Nonparametric Shrinkage Estimation for Aalen's Additive Hazards Model

Aalen's nonparametric additive model in which the regression coefficients are assumed to be unspecified functions of time is a flexible alternative to Cox's proportional hazards model when the proportionality assumption is in doubt. In this paper, we incorporate a general linear hypothesis into the estimation of the time‐varying regression coefficients. We combine unrestricted least squares estimators and estimators that are restricted by the linear hypothesis and produce James‐Stein‐type shrinkage estimators of the regression coefficients. We develop the asymptotic joint distribution of such restricted and unrestricted estimators and use this to study the relative performance of the proposed estimators via their integrated asymptotic distributional risks. We conduct Monte Carlo simulations to examine the relative performance of the estimators in terms of their integrated mean square errors. We also compare the performance of the proposed estimators with a recently devised LASSO estimator as well as with ridge‐type estimators both via simulations and data on the survival of primary billiary cirhosis patients.     

Analysis of cointegrated models with measurement errors

We study the asymptotic properties of the reduced-rank estimator of error correction models of vector processes observed with measurement errors. Although it is well known that there is no asymptotic measurement error bias when predictor variables are integrated processes in regression models [Phillips BCB, Durlauf SN. Multiple time series regression with integrated processes. Rev Econom Stud. 1986;53:473–495], we systematically investigate the effects of the measurement errors (in the dependent variables as well as in the predictor variables) on the estimation of not only cointegrating vectors but also the speed of the adjustment matrix. Furthermore, we present the asymptotic properties of the estimators. We also obtain the asymptotic distribution of the likelihood ratio test for the cointegrating ranks. We investigate the effects of the measurement errors on estimation and test through a Monte Carlo simulation study.     

Some ridge regression estimators for the zero-inflated Poisson model

The zero-inflated Poisson regression model is commonly used when analyzing economic data that come in the form of non-negative integers since it accounts for excess zeros and overdispersion of the dependent variable. However, a problem often encountered when analyzing economic data that has not been addressed for this model is multicollinearity. This paper proposes ridge regression (RR) estimators and some methods for estimating the ridge parameter k for a non-negative model. A simulation study has been conducted to compare the performance of the estimators. Both mean squared error and mean absolute error are considered as the performance criteria. The simulation study shows that some estimators are better than the commonly used maximum-likelihood estimator and some other RR estimators. Based on the simulation study and an empirical application, some useful estimators are recommended for practitioners.     

Preliminary test almost unbiased two-parameter estimators with student’s t errors and conflicting test statistics

Abstract

In this paper, we consider the preliminary test approach to the estimation of the regression parameter in a multiple regression model under multicollinearity situation. The preliminary test almost unbiased two-parameter estimators based on the Wald, the Likelihood ratio, and the Lagrangian multiplier tests are given, when it is suspected that the regression parameter may be restricted to a subspace and the regression error is distributed with multivariate Student’s t errors. The bias and quadratic risk of the proposed estimators are derived and compared. Furthermore, a Monte Carlo simulation is provided to illustrate some of the theoretical results.     

Bayesian mismeasurement t-models for censored responses

This paper aims at introducing a Bayesian robust error-in-variable regression model in which the dependent variable is censored. We extend previous works by assuming a multivariate t distribution for jointly modelling the behaviour of the errors and the latent explanatory variable. Inference is done under the Bayesian paradigm. We use a data augmentation approach and develop a Markov chain Monte Carlo algorithm to sample from the posterior distributions. We run a Monte Carlo study to evaluate the efficiency of the posterior estimators in different settings. We compare the proposed model to three other models previously discussed in the literature. As a by-product we also provide a Bayesian analysis of the t-tobit model. We fit all four models to analyse the 2001 Medical Expenditure Panel Survey data.     

Empirical likelihood estimation for linear regression models with AR(p) error terms with numerical examples

Linear regression models are useful statistical tools to analyze data sets in different fields. There are several methods to estimate the parameters of a linear regression model. These methods usually perform under normally distributed and uncorrelated errors. If error terms are correlated the Conditional Maximum Likelihood (CML) estimation method under normality assumption is often used to estimate the parameters of interest. The CML estimation method is required a distributional assumption on error terms. However, in practice, such distributional assumptions on error terms may not be plausible. In this paper, we propose to estimate the parameters of a linear regression model with autoregressive error term using Empirical Likelihood (EL) method, which is a distribution free estimation method. A small simulation study is provided to evaluate the performance of the proposed estimation method over the CML method. The results of the simulation study show that the proposed estimators based on EL method are remarkably better than the estimators obtained from CML method in terms of mean squared errors (MSE) and bias in almost all the simulation configurations. These findings are also confirmed by the results of the numerical and real data examples.     

Sampling properties of estimators of a Weibull distribution of use in the lumber industry

We consider maximum-likelihood estimators of the three parameters in the Weibull distribution. Motivated by an application regarding the determination of a lower percentile of the strength of dimension lumber, we investigate the sampling properties of these estimators. Consistency is established when the shape parameter is greater than one, including some nonregular cases encountered in fitting lumber data. The joint distribution of the estimators is studied by a Monte Carlo approach. The maximum-likelihood estimator of the 5th percentile is compared with the sample 5th percentile. For the cases considered it was generally found that with sample size 70 the usual asymptotic normality does not hold.     

Application of shrinkage estimation in linear regression models with autoregressive errors

In this paper, we consider the shrinkage and penalty estimation procedures in the linear regression model with autoregressive errors of order p when it is conjectured that some of the regression parameters are inactive. We develop the statistical properties of the shrinkage estimation method including asymptotic distributional biases and risks. We show that the shrinkage estimators have a significantly higher relative efficiency than the classical estimator. Furthermore, we consider the two penalty estimators: least absolute shrinkage and selection operator (LASSO) and adaptive LASSO estimators, and numerically compare their relative performance with that of the shrinkage estimators. A Monte Carlo simulation experiment is conducted for different combinations of inactive predictors and the performance of each estimator is evaluated in terms of the simulated mean-squared error. This study shows that the shrinkage estimators are comparable to the penalty estimators when the number of inactive predictors in the model is relatively large. The shrinkage and penalty methods are applied to a real data set to illustrate the usefulness of the procedures in practice.     

Performance of some new preliminary test ridge regression estimators and their properties

The problem of estimation of the regression coefficients in a multiple regression model is considered under multicollinearity situation when it is suspected that the regression coefficients may be restricted to a subspace. We present the estimators of the regression coefficients combining the idea of preliminary test and ridge regression methodology. Accordingly, we consider three estimators, namely, the unrestricted ridge regression estimator (URRE), the restricted ridge regression estimator (RRRE), and finally, the preliminary test ridge regression estimator (PTRRE). The biases, variancematrices and mean square errors (mse) of the estimators are derived and compared with the usual estimators. Regions of optimality of the estimators are determined by studying the mse criterion. The conditions of superiority of the estimators over the traditional estimators as in Saleh and Han (1990) and Ali and Saleh (1991) have also been discussed.     

Non-parametric Maximum-Likelihood Estimation in a Semiparametric Mixture Model for Competing-Risks Data

Abstract.  This paper describes our studies on non-parametric maximum-likelihood estimators in a semiparametric mixture model for competing-risks data, in which proportional hazards models are specified for failure time models conditional on cause and a multinomial model is specified for the marginal distribution of cause conditional on covariates. We provide a verifiable identifiability condition and, based on it, establish an asymptotic profile likelihood theory for this model. We also provide efficient algorithms for the computation of the non-parametric maximum-likelihood estimate and its asymptotic variance. The success of this method is demonstrated in simulation studies and in the analysis of Taiwan severe acute respiratory syndrome data.     

Estimation in Varying-Coefficient Errors-in-Variables Models with Missing Response Variables

This article is concerned with the estimation of a varying-coefficient regression model when the response variable is sometimes missing and some of the covariates are measured with additive errors. We propose a class of estimators for the coefficient functions, as well as for the population mean and the error variance. The resulting estimators are shown to be asymptotically normal. Simulation studies are conducted to illustrate our approach.     

Sparsity identification for high-dimensional partially linear model with measurement error

In this article, we studied the identification of significant predictors in partially linear model in which some regressors are contaminated with random errors. Moreover, the dimension of parametric component is divergent and the regression coefficients are sparse. We applied difference technique to remove the nonparametric component for circumventing the selection of bandwidth, and constructed a bias-corrected shrinking estimator for the coefficient by using smoothly clipped absolute deviation (SCAD) penalty. Then, we derived the estimating and selecting consistency and established the asymptotic distribution for the identified significant estimators. Finally, Monte Carlo studies illustrate the performance of our approach.     

A new model selection procedure based on dynamic quantile regression

In this article, we propose a novel robust data-analytic procedure, dynamic quantile regression (DQR), for model selection. It is robust in the sense that it can simultaneously estimate the coefficients and the distribution of errors over a large collection of error distributions even those that are heavy-tailed and may not even possess variances or means; and DQR is easy to implement in the sense that it does not need to decide in advance which quantile(s) should be gathered. Asymptotic properties of related estimators are derived. Simulations and illustrative real examples are also given.     

Preliminary test Liu-type estimators based on W,LR, and LM test statistics in a regression model

This article discusses the preliminary test approach for the regression parameter in multiple regression model. The preliminary test Liu-type estimators based on the Wald (W), Likelihood ratio (LR), and Lagrangian multiplier(LM) tests are presented, when it is supposed that the regression parameter may be restricted to a subspace. We also give the bias and mean squared error of the proposed estimators and the superior of the proposed estimators is also discussed.     

Autocovariance Estimation in Regression with a Discontinuous Signal and m‐Dependent Errors: A Difference‐Based Approach

We discuss a class of difference‐based estimators for the autocovariance in nonparametric regression when the signal is discontinuous and the errors form a stationary m‐dependent process. These estimators circumvent the particularly challenging task of pre‐estimating such an unknown regression function. We provide finite‐sample expressions of their mean squared errors for piecewise constant signals and Gaussian errors. Based on this, we derive biased‐optimized estimates that do not depend on the unknown autocovariance structure. Notably, for positively correlated errors, that part of the variance of our estimators that depend on the signal is minimal as well. Further, we provide sufficient conditions for ‐consistency; this result is extended to piecewise Hölder regression with non‐Gaussian errors. We combine our biased‐optimized autocovariance estimates with a projection‐based approach and derive covariance matrix estimates, a method that is of independent interest. An R package, several simulations and an application to biophysical measurements complement this paper.     

Low order approximations in deconvolution and regression with errors in variables

Summary.  We suggest two new methods, which are applicable to both deconvolution and regression with errors in explanatory variables, for nonparametric inference. The two approaches involve kernel or orthogonal series methods. They are based on defining a low order approximation to the problem at hand, and proceed by constructing relatively accurate estimators of that quantity rather than attempting to estimate the true target functions consistently. Of course, both techniques could be employed to construct consistent estimators, but in many contexts of importance (e.g. those where the errors are Gaussian) consistency is, from a practical viewpoint, an unattainable goal. We rephrase the problem in a form where an explicit, interpretable, low order approximation is available. The information that we require about the error distribution (the error-in-variables distribution, in the case of regression) is only in the form of low order moments and so is readily obtainable by a rudimentary analysis of indirect measurements of errors, e.g. through repeated measurements. In particular, we do not need to estimate a function, such as a characteristic function, which expresses detailed properties of the error distribution. This feature of our methods, coupled with the fact that all our estimators are explicitly defined in terms of readily computable averages, means that the methods are particularly economical in computing time.     

Analysis of growth curves under measurement errors

In this paper, we propose a method for the analysis of growth curve models when also the regressor variable may be measured with errors. Two classes of structure for errors in regressors are discussed. For complete and balanced data, estimators for the model parameters are derived under the maximum-likelihood framework. Numerical examples are provided to illustrate the proposed technique.     

Robust Linear Calibration

We regard the simple linear calibration problem where only the response y of the regression line y = β₀ + β₁ t is observed with errors. The experimental conditions t are observed without error. For the errors of the observations y we assume that there may be some gross errors providing outlying observations. This situation can be modeled by a conditionally contaminated regression model. In this model the classical calibration estimator based on the least squares estimator has an unbounded asymptotic bias. Therefore we introduce calibration estimators based on robust one-step-M-estimators which have a bounded asymptotic bias. For this class of estimators we discuss two problems: The optimal estimators and their corresponding optimal designs. We derive the locally optimal solutions and show that the maximin efficient designs for non-robust estimation and robust estimation coincide.     

Small-area estimation by combining time-series and cross-sectional data

A model involving autocorrelated random effects and sampling errors is proposed for small-area estimation, using both time-series and cross-sectional data. The sampling errors are assumed to have a known block-diagonal covariance matrix. This model is an extension of a well-known model, due to Fay and Herriot (1979), for cross-sectional data. A two-stage estimator of a small-area mean for the current period is obtained under the proposed model with known autocorrelation, by first deriving the best linear unbiased prediction estimator assuming known variance components, and then replacing them with their consistent estimators. Extending the approach of Prasad and Rao (1986, 1990) for the Fay-Herriot model, an estimator of mean squared error (MSE) of the two-stage estimator, correct to a second-order approximation for a small or moderate number of time points, T, and a large number of small areas, m, is obtained. The case of unknown autocorrelation is also considered. Limited simulation results on the efficiency of two-stage estimators and the accuracy of the proposed estimator of MSE are présentés.