Showing papers on "Semiparametric model published in 2002"

An adaptive estimation of dimension reduction space

Abstract: Summary. Searching for an effective dimension reduction space is an important problem in regression, especially for high dimensional data. We propose an adaptive approach based on semiparametric models, which we call the (conditional) minimum average variance estimation (MAVE) method, within quite a general setting. The MAVE method has the following advantages. Most existing methods must undersmooth the nonparametric link function estimator to achieve a faster rate of consistency for the estimator of the parameters (than for that of the nonparametric function). In contrast, a faster consistency rate can be achieved by the MAVE method even without undersmoothing the nonparametric link function estimator. The MAVE method is applicable to a wide range of models, with fewer restrictions on the distribution of the covariates, to the extent that even time series can be included. Because of the faster rate of consistency for the parameter estimators, it is possible for us to estimate the dimension of the space consistently. The relationship of the MAVE method with other methods is also investigated. In particular, a simple outer product gradient estimator is proposed as an initial estimator. In addition to theoretical results, we demonstrate the efficacy of the MAVE method for high dimensional data sets through simulation. Two real data sets are analysed by using the MAVE approach.

Estimation of Semiparametric Models when the Criterion Function Is Not Smooth

Abstract: We provide easy to verify sufficient conditions for the consistency and asymptotic normality of a class of semiparametric optimization estimators where the criterion function does not obey standard smoothness conditions and simultaneously depends on some nonparametric estimators that can themselves depend on the parameters to be estimated. Our results extend existing theories such as those of Pakes and Pollard (1989), Andrews (1994a), and Newey (1994). We also show that bootstrap provides asymptotically correct confidence regions for the finite dimensional parameters. We apply our results to two examples: a 'hit rate' and a partially linear median regression with some endogenous regressors.

Semiparametric Smooth Coefficient Models

Abstract: In this article, we propose a semiparametric smooth coefficient model as a useful yet flexible specification for studying a general regression relationship with varying coefficients. The article proposes a local least squares method with a kernel weight function to estimate the smooth coefficient function. The consistency of the estimator and its asymptotic normality are established. A simple statistic for testing a parametric model versus the semiparametric smooth coefficient model is proposed. An empirical application of the proposed method is presented with an estimation of the production function of the nonmetal mineral industry in China. The empirical findings show that the intermediate production and management expense has played a vital role and is an unbalanced determinant of the labor and capital elasticities of output in production.

Semiparametric analysis of transformation models with censored data

Abstract: SUMMARY A unified estimation procedure is proposed for the analysis of censored data using linear transformation models, which include the proportional hazards model and the proportional odds model as special cases. This procedure is easily implemented numerically and its validity does not rely on the assumption of independence between the covariates and the censoring variable. The estimator is the same as the Cox partial likelihood estimator in the case of the proportional hazards model. Moreover, the asymptotic variance of the proposed estimator has a closed form and its variance estimator is easily obtained by plug-in rules. The method is illustrated by simulation and is applied to the Veterans' Administration lung cancer data.

Estimation in a semiparametric model for longitudinal data with unspecified dependence structure

Abstract: This paper considers an extension of M-estimators in semiparametric models for independent observations to the case of longitudinal data. We approximate the nonparametric function by a regression spline, and any M-estimation algorithm for the usual linear models can then be used to obtain consistent estimators of the model and valid largesample inferences about the regression parameters without any specification of the error distribution and the covariance structure. Included as special cases are the analysis of the conditional mean and median functions for longitudinal data.

Semiparametric Bayesian Inference in Autoregressive Panel Data Models

Abstract: This paper develops Bayesian methods for inference in dynamic panel data models with individual effects, and applies them to study longitudinal data on earnings from the Panel Study of Income Dynamics (PSID). We study semiparametric versions of commonly used random effects autoregressive models, in which the distribution of the disturbances is not restricted to fall in a parametric class. To model the unknown distributions without resorting to strong parametric assumptions, we draw upon recent advances in the theory and computation of nonparametric Bayesian models using Dirichlet process priors. The overall approach can be viewed as an application of the general semiparametric Bayesian approach of West, Muller, and Escobar (1994) which in turn makes use of results on Bayesian density estimation by Escobar (1994) and Escobar and West (1995). Most conventional panel data methods (e.g., Anderson and Hsiao (1981), MaCurdy

A Computational Approach for Full Nonparametric Bayesian Inference Under Dirichlet Process Mixture Models

Abstract: Widely used parametric generalized linear models are, unfortunately, a somewhat limited class of specifications. Nonparametric aspects are often introduced to enrich this class, resulting in semiparametric models. Focusing on single or k-sample problems, many classical nonparametric approaches are limited to hypothesis testing. Those that allow estimation are limited to certain functionals of the underlying distributions. Moreover, the associated inference often relies upon asymptotics when nonparametric specifications are often most appealing for smaller sample sizes. Bayesian nonparametric approaches avoid asymptotics but have, to date, been limited in the range of inference. Working with Dirichlet process priors, we overcome the limitations of existing simulation-based model fitting approaches which yield inference that is confined to posterior moments of linear functionals of the population distribution. This article provides a computational approach to obtain the entire posterior distribution for mor...

Semi-parametric modelling in finance: theoretical foundations

Abstract: The benchmark theory of mathematical finance is the Black-Scholes-Merton theory, based on Brownian motion as the driving noise process for asset prices. Here the distributions of returns of the assets in a portfolio are multivariate normal. The two most obvious limitations here concern symmetry and thin tails, neither being consistent with real data. The most common replacements for the multinormal are parametric—stable, generalized hyperbolic, variance gamma. In this paper we advocate the use of semi-parametric models for distributions, where the mean vector μ and covariance Σ are parametric components and the so-called density generator (function) g is the non-parametric component. We work mainly within the family of elliptically contoured distributions, focusing particularly on normal variance mixtures with self-decomposable mixing distributions. We show how the parametric cases can be treated in a unified, systematic way within the non-parametric framework and obtain the density generators fo...

Flexible smoothing with P-splines: a unified approach

Abstract: We consider the application of P-splines (Eilers and Marx, 1996) to three classes of models with smooth components: semiparametric models, models with serially correlated errors, and models with heteroscedastic errors. We show that P-splines provide a common approach to these problems. We set out a simple nonparametric strategy for the choice of the P-spline parameters (the number of knots, the degree of the P-spline, and the order of the penalty) and use mixed model (REML) methods for smoothing parameter selection. We give an example of a model in each of the three classes and analyse appropriate data sets.

Efficient Estimation of Fixed and Time‐varying Covariate Effects in Multiplicative Intensity Models

Abstract: The proportional hazards assumption of the Cox model does sometimes not hold in practise. An example is a treatment effect that decreases with time. We study a general multiplicative intensity model allowing the influence of each covariate to vary non-parametrically with time. An efficient estimation procedure for the cumulative parameter functions is developed. Its properties are studied using the martingale structure of the problem. Furthermore, we introduce a partly parametric version of the general non-parametric model in which the influence of some of the covariates varies with time while the effects of the remaining covariates are constant. This semiparametric model has not been studied in detail before. An efficient procedure for estimating the parametric as well as the non-parametric components of this model is developed. Again the martingale structure of the model allows us to describe the asymptotic properties of the suggested estimators. The approach is applied to two different data sets, and a Monte Carlo simulation is presented.

Efficient estimation in additive hazards regression with current status data

Abstract: SUMMARY Current status data arise when the exact timing of an event is unobserved, and it is only known at a given point in time whether or not the event has occurred. Recently Lin et al. (1998) studied the additive semiparametric hazards model for current status data. They showed that the analysis of current status data under the additive hazards model reduces to ordinary Cox regression under the assumption that a proportional hazards model may be used to describe the monitoring intensity. This analysis does not make efficient use of data, and in some cases it may not be appropriate to assume a proportional hazards model for the monitoring times. We study the semiparametric hazards model for current status data but make use of the semiparametric efficient score function. The suggested approach has the advantages that it is efficient in that it reaches the semiparametric information bound, and it does not involve any modelling of the monitoring times.

Detecting the Presence of Mixing with Multiscale Maximum Likelihood

Abstract: A test of homogeneity tries to decide whether observations come from a single distribution or from a mixture of several distributions. A powerful theory has been developed for the case where the component distributions are members of an exponential family. When no parametric assumptions are appropriate, the standard approach is to test for bimodality, which is known not to be very sensitive for detecting heterogeneity. To develop a more sensitive procedure, this article builds on an approach employed in sampling literature and models the component distributions as logarithmically concave densities. It is shown how this leads to a special semiparametric model, in which the homogeneity problem is equivalent to testing whether a parameter c equals zero, versus the alternative that c > 0. This setup leads naturally to a novel multiscale maximum likelihood procedure, where the multiscale character reflects the desirable property of adaptivity to the unknown value of the parameter c under the alternative, to en...

Asymptotic normality of semiparametric and nonparametric posterior distributions

Abstract: In this article it is shown that the marginal semiparametric and nonparametric posterior distributions for a parameter of interest behave like an ordinary parametric posterior distribution. This in practice provides support of the utility of marginal semiparametric and nonparametric posterior distributions. In particular, the marginal semiparametric and nonparametric posterior distributions are asymptotically normal and centered at the corresponding maximum likelihood estimates (MLEs) or posterior means, with covariance matrix the inverse of the Fisher information. Additionally, the semiparametric and nonparametric MLEs for the parameter of interest and the marginal posterior means are asymptotically normal and centered at the true parameter, with the same covariance matrix. The results are a semiparametric version and a nonparametric version of the parametric Bayesian central limit theorem that establish a connection between the semiparametric and the nonparametric Bayesian inference and their frequentis...

Semiparametric Receiver Operating Characteristic Analysis to Evaluate Biomarkers for Disease

Abstract: The receiver operating characteristic (ROC) curve is a popular method for characterizing the accuracy of diagnostic tests when test results are not binary. Various methodologies for estimating and comparing ROC curves have been developed. One approach, due to Pepe, uses a parametric regression model ROCx(u) = g(h0(u) + θ'0x) with the baseline function h0(u) specified up to a finite-dimensional parameter. In this article we extend the regression models by allowing arbitrary nonparametric baseline functions. We also provide asymptotic distribution theory and procedures for making statistical inference. We illustrate our approach with dataset from a prostate cancer biomarker study. Simulation studies suggest that the extra flexibility inherent in the semiparametric method is gained with little loss in statistical efficiency.

Semi-parametric Estimation of the Second Order Parameter in Statistics of Extremes

Abstract: We present a class of semi-parametric estimators for the second order parameter related to a probability distribution with a regularly varying tail. The second order parameter plays an important role whenever dealing with optimization problems in statistics of extreme values. Consistency and asymptotic normality are proven under appropriate conditions.

A semiparametric empirical likelihood method for data from an outcome-dependent sampling scheme with a continuous outcome.

Abstract: SUMMARY. Outcome-dependent sampling (ODS) schemes can be a cost effective way to enhance study efficiency. The case-control design has been widely used in epidemiologic studies. However, when the outcome is measured on a continuous scale, dichotomizing the outcome could lead to a loss of efficiency. Recent epidemiologic studies have used ODS sampling schemes where, in addition to an overall random sample, there are also a number of supplemental samples that are collected based on a continuous outcome variable. We consider a semiparametric empirical likelihood inference procedure in which the underlying distribution of covariates is treated as a nuisance parameter and is left unspecified. The proposed estimator has asymptotic normality properties. The likelihood ratio statistic using the semiparametric empirical likelihood function has Wilks-type properties in that, under the null, it follows a chi-square distribution asymptotically and is independent of the nuisance parameters. Our simulation results indicate that, for data obtained using an ODS design, the semiparametric empirical likelihood estimator is more efficient than conditional likelihood and probability weighted pseudolikelihood estimators and that ODS designs (along with the proposed estimator) can produce more efficient estimates than simple random sample designs of the same size. We apply the proposed method to analyze a data set from the Collaborative Perinatal Project (CPP), an ongoing environmental epidemiologic study, to assess the relationship between maternal polychlorinated biphenyl (PCB) level and children's IQ test performance.

Predicting Spatial Patterns of House Prices Using LPR and Bayesian Smoothing

Abstract: This article is motivated by the limited ability of standard hedonic price equations to deal with spatial variation in house prices. Spatial patterns of house prices can be viewed as the sum of many causal factors: Access to the central business district is associated with a house price gradient; access to decentralized employment subcenters causes more localized changes in house prices; in addition, neighborhood amenities (and disamenities) can cause house prices to change rapidly over relatively short distances. Spatial prediction (e.g., for an automated valuation system) requires models that can deal with all of these sources of spatial variation. We propose to accommodate these factors using a standard hedonic framework but incoporating a semiparametric model with structure in the residuals modeled with a partially Bayesian approach. The Bayesian framework enables us to provide complete inference in the form of a posterior distribution for each model parameter. Our model allows prediction at sampled or unsampled locations as well as prediction interval estimates. The nonparametric part of our model allows sufficient flexibility to find substantial spatial variation in house values. The parameters of the kriging model provide further insights into spatial patterns. Out–of–sample mean squared error and related statistics validate the proposed methods and justify their use for spatial prediction of house values.

Influence diagnostics and outlier tests for semiparametric mixed models

Abstract: Summary. Semiparametric mixed models are useful in biometric and econometric applications, especially for longitudinal data. Maximum penalized likelihood estimators (MPLEs) have been shown to work well by Zhang and co-workers for both linear coefficients and nonparametric functions. This paper considers the role of influence diagnostics in the MPLE by extending the case deletion and subject deletion analysis of linear models to accommodate the inclusion of a nonparametric component. We focus on influence measures for the fixed effects and provide formulae that are analogous to those for simpler models and readily computable with the MPLE algorithm. We also establish an equivalence between the case or subject deletion model and a mean shift outlier model from which we derive tests for outliers. The influence diagnostics proposed are illustrated through a longitudinal hormone study on progesterone and a simulated example.

The study of long-term HIV dynamics using semi-parametric non-linear mixed-effects models

Abstract: Modelling HIV dynamics has played an important role in understanding the pathogenesis of HIV infection in the past several years. Non-linear parametric models, derived from the mechanisms of HIV infection and drug action, have been used to fit short-term clinical data from AIDS clinical trials. However, it is found that the parametric models may not be adequate to fit long-term HIV dynamic data. To preserve the meaningful interpretation of the short-term HIV dynamic models as well as to characterize the long-term dynamics, we introduce a class of semi-parametric non-linear mixed-effects (NLME) models. The models are non-linear in population characteristics (fixed effects) and individual variations (random effects), both of which are modelled semi-parametrically. A basis-based approach is proposed to fit the models, which transforms a general semi-parametric NLME model into a set of standard parametric NLME models, indexed by the bases used. The bases that we employ are natural cubic splines for easy implementation. The resulting standard NLME models are low-dimensional and easy to solve. Statistical inferences that include testing parametric against semi-parametric mixed-effects are investigated. Innovative bootstrap procedures are developed for simulating the empirical distributions of the test statistics. Small-scale simulation and bootstrap studies show that our bootstrap procedures work well. The proposed approach and procedures are applied to long-term HIV dynamic data from an AIDS clinical study.

Semiparametric estimation of partially linear models for dependent data with generated regressors

Abstract: In this paper we consider the problem of estimating a semiparametric partially linear model for dependent data with generated regressors This type of model comes naturally from various econometric models such as a semiparametric rational expectation model when the surprise term enters the model nonparametrically, or a semiparametric type-3 Tobit model when the error distributions are of unknown forms, or a semiparametric error correction model Using the nonparametric kernel method and under primitive conditions, we show that the [square root]n-consistent estimation results of the finite-dimensional parameter in a partially linear model can be generalized to the case of generated regressors with weakly dependent data The regularity conditions we use are quite weak, and they are similar to those used in Robinson (1988, Econometrica 56, 931–954) for independent and observed data

Mixture models in measurement error problems, with reference to epidemiological studies

Abstract: Summary The paper focuses on a Bayesian treatment of measurement error problems and on the question of the specification of the prior distribution of the unknown covariates It presents a flexible semiparametric model for this distribution based on a mixture of normal distributions with an unknown number of components Implementation of this prior model as part of a full Bayesian analysis of measurement error problems is described in classical set-ups that are encountered in epidemiological studies: logistic regression between unknown covariates and outcome, with a normal or log-normal error model and a validation group The feasibility of this combined model is tested and its performance is demonstrated in a simulation study that includes an assessment of the influence of misspecification of the prior distribution of the unknown covariates and a comparison with the semiparametric maximum likelihood method of Roeder, Carroll and Lindsay Finally, the methodology is illustrated on a data set on coronary heart disease and cholesterol levels in blood

Semiparametric regression for count data

Abstract: We introduce a class of Bayesian semiparametric models for regression problems in which the response variable is a count. Our goal is to provide a flexible, easy-to-implement and robust extension of generalised linear models, for datasets of moderate or large size. Our approach is based on modelling the distribution of the response variable using a Dirichlet process, whose mean distribution function is itself random and is given a parametric form, such as a generalised linear model. The effects of the explanatory variables on the response are modelled via both the parameters of the mean distribution function of the Dirichlet process and the total mass parameter. We discuss modelling options and relationships with other approaches. We derive in closed form the marginal posterior distribution of the regression coefficients and discuss its use in inference and computing. We illustrate the benefits of our approach with a prognostic model for early breast cancer patients.

On empirical likelihood for a semiparametric mixture model

Abstract: SUMMARY Plant and animal studies of quantitative trait loci provide data which arise from mixtures of distributions with known mixing proportions. Previous approaches to estimation involve modelling the distributions parametrically. We propose a semiparametric alternative which assumes that the log ratio of the component densities satisfies a linear model, with the baseline density unspecified. It is demonstrated that a constrained empirical likelihood has an irregularity under the null hypothesis that the two densities are equal. A factorisation of the likelihood suggests a partial empirical likelihood which permits unconstrained estimation of the parameters, and which is shown to give consistent and asymptotically normal estimators, regardless of the null. The asymptotic null distribution of the log partial likelihood ratio is chi-squared. Theoretical calculations show that the procedure may be as efficient as the full empirical likelihood in the regular set-up. The usefulness of the robust methodology is illustrated with a rat study of breast cancer resistance genes.

Semiparametric method for estimating paleodemographic profiles from age indicator data.

Abstract: This paper addresses the problem of estimating an age-at-death distribution or paleodemographic profile from osteological data. It is demonstrated that the classical two-stage procedure whereby one first constructs estimates of age-at-death of individual skeletons and then uses these age estimates to obtain a paleodemographic profile is not a correct approach. This is a consequence of Bayes' theorem. Instead, we demonstrate a valid approach that proceeds from the opposite starting point: given skeletal age-at-death, one first estimates the probability of assigning the skeleton into a specific osteological age-indicator stage. We show that this leads to a statistically valid method for obtaining a paleodemographic profile, and moreover, that valid individual age estimation itself requires a demographic profile and therefore is done subsequent to its construction. Individual age estimation thus becomes the last rather than the first step in the estimation procedure. A central concept of our statistical approach is that of a weight function. A weight function is associated with each osteological age-indicator stage or category, and provides the probability that a specific age indicator stage is observed, given age-at-death of the individual. We recommend that weight functions be estimated nonparametrically from a reference data set. In their entirety, the weight functions characterize the relevant stochastic properties of a chosen age indicator. For actual estimation of the paleodemographic profile, a parametric age distribution in the target sample is assumed. The maximum likelihood method is used to identify the unknown parameters of this distribution. As some components are estimated nonparametrically, one then has a semiparametric model. We show how to obtain valid estimates of individual age-at-death, confidence regions, and goodness-of-fit tests. The methods are illustrated with both real and simulated data.

Double-semiparametric method for missing covariates in Cox regression models

Abstract: The problem of nuisance covariate model specification is considered in Cox regression where the maximum semiparametric likelihood method is used to handle the missing covariates. A component of the covariates is modeled nonparametrically to achieve robustness against covariate model misspecification and to reduce the number of possibly intractable integrations involved in the parametric modeling of the covariates. The statistical properties of the proposed method are examined. It is found that in some important situations, the maximum semiparametric likelihood can be applied without making any additional parametric model assumptions on covariates. The proposed method can yield a more efficient estimator than the nonparametric imputation methods and does not require specification of the missingness mechanism when compared with the inverse probability weighting method. A real data example is analyzed to demonstrate use of the proposed method.

Conditions for the Asymptotic Semiparametric Efficiency of an Omnibus Estimator of Dependence Parameters in Copula Models

Abstract: Oakes (1994) described in broad terms an omnibus semiparametric procedure for estimating the dependence parameter in a copula model when marginal distributions are treated as (infinite-dimensional) nuisance parameters. The resulting estimator was subsequently shown to be consistent and normally distributed asymptotically (Genest et al. 1995, Shih and Louis 1995). Conditions under which it is also semiparametrically efficient in large samples are given. While these requirements are met for the normal copula model (Klaassen and Wellner 1997), it is argued that this is an exception rather than the norm.

Semiparametric Mixed-Effects Models for Clustered Failure Time Data

Abstract: The Cox proportional hazards model with a random effect has been proposed for the analysis of data which consist of a large number of small clusters of correlated failure time observations. The class of linear transformation models provides many useful alternatives to the Cox model for analyzing univariate failure time data. In this article, we generalize these models by incorporating random effects, which generate the dependence among the failure times within the cluster, to handle correlated data. Inference and prediction procedures for such random effects models are proposed. They are relatively simple compared with the methods based on the nonparametric maximum likelihood estimators for the Cox frailty model in the literature. Our proposals are illustrated with a data set from a well-known eye study. Extensive numerical studies are conducted to evaluate various robustness properties of the new procedures.