A SEMIPARAMETRIC HIERARCHICAL METHOD FOR A REGRESSION MODEL WITH AN INTERVAL-CENSORED COVARIATE

A Bayesian framework is proposed for analysing regression models in which one of the covariates is interval‐censored. Such a situation was encountered in an AIDS clinical trial in which the goal was to examine the association between delays in initiating a new treatment after Indinavir failure and the subsequent viral load level of patients at the time of enrolment into the new treatment. The new method uses a mixture of Dirichlet processes allowing all the components in the model to be specified parametrically, except for the distribution of the interval‐censored covariate, which is treated non‐parametrically. The paper explains the proposed method for the linear regression model in detail. The performance of the method is assessed by simulations and illustrated using the AIDS clinical trial.     

On Semiparametric EV Models with Serially Correlated Errors in Both Regression Models and Mismeasured Covariates

Abstract.  We consider inference for a semiparametric regression model where some covariates are measured with errors, and the errors in both the regression model and the mismeasured covariates are serially correlated. We propose a weighted estimating equations-based estimator (WEEBE) for the regression coefficients. We show that the WEEBE is asymptotically more efficient than the estimators that neglect the serial correlations. This is an interesting new finding since earlier results in the statistical literature have shown that the weighted estimation is not as efficient as the unweighted estimation when the measurement errors and serially correlated errors of the regression models exist simultaneously (Biometrics, 49, 1993, 1262; Technometrics, 42, 2000, 137). The proposed WEEBE does not require undersmoothing the regressor functions in order to make it attain the root- n consistency. Simulation studies show that the proposed estimator has nice finite sample properties. A real data set is used to illustrate the proposed method.     

Efficient Estimation of Models with Conditional Moment Restrictions Containing Unknown Functions

We propose an estimation method for models of conditional moment restrictions, which contain finite dimensional unknown parameters (θ) and infinite dimensional unknown functions (h). Our proposal is to approximate h with a sieve and to estimate θ and the sieve parameters jointly by applying the method of minimum distance. We show that: (i) the sieve estimator of h is consistent with a rate faster than n^‐1/4 under certain metric; (ii) the estimator of θ is √n consistent and asymptotically normally distributed; (iii) the estimator for the asymptotic covariance of the θ estimator is consistent and easy to compute; and (iv) the optimally weighted minimum distance estimator of θ attains the semiparametric efficiency bound. We illustrate our results with two examples: a partially linear regression with an endogenous nonparametric part, and a partially additive IV regression with a link function.     

经济转型时期中国的非线性菲利普斯曲线

本文应用半参数模型识别经济转型时期中国的非线性菲利普斯曲线,这种非线性关系表现为三次多项式函数,拟合效果较好,统计检验显著。此外,本文给出了非线性菲利普斯曲线的经济学阐释,并用于预测2007-2010中国的通货膨胀趋势。     

SHRINKAGE, PRETEST AND ABSOLUTE PENALTY ESTIMATORS IN PARTIALLY LINEAR MODELS

We consider a partially linear model in which the vector of coefficients β in the linear part can be partitioned as ( β ₁, β ₂) , where β ₁ is the coefficient vector for main effects (e.g. treatment effect, genetic effects) and β ₂ is a vector for ‘nuisance’ effects (e.g. age, laboratory). In this situation, inference about β ₁ may benefit from moving the least squares estimate for the full model in the direction of the least squares estimate without the nuisance variables (Steinian shrinkage), or from dropping the nuisance variables if there is evidence that they do not provide useful information (pretesting). We investigate the asymptotic properties of Stein‐type and pretest semiparametric estimators under quadratic loss and show that, under general conditions, a Stein‐type semiparametric estimator improves on the full model conventional semiparametric least squares estimator. The relative performance of the estimators is examined using asymptotic analysis of quadratic risk functions and it is found that the Stein‐type estimator outperforms the full model estimator uniformly. By contrast, the pretest estimator dominates the least squares estimator only in a small part of the parameter space, which is consistent with the theory. We also consider an absolute penalty‐type estimator for partially linear models and give a Monte Carlo simulation comparison of shrinkage, pretest and the absolute penalty‐type estimators. The comparison shows that the shrinkage method performs better than the absolute penalty‐type estimation method when the dimension of the β ₂ parameter space is large.     

Robustness,Infinitesimal Neighborhoods,and Moment Restrictions

This paper is concerned with robust estimation under moment restrictions. A moment restriction model is semiparametric and distribution‐free; therefore it imposes mild assumptions. Yet it is reasonable to expect that the probability law of observations may have some deviations from the ideal distribution being modeled, due to various factors such as measurement errors. It is then sensible to seek an estimation procedure that is robust against slight perturbation in the probability measure that generates observations. This paper considers local deviations within shrinking topological neighborhoods to develop its large sample theory, so that both bias and variance matter asymptotically. The main result shows that there exists a computationally convenient estimator that achieves optimal minimax robust properties. It is semiparametrically efficient when the model assumption holds, and, at the same time, it enjoys desirable robust properties when it does not.     

Propriety of posteriors in structured additive regression models: Theory and empirical evidence

Structured additive regression comprises many semiparametric regression models such as generalized additive (mixed) models, geoadditive models, and hazard regression models within a unified framework. In a Bayesian formulation, non-parametric functions, spatial effects and further model components are specified in terms of multivariate Gaussian priors for high-dimensional vectors of regression coefficients. For several model terms, such as penalized splines or Markov random fields, these Gaussian prior distributions involve rank-deficient precision matrices, yielding partially improper priors. Moreover, hyperpriors for the variances (corresponding to inverse smoothing parameters) may also be specified as improper, e.g. corresponding to Jeffreys prior or a flat prior for the standard deviation. Hence, propriety of the joint posterior is a crucial issue for full Bayesian inference in particular if based on Markov chain Monte Carlo simulations. We establish theoretical results providing sufficient (and sometimes necessary) conditions for propriety and provide empirical evidence through several accompanying simulation studies.     

Frequency Domain Tests of Semiparametric Hypotheses for Locally Stationary Processes

Abstract.  Many time series in applied sciences obey a time-varying spectral structure. In this article, we focus on locally stationary processes and develop tests of the hypothesis that the time-varying spectral density has a semiparametric structure, including the interesting case of a time-varying autoregressive moving-average (tvARMA) model. The test introduced is based on a L ₂ -distance measure of a kernel smoothed version of the local periodogram rescaled by the time-varying spectral density of the estimated semiparametric model. The asymptotic distribution of the test statistic under the null hypothesis is derived. As an interesting special case, we focus on the problem of testing for the presence of a tvAR model. A semiparametric bootstrap procedure to approximate more accurately the distribution of the test statistic under the null hypothesis is proposed. Some simulations illustrate the behaviour of our testing methodology in finite sample situations.     

Statistical inference for multivariate partially linear regression models

In this paper we study a class of multivariate partially linear regression models. Various estimators for the parametric component and the nonparametric component are constructed and their asymptotic normality established. In particular, we propose an estimator of the contemporaneous correlation among the multiple responses and develop a test for detecting the existence of such contemporaneous correlation without using any nonparametric estimation. The performance of the proposed estimators and test is evaluated through some simulation studies and an analysis of a real data set is used to illustrate the developed methodology. The Canadian Journal of Statistics 41: 1–22; 2013 © 2013 Statistical Society of Canada