Randomly censored partially linear single-index models

This paper proposes a method for estimation of a class of partially linear single-index models with randomly censored samples. The method provides a flexible way for modelling the association between a response and a set of predictor variables when the response variable is randomly censored. It presents a technique for “dimension reduction” in semiparametric censored regression models and generalizes the existing accelerated failure-time models for survival analysis. The estimation procedure involves three stages: first, transform the censored data into synthetic data or pseudo-responses unbiasedly; second, obtain quasi-likelihood estimates of the regression coefficients in both linear and single-index components by an iteratively algorithm; finally, estimate the unknown nonparametric regression function using techniques for univariate censored nonparametric regression. The estimators for the regression coefficients are shown to be jointly root-n consistent and asymptotically normal. In addition, the estimator for the unknown regression function is a local linear kernel regression estimator and can be estimated with the same efficiency as all the parameters are known. Monte Carlo simulations are conducted to illustrate the proposed methodology.     

Minimum f-divergence estimators and quasi-likelihood functions

Maximum quasi-likelihood estimators have several nice asymptotic properties. We show that, in many situations, a family of estimators, called the minimum f-divergence estimators, can be defined such that each estimator has the same asymptotic properties as the maximum quasi-likelihood estimator. The family of minimum f-divergence estimators include the maximum quasi-likelihood estimators as a special case. When a quasi-likelihood is the log likelihood from some exponential family, Amari's dual geometries can be used to study the maximum likelihood estimator. A dual geometric structure can also be defined for more general quasi-likelihood functions as well as for the larger family of minimum f-divergence estimators. The relationship between the f-divergence and the quasi-likelihood function and the relationship between the f-divergence and the power divergence is discussed.This work was supported by National Science Foundation grant DMS 88-03584.     

Iteratively Reweighted Generalized Least Squares for Estimation and Testing With Correlated Data: An Inference Function Framework

The focus of this article is on fitting regression models and testing of general linear hypotheses for correlated data using quasi-likelihood based techniques. The class of generalized method of moments or GMMs provides an elegant approach for estimating a vector of regression parameters from a set of score functions. Extending the principle of the GMMs, in the generalized estimating equation framework, leads to a quadratic inference function or QIF approach for the analysis of correlated data. We derive an iteratively reweighted generalized least squares or IRGLS algorithm for finding the QIF estimator and establish its convergence properties. A software library implementing the techniques is demonstrated through several datasets.     

Dimension Reduction in Regressions With Exponential Family Predictors

We present first methodology for dimension reduction in regressions with predictors that, given the response, follow one-parameter exponential families. Our approach is based on modeling the conditional distribution of the predictors given the response, which allows us to derive and estimate a sufficient reduction of the predictors. We also propose a method of estimating the forward regression mean function without requiring an explicit forward regression model. Whereas nearly all existing estimators of the central subspace are limited to regressions with continuous predictors only, our proposed methodology extends estimation to regressions with all categorical or a mixture of categorical and continuous predictors. Supplementary materials including the proofs and the computer code are available from the JCGS website.     

Nonparametric estimation in generalized varying-coefficient models based on iterative weighted quasi-likelihood method

This paper focuses on the estimation of the coefficient functions, which is of primary interest, in generalized varying-coefficient models with non-exponential family error. The local weighted quasi-likelihood method which results from local polynomial regression techniques is presented. The nonparametric estimator based on iterative weighted quasi-likelihood method is obtained to estimate coefficient functions. The asymptotic efficiency of the proposed estimator is given. Furthermore, some simulations are carried out to evaluate the finite sample performance of the proposed method, which show that it possesses some advantages to the previous methods. Finally, a real data example is used to illustrate the proposed methodology.     

Quasi-likelihood estimation of average treatment effects based on model information

In this paper, the estimation of average treatment effects is considered when we have the model information of the conditional mean and conditional variance for the responses given the covariates. The quasi-likelihood method adapted to treatment effects data is developed to estimate the parameters in the conditional mean and conditional variance models. Based on the model information, we define three estimators by imputation, regression and inverse probability weighted methods. All the estimators are shown asymptotically normal. Our simulation results show that by using the model information, the substantial efficiency gains are obtained which are comparable with the existing estimators.     

Adaptive quasi-likelihood estimate in generalized linear models

This paper gives a thorough theoretical treatment on the adaptive quasi-likelihood estimate of the parameters in the generalized linear models. The unknown covariance matrix of the response variable is estimated by the sample. It is shown that the adaptive estimator defined in this paper is asymptotically most efficient in the sense that it is asymptotic normal, and the covariance matrix of the limit distribution coincides with the one for the quasi-likelihood estimator for the case that the covariance matrix of the response variable is completely known.     

协方差结构分析中拟似然估计的对偶几何方法

本文应用几何方法研究了协方差结构分析中的拟似然估计.对于该模型引进了对偶几何,在此基础上得到了拟似然估计的二阶渐近性质.通过对偶曲率给出了拟似然估计的偏差、方差和信息损失,并且给出了反映拟观察信息和拟期望信息之间关系的一个极限定理.     

Empirical likelihood-based dimension reduction inference for linear error-in-responses models with validation study

In this paper, linear errors-in-response models are considered in the presence of validation data on the responses. A semiparametric dimension reduction technique is employed to define an estimator of β with asymptotic normality, the estimated empirical loglikelihoods and the adjusted empirical loglikelihoods for the vector of regression coefficients and linear combinations of the regression coefficients, respectively. The estimated empirical log-likelihoods are shown to be asymptotically distributed as weighted sums of independent x12 and the adjusted empirical loglikelihoods are proved to be asymptotically distributed as standard chi-squares, respectively.     

Approximate Bayesian shrinkage estimation

A Bayesian shrinkage estimate for the mean in the generalized linear empirical Bayes model is proposed. The posterior mean under the empirical Bayes model has a shrinkage pattern. The shrinkage factor is estimated by using a Bayesian method with the regression coefficients to be fixed at the maximum extended quasi-likelihood estimates. This approach develops a Bayesian shrinkage estimate of the mean which is numerically quite tractable. The method is illustrated with a data set, and the estimate is compared with an earlier one based on an empirical Bayes method. In a special case of the homogeneous model with exchangeable priors, the performance of the Bayesian estimate is illustrated by computer simulations. The simulation result shows as improvement of the Bayesian estimate over the empirical Bayes estimate in some situations.     

Successive direction extraction for estimating the central subspace in a multiple-index regression

In this paper we propose a dimension reduction method for estimating the directions in a multiple-index regression based on information extraction. This extends the recent work of Yin and Cook [X. Yin, R.D. Cook, Direction estimation in single-index regression, Biometrika 92 (2005) 371-384] who introduced the method and used it to estimate the direction in a single-index regression. While a formal extension seems conceptually straightforward, there is a fundamentally new aspect of our extension: We are able to show that, under the assumption of elliptical predictors, the estimation of multiple-index regressions can be decomposed into successive single-index estimation problems. This significantly reduces the computational complexity, because the nonparametric procedure involves only a one-dimensional search at each stage. In addition, we developed a permutation test to assist in estimating the dimension of a multiple-index regression.     

Asymptotic distributions of two “synthetic data” estimators for censored single-index models

The censored single-index model provides a flexible way for modelling the association between a response and a set of predictor variables when the response variable is randomly censored and the link function is unknown. It presents a technique for “dimension reduction” in semiparametric censored regression models and generalizes the existing accelerated failure time models for survival analysis. This paper proposes two methods for estimation of single-index models with randomly censored samples. We first transform the censored data into synthetic data or pseudo-responses unbiasedly, then obtain estimates of the index coefficients by the rOPG or rMAVE procedures of Xia (2006) [1]. Finally, we estimate the unknown nonparametric link function using techniques for univariate censored nonparametric regression. The estimators for the index coefficients are shown to be root-n consistent and asymptotically normal. In addition, the estimator for the unknown regression function is a local linear kernel regression estimator and can be estimated with the same efficiency as the parameters are known. Monte Carlo simulations are conducted to illustrate the proposed methodologies.     

ITERATIVE QUASI-LIKELIHOOD FOR SEEMINGLY UNRELATED REGRESSION SYSTEMS

In the seemingly unrelated regression systems, the existing quasi-likelihood is always involved in the difficult problem of calculating inverse of a high order matrix specially for large systems. To avoid this problem, the new quasi-likelihood proposed in this paper is based mainly on a linearly iterative process of some unbiased estimating functions. Some finite sample properties and asymptotic behaviours for this new quasi-likelihood are investigated. These results show that the new quasi-likelihood for parameter of interest is E-sufficient, iteratively efficient and approximately efficient. Some examples are given to illustrate the theoretical results.     

On spline approximation of sliced inverse regression

The dimension reduction is helpful and often necessary in exploring the nonparametric regression structure.In this area,Sliced inverse regression (SIR) is a promising tool to estimate the central dimension reduction (CDR) space.To estimate the kernel matrix of the SIR,we herein suggest the spline approximation using the least squares regression.The heteroscedasticity can be incorporated well by introducing an appropriate weight function.The root-n asymptotic normality can be achieved for a wide range choice of knots.This is essentially analogous to the kernel estimation.Moreover, we also propose a modified Bayes information criterion (BIC) based on the eigenvalues of the SIR matrix.This modified BIC can be applied to any form of the SIR and other related methods.The methodology and some of the practical issues are illustrated through the horse mussel data.Empirical studies evidence the performance of our proposed spline approximation by comparison of the existing estimators.     

A graphical tool for selecting the number of slices and the dimension of the model in SIR and SAVE approaches

Sliced inverse regression (SIR) and related methods were introduced in order to reduce the dimensionality of regression problems. In general semiparametric regression framework, these methods determine linear combinations of a set of explanatory variables X related to the response variable Y, without losing information on the conditional distribution of Y given X. They are based on a “slicing step” in the population and sample versions. They are sensitive to the choice of the number H of slices, and this is particularly true for SIR-II and SAVE methods. At the moment there are no theoretical results nor practical techniques which allows the user to choose an appropriate number of slices. In this paper, we propose an approach based on the quality of the estimation of the effective dimension reduction (EDR) space: the square trace correlation between the true EDR space and its estimate can be used as goodness of estimation. We introduce a na?ve bootstrap estimation of the square trace correlation criterion to allow selection of an “optimal” number of slices. Moreover, this criterion can also simultaneously select the corresponding suitable dimension K (number of the linear combination of X). From a practical point of view, the choice of these two parameters H and K is essential. We propose a 3D-graphical tool, implemented in R, which can be useful to select the suitable couple (H, K). An R package named “edrGraphicalTools” has been developed. In this article, we focus on the SIR-I, SIR-II and SAVE methods. Moreover the proposed criterion can be use to determine which method seems to be efficient to recover the EDR space, that is the structure between Y and X. We indicate how the proposed criterion can be used in practice. A simulation study is performed to illustrate the behavior of this approach and the need for selecting properly the number H of slices and the dimension K. A short real-data example is also provided.     

拟似然非线性模型的某些渐近推断:几何方法

Abstract. A modified Bates and Watts geometric framework is proposed for quasi-likelihoodnonlinear models in Euclidean inner product space. Based on the modified geometric framework,some asymptotic inference in terms of curvatures for quasi-likelihood nonlinear models is stud-ied. Several previous results for nonlinear regression models and exponential family nonlinearmodels etc. are extended to quasi-likelihood nonlinear models.     

Sufficient dimension reduction for the conditional mean with a categorical predictor in multivariate regression

Recent sufficient dimension reduction methodologies in multivariate regression do not have direct application to a categorical predictor. For this, we define the multivariate central partial mean subspace and propose two methodologies to estimate it. The first method uses the ordinary least squares. Chi-squared distributed statistics for dimension tests are constructed, and an estimate of the target subspace is consistent and efficient. Moreover, the effects of continuous predictors can be tested without assuming any model. The second method extends Iterative Hessian Transformation to this context. For dimension estimation, permutation tests are used. Simulated and real data examples for illustrating various properties of the proposed methods are presented.     

Censored multiple regression by the method of average derivatives

This paper proposes a technique [termed censored average derivative estimation (CADE)] for studying estimation of the unknown regression function in nonparametric censored regression models with randomly censored samples. The CADE procedure involves three stages: firstly-transform the censored data into synthetic data or pseudo-responses using the inverse probability censoring weighted (IPCW) technique, secondly estimate the average derivatives of the regression function, and finally approximate the unknown regression function by an estimator of univariate regression using techniques for one-dimensional nonparametric censored regression. The CADE provides an easily implemented methodology for modelling the association between the response and a set of predictor variables when data are randomly censored. It also provides a technique for “dimension reduction” in nonparametric censored regression models. The average derivative estimator is shown to be root-n consistent and asymptotically normal. The estimator of the unknown regression function is a local linear kernel regression estimator and is shown to converge at the optimal one-dimensional nonparametric rate. Monte Carlo experiments show that the proposed estimators work quite well.     

Sufficient Dimension Reduction and Graphics in Regression

In this article, we review, consolidate and extend a theory for sufficient dimension reduction in regression settings. This theory provides a powerful context for the construction, characterization and interpretation of low-dimensional displays of the data, and allows us to turn graphics into a consistent and theoretically motivated methodological body. In this spirit, we propose an iterative graphical procedure for estimating the meta-parameter which lies at the core of sufficient dimension reduction; namely, the central dimension-reduction subspace.     

Estimation for mixtures of Markov processes

Finite mixtures of Markov processes with densities belonging to exponential families are introduced. Quasi-likelihood and maximum likelihood methods are used to estimate the parameters of the mixing distributions and of the component distributions. The E-M algorithm is used to compute the ML estimates. Mixture of Autoregressive processes and of two-state Markov chains are discussed as specific examples. Simulation results on the comparison of quasi-likelihood and ML estimates are reported.