vaccine trials

SUMMARY The following problem is treated: given s possibly selection biased samples from an unknown distribution function, and assuming that the sampling rule weight functions for each of the samples are mathematically specified up to a common unknown finite-dimensional parameter, how can we use the data to estimate the unknown parameters? We propose a simple maximum partial likelihood method for deriving the semiparametric maximum likelihood estimator. A discussion of assumptions under which the selection bias model is identifiable and uniquely estimable is presented. We motivate the need for the methodology by discussing the generalised logistic regression model (Gilbert, Self & Ashby, 1998), a semiparametric selection bias model which is useful for assessing from vaccine trial data how the efficacy of an HIV vaccine varies with characteristics of the exposing virus. We show through simulations and an example that the maximum likelihood estimator in the generalised logistic regression model has satisfactory finite-sample properties.

Structural Latent Class Models

Linear logistic latent class analysis (LCA) relates the item latent probabilities of LCA to basic parameters representing the effects of explanatory variables. Applications of this model to dichotomous data comprise paired comparisons (the compared objects are the explanatory variables); the measurement of change due to, for example, therapeutic interventions (the treatments are the explanatory variables); simple scaling models of the Rasch type (the item difficulties and the person abilities are the explanatory variables); and similar models that try to explain the item difficulties by a hypothesized item structure (the cognitive operations needed to solve the items are the explanatory variables). The two latter types of models approximate the Rasch model and the linear logistic test model (LLTM), respectively, and become equivalent to these models for a sufficiently large number of classes. The same principle of semiparametric maximum likelihood estimation allows one to approximate the mixed Rasch model with its class-specific item difficulties and the mixed LLTM with its class-specific operation difficulties. Generalizations of linear logistic LCA include models for dichotomous items having varying discriminatory power and/or being affected by guessing (three-parameter linear logistic LCA) and models for the analysis of polytomous data.

A Comparison of Poisson, Negative Binomial, and Semiparametric Mixed Poisson Regression Models

Specifications and moment properties of the univariate Poisson and negative binomial distributions are briefly reviewed and illustrated. Properties and limitations of the corresponding poisson and negative binomial (gamma mixtures of Poissons) regression models are described. It is shown how a misspecification of the mixing distribution of a mixed Poisson model to accommodate hidden heterogeneity ascribable to unobserved variables—although not affecting the consistency of maximum likelihood estimators of the Poisson mean rate parameter or its regression parameterization—can lead to inflated t ratios of regression coefficients and associated incorrect inferences. Then the recently developed semiparametric maximum likelihood estimator for regression models composed of arbitrary mixtures of Poisson processes is specified and further developed. It is concluded that the semiparametric mixed Poisson regression model adds considerable flexibility to Poisson-family regression models and provides opportunities for interpretation of empirical patterns not available in the conventional approaches.

Semiparametric Estimation in a Three-State Duration-Dependent Markov Model from Interval-Censored Observations with Application to AIDS Data

The semiparametric maximum likelihood estimation is considered in a three-state duration dependent Markov process when times of the intermediate transition (e.g., onset of a disease) are interval censored and the times of transitions to an absorbing state (e.g., death) are known exactly or are right censored. It is assumed that the intensity of the transition to an absorbing state depends both on chronological time and duration in the intermediate state. De Gruttola and Lagakos (1989, Biometrics 45, 1-11) and Frydman (1992, Journal of the Royal Statistical Society, Series B 54, 853-866) discussed non-parametric estimation from the same sampling scheme under the assumption that the intensity of transition to an absorbing state depends only on the duration in the intermediate state or only on the chronological time respectively. The approach taken here generalizes, but in discrete time framework, the results from Frydman (1992). The distribution of the time to the intermediate transition is modelled nonparametrically and the intensity of onset of terminal condition semiparametrically. The algorithm is developed for the computation of the estimators. The methods are illustrated with AIDS data.

Semiparametric maximum likelihood estimation of polychotomous and sequential choice models

This article considers semiparametric estimation of discrete choice models. The estimation method is a semiparametric maximum likelihood method which generalizes Klein and Spady (1993) to the estimation of polychotomous choice and sequential choice models. Special emphasis is on the estimation of models with indices of which the density function is not bounded away from zero, issues on trimming indices, the correction of asymptotic bias, and issues of negative kernel density estimation. A semiparametric efficiency bound is derived for the polychotomous choice model with index restrictions. Our estimator is shown to be √ n-consistent, asymptotically normal, and almost asymptotically efficient. Monte Carlo experiments are provided to evaluate the finite-sample performance of the estimator.