SEMIPARAMETRIC ESTIMATION OF THE ERROR DISTRIBUTION IN MULTIVARIATE REGRESSION USING COPULAS

A semiparametric method is developed to estimate the dependence parameter and the joint distribution of the error term in the multivariate linear regression model. The nonparametric part of the method treats the marginal distributions of the error term as unknown, and estimates them using suitable empirical distribution functions. Then the dependence parameter is estimated by either maximizing a pseudolikelihood or solving an estimating equation. It is shown that this estimator is asymptotically normal, and a consistent estimator of its large sample variance is given. A simulation study shows that the proposed semiparametric method is better than the parametric ones available when the error distribution is unknown, which is almost always the case in practice. It turns out that there is no loss of asymptotic efficiency as a result of the estimation of regression parameters. An empirical example on portfolio management is used to illustrate the method.     

Bayesian spectral analysis models for quantile regression with Dirichlet process mixtures

This paper presents a Bayesian analysis of partially linear additive models for quantile regression. We develop a semiparametric Bayesian approach to quantile regression models using a spectral representation of the nonparametric regression functions and the Dirichlet process (DP) mixture for error distribution. We also consider Bayesian variable selection procedures for both parametric and nonparametric components in a partially linear additive model structure based on the Bayesian shrinkage priors via a stochastic search algorithm. Based on the proposed Bayesian semiparametric additive quantile regression model referred to as BSAQ, the Bayesian inference is considered for estimation and model selection. For the posterior computation, we design a simple and efficient Gibbs sampler based on a location-scale mixture of exponential and normal distributions for an asymmetric Laplace distribution, which facilitates the commonly used collapsed Gibbs sampling algorithms for the DP mixture models. Additionally, we discuss the asymptotic property of the sempiparametric quantile regression model in terms of consistency of posterior distribution. Simulation studies and real data application examples illustrate the proposed method and compare it with Bayesian quantile regression methods in the literature.     

BAYESIAN BETA REGRESSION: APPLICATIONS TO HOUSEHOLD EXPENDITURE DATA AND GENETIC DISTANCE BETWEEN FOOT-AND-MOUTH DISEASE VIRUSES

There is considerable interest in understanding how factors such as time and geographic distance between isolates might influence the evolutionary direction of foot‐and‐mouth disease. Genetic differences between viruses can be measured as the proportion of nucleotides that differ for a given sequence or gene. We present a Bayesian hierarchical regression model for the statistical analysis of continuous data with sample space restricted to the interval (0, 1). The data are modelled using beta distributions with means that depend on covariates through a link function. We discuss methodology for: (i) the incorporation of informative prior information into an analysis; (ii) fitting the model using Markov chain Monte Carlo sampling; (iii) model selection using Bayes factors; and (iv) semiparametric beta regression using penalized splines. The model was applied to two different datasets.     

Locally Efficient Semiparametric Estimators for Proportional Hazards Models with Measurement Error

We propose a new class of semiparametric estimators for proportional hazards models in the presence of measurement error in the covariates, where the baseline hazard function, the hazard function for the censoring time, and the distribution of the true covariates are considered as unknown infinite dimensional parameters. We estimate the model components by solving estimating equations based on the semiparametric efficient scores under a sequence of restricted models where the logarithm of the hazard functions are approximated by reduced rank regression splines. The proposed estimators are locally efficient in the sense that the estimators are semiparametrically efficient if the distribution of the error‐prone covariates is specified correctly and are still consistent and asymptotically normal if the distribution is misspecified. Our simulation studies show that the proposed estimators have smaller biases and variances than competing methods. We further illustrate the new method with a real application in an HIV clinical trial.     

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

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.     

M-estimation Under a Two-Sample Semiparametric Model

We consider M -estimation under a two-sample semiparametric model in which the log ratio of two unknown density functions has a known parametric form. This two-sample semiparametric model, arising naturally from case-control studies and logistic discriminant analysis, can be regarded as a biased sampling model. A new class of M -estimators are constructed on the basis of the maximum semiparametric likelihood estimator of the underlying distribution function. It is shown that the proposed M -estimators are consistent and asymptotically normally distributed. A simulation study is presented to demonstrate the performance of the proposed M -estimators.     

On the asymptotic behaviour of the recursive Nadaraya–Watson estimator associated with the recursive sliced inverse regression method

We investigate the asymptotic behaviour of the recursive Nadaraya–Watson estimator for the estimation of the regression function in a semiparametric regression model. On the one hand, we make use of the recursive version of the sliced inverse regression method for the estimation of the unknown parameter of the model. On the other hand, we implement a recursive Nadaraya–Watson procedure for the estimation of the regression function which takes into account the previous estimation of the parameter of the semiparametric regression model. We establish the almost sure convergence as well as the asymptotic normality for our Nadaraya–Watson estimate. We also illustrate our semiparametric estimation procedure on simulated data.     

SEMIPARAMETRIC REGRESSION AND GRAPHICAL MODELS

Semiparametric regression models that use spline basis functions with penalization have graphical model representations. This link is more powerful than previously established mixed model representations of semiparametric regression, as a larger class of models can be accommodated. Complications such as missingness and measurement error are more naturally handled within the graphical model architecture. Directed acyclic graphs, also known as Bayesian networks, play a prominent role. Graphical model-based Bayesian 'inference engines', such as bugs and vibes , facilitate fitting and inference. Underlying these are Markov chain Monte Carlo schemes and recent developments in variational approximation theory and methodology.     

Nonlinear semiparametric autoregressive model with finite mixtures of scale mixtures of skew normal innovations

We propose data generating structures which can be represented as the nonlinear autoregressive models with single and finite mixtures of scale mixtures of skew normal innovations. This class of models covers symmetric/asymmetric and light/heavy-tailed distributions, so provide a useful generalization of the symmetrical nonlinear autoregressive models. As semiparametric and nonparametric curve estimation are the approaches for exploring the structure of a nonlinear time series data set, in this article the semiparametric estimator for estimating the nonlinear function of the model is investigated based on the conditional least square method and nonparametric kernel approach. Also, an Expectation–Maximization-type algorithm to perform the maximum likelihood (ML) inference of unknown parameters of the model is proposed. Furthermore, some strong and weak consistency of the semiparametric estimator in this class of models are presented. Finally, to illustrate the usefulness of the proposed model, some simulation studies and an application to real data set are considered.     

Estimation of regression parameters in a semiparametric transformation model

A semiparametric approach to model skewed/heteroscedastic regression data is discussed. We work with a semiparametric transform-both-sides regression model, which contains a parametric regression function and a nonparametric transformation. This model is adequate when the relationship between the median response and the explanatory variable has been specified by a theoretical result or a previous empirical study. The transform-both-sides model with a parametric transformation has been studied extensively and applied successfully to a number data sets. Allowing a nonparametric transformation function increases the flexibility of the model. In this article, we estimate the nonparametric transformation function by the conditional kernel density approach developed by Wang and Ruppert (1995), and then use a pseudo-maximum likelihood estimator to estimate the regression parameters. This estimate of the regression parameters has not been studied previously. In this article, the asymptotic distribution of this pseudo-MLE is derived. We also show that when σ, the standard deviation of the error, goes to zero (small σ asymptotics), this estimator is adaptive. Adaptive means that the regression parameters are estimated as precisely as when the transformation is known exactly. A similar result holds in the parametric approaches of Carroll and Ruppert (1984) and Ruppert and Aldershof (1989). Simulated and real examples are provided to illustrate the performance of the proposed estimator for finite sample size.     

A Spline‐Based Semiparametric Maximum Likelihood Estimation Method for the Cox Model with Interval‐Censored Data

Abstract. We propose a spline‐based semiparametric maximum likelihood approach to analysing the Cox model with interval‐censored data. With this approach, the baseline cumulative hazard function is approximated by a monotone B‐spline function. We extend the generalized Rosen algorithm to compute the maximum likelihood estimate. We show that the estimator of the regression parameter is asymptotically normal and semiparametrically efficient, although the estimator of the baseline cumulative hazard function converges at a rate slower than root‐n. We also develop an easy‐to‐implement method for consistently estimating the standard error of the estimated regression parameter, which facilitates the proposed inference procedure for the Cox model with interval‐censored data. The proposed method is evaluated by simulation studies regarding its finite sample performance and is illustrated using data from a breast cosmesis study.     

SEMIPARAMETRIC COMPARISON OF REGRESSION CURVES VIA NORMAL LIKELIHOODS

Härdle & Marron (1990) treated the problem of semiparametric comparison of nonparametric regression curves by proposing a kernel-based estimator derived by minimizing a version of weighted integrated squared error. The resulting estimators of unknown transformation parameters are n-consistent, which prompts a consideration of issues. of optimality. We show that when the unknown mean function is periodic, an optimal nonparametric estimator may be motivated by an elegantly simple argument based on maximum likelihood estimation in a parametric model with normal errors. Strikingly, the asymptotic variance of an optimal estimator of θ does not depend at all on the manner of estimating error variances, provided they are estimated n-consistently. The optimal kernel-based estimator derived via these considerations is asymptotically equivalent to a periodic version of that suggested by Härdle & Marron, and so the latter technique is in fact optimal in this sense. We discuss the implications of these conclusions for the aperiodic case.     

Parametric and semiparametric hypotheses in the linear model

The independent additive errors linear model consists of a structure for the mean and a separate structure for the error distribution. The error structure may be parametric or it may be semiparametric. Under alternative values of the mean structure, the best fitting additive errors model has an error distribution which can be represented as the convolution of the actual error distribution and the marginal distribution of a misspecification term. The model misspecification term results from the covariates' distribution. Conditions are developed to distinguish when the semiparametric model yields sharper inference than the parametric model and vice versa. The main conditions concern the actual error distribution and the covariates' distribution. The theoretical results explain a paradoxical finding in semiparametric Bayesian modelling, where the posterior distribution under a semiparametric model is found to be more concentrated than is the posterior distribution under a corresponding parametric model. The paradox is illustrated on a set of allometric data. The Canadian Journal of Statistics 39: 165–180; 2011 ©2011 Statistical Society of Canada     

Semiparametric Regression with Kernel Error Model

Abstract.  We propose and study a class of regression models, in which the mean function is specified parametrically as in the existing regression methods, but the residual distribution is modelled non-parametrically by a kernel estimator, without imposing any assumption on its distribution. This specification is different from the existing semiparametric regression models. The asymptotic properties of such likelihood and the maximum likelihood estimate (MLE) under this semiparametric model are studied. We show that under some regularity conditions, the MLE under this model is consistent (when compared with the possibly pseudo-consistency of the parameter estimation under the existing parametric regression model), is asymptotically normal with rate and efficient. The non-parametric pseudo-likelihood ratio has the Wilks property as the true likelihood ratio does. Simulated examples are presented to evaluate the accuracy of the proposed semiparametric MLE method.     

Feasible robust estimator in restricted semiparametric regression models based on the LTS approach

Under some nonstochastic linear restrictions based on either additional information or prior knowledge in a semiparametric regression model, a family of feasible generalized robust estimators for the regression parameter is proposed. The least trimmed squares (LTS) method proposed by Rousseeuw as a highly robust regression estimator is a statistical technique for fitting a regression model based on the subset of h observations (out of n) whose least-square fit possesses the smallest sum of squared residuals. The coverage h may be set between n/2 and n. The LTS estimator involves computing the hyperplane that minimizes the sum of the smallest h squared residuals. For practical purpose, it is assumed that the covariance matrix of the error term is unknown and thus feasible estimators are replaced. Then, we develop an algorithm for the LTS estimator based on feasible methods. Through the Monte Carlo simulation studies and a real data example, performance of the feasible type of robust estimators is compared with the classical ones in restricted semiparametric regression models.     

New Ridge Regression Estimator in Semiparametric Regression Models

In the context of ridge regression, the estimation of shrinkage parameter plays an important role in analyzing data. Many efforts have been put to develop the computation of risk function in different full-parametric ridge regression approaches using eigenvalues and then bringing an efficient estimator of shrinkage parameter based on them. In this respect, the estimation of shrinkage parameter is neglected for semiparametric regression model. Not restricted, but the main focus of this approach is to develop necessary tools for computing the risk function of regression coefficient based on the eigenvalues of design matrix in semiparametric regression. For this purpose the differencing methodology is applied. We also propose a new estimator for shrinkage parameter which is of harmonic type mean of ridge estimators. It is shown that this estimator performs better than all the existing ones for the regression coefficient. For our proposal, a Monte Carlo simulation study and a real dataset analysis related to housing attributes are conducted to illustrate the efficiency of shrinkage estimators based on the minimum risk and mean squared error criteria.     

An asymptotic theory for semiparametric generalized least squares estimation in partially linear regression models

Consider a partially linear regression model with an unknown vector parameter β, an unknown functiong(·), and unknown heteroscedastic error variances. In this paper we develop an asymptotic semiparametric generalized least squares estimation theory under some weak moment conditions. These moment conditions are satisfied by many of the error distributions encountered in practice, and our theory does not require the number of replications to go to infinity.     

On Semiparametric Mode Regression Estimation

It has been found that, for a variety of probability distributions, there is a surprising linear relation between mode, mean, and median. In this article, the relation between mode, mean, and median regression functions is assumed to follow a simple parametric model. We propose a semiparametric conditional mode (mode regression) estimation for an unknown (unimodal) conditional distribution function in the context of regression model, so that any m-step-ahead mean and median forecasts can then be substituted into the resultant model to deliver m-step-ahead mode prediction. In the semiparametric model, Least Squared Estimator (LSEs) for the model parameters and the simultaneous estimation of the unknown mean and median regression functions by the local linear kernel method are combined to infer about the parametric and nonparametric components of the proposed model. The asymptotic normality of these estimators is derived, and the asymptotic distribution of the parameter estimates is also given and is shown to follow usual parametric rates in spite of the presence of the nonparametric component in the model. These results are applied to obtain a data-based test for the dependence of mode regression over mean and median regression under a regression model.     

Semiparametric Regression Analysis of Longitudinal Skewed Data

In this paper, we develop a semiparametric regression model for longitudinal skewed data. In the new model, we allow the transformation function and the baseline function to be unknown. The proposed model can provide a much broader class of models than the existing additive and multiplicative models. Our estimators for regression parameters, transformation function and baseline function are asymptotically normal. Particularly, the estimator for the transformation function converges to its true value at the rate n ^{? 1 ∕ 2}, the convergence rate that one could expect for a parametric model. In simulation studies, we demonstrate that the proposed semiparametric method is robust with little loss of efficiency. Finally, we apply the new method to a study on longitudinal health care costs.     

半参数空间变系数回归模型的两步估计方法及其数值模拟

文章提出了关于半参数空间变系数回归模型的两步估计方法,该方法可得到模型中常值系数估计量的精确解析表达式,广泛的数值模拟表明所提出的估计方法对估计常值系数具有满意的精度和稳定性。