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.     

Penalized and Shrinkage Estimation in the Cox Proportional Hazards Model

This article considers the shrinkage estimation procedure in the Cox's proportional hazards regression model when it is suspected that some of the parameters may be restricted to a subspace. We have developed the statistical properties of the shrinkage estimators including asymptotic distributional biases and risks. The shrinkage estimators have much higher relative efficiency than the classical estimator, furthermore, we consider two penalty estimators—the LASSO and adaptive LASSO—and compare their relative performance with that of the shrinkage estimators numerically. A Monte Carlo simulation experiment is conducted for different combinations of irrelevant predictors and the performance of each estimator is evaluated in terms of simulated mean squared error. Simulation study shows that the shrinkage estimators are comparable to the penalty estimators when the number of irrelevant predictors in the model is relatively large. The shrinkage and penalty methods are applied to two real data sets to illustrate the usefulness of the procedures in practice.     

Shrinkage estimation in lognormal regression model for censored data

We introduce in this paper, the shrinkage estimation method in the lognormal regression model for censored data involving many predictors, some of which may not have any influence on the response of interest. We develop the asymptotic properties of the shrinkage estimators (SEs) using the notion of asymptotic distributional biases and risks. We show that if the shrinkage dimension exceeds two, the asymptotic risk of the SEs is strictly less than the corresponding classical estimators. Furthermore, we study the penalty (LASSO and adaptive LASSO) estimation methods and compare their relative performance with the SEs. A simulation study for various combinations of the inactive predictors and censoring percentages shows that the SEs perform better than the penalty estimators in certain parts of the parameter space, especially when there are many inactive predictors in the model. It also shows that the shrinkage and penalty estimators outperform the classical estimators. A real-life data example using Worcester heart attack study is used to illustrate the performance of the suggested estimators.     

Model selection and parameter estimation of a multinomial logistic regression model

In the multinomial regression model, we consider the methodology for simultaneous model selection and parameter estimation by using the shrinkage and LASSO (least absolute shrinkage and selection operation) [R. Tibshirani, Regression shrinkage and selection via the LASSO, J. R. Statist. Soc. Ser. B 58 (1996), pp. 267–288] strategies. The shrinkage estimators (SEs) provide significant improvement over their classical counterparts in the case where some of the predictors may or may not be active for the response of interest. The asymptotic properties of the SEs are developed using the notion of asymptotic distributional risk. We then compare the relative performance of the LASSO estimator with two SEs in terms of simulated relative efficiency. A simulation study shows that the shrinkage and LASSO estimators dominate the full model estimator. Further, both SEs perform better than the LASSO estimators when there are many inactive predictors in the model. A real-life data set is used to illustrate the suggested shrinkage and LASSO estimators.     

LASSO and shrinkage estimation in Weibull censored regression models

In this paper we address the problem of estimating a vector of regression parameters in the Weibull censored regression model. Our main objective is to provide natural adaptive estimators that significantly improve upon the classical procedures in the situation where some of the predictors may or may not be associated with the response. In the context of two competing Weibull censored regression models (full model and candidate submodel), we consider an adaptive shrinkage estimation strategy that shrinks the full model maximum likelihood estimate in the direction of the submodel maximum likelihood estimate. We develop the properties of these estimators using the notion of asymptotic distributional risk. The shrinkage estimators are shown to have higher efficiency than the classical estimators for a wide class of models. Further, we consider a LASSO type estimation strategy and compare the relative performance with the shrinkage estimators. Monte Carlo simulations reveal that when the true model is close to the candidate submodel, the shrinkage strategy performs better than the LASSO strategy when, and only when, there are many inactive predictors in the model. Shrinkage and LASSO strategies are applied to a real data set from Veteran's administration (VA) lung cancer study to illustrate the usefulness of the procedures in practice.     

Non-penalty shrinkage estimation of random effect models for longitudinal data with AR(1) errors

In this paper, we consider the non-penalty shrinkage estimation method of random effect models with autoregressive errors for longitudinal data when there are many covariates and some of them may not be active for the response variable. In observational studies, subjects are followed over equally or unequally spaced visits to determine the continuous response and whether the response is associated with the risk factors/covariates. Measurements from the same subject are usually more similar to each other and thus are correlated with each other but not with observations of other subjects. To analyse this data, we consider a linear model that contains both random effects across subjects and within-subject errors that follows autoregressive structure of order 1 (AR(1)). Considering the subject-specific random effect as a nuisance parameter, we use two competing models, one includes all the covariates and the other restricts the coefficients based on the auxiliary information. We consider the non-penalty shrinkage estimation strategy that shrinks the unrestricted estimator in the direction of the restricted estimator. We discuss the asymptotic properties of the shrinkage estimators using the notion of asymptotic biases and risks. A Monte Carlo simulation study is conducted to examine the relative performance of the shrinkage estimators with the unrestricted estimator when the shrinkage dimension exceeds two. We also numerically compare the performance of the shrinkage estimators to that of the LASSO estimator. A longitudinal CD4 cell count data set will be used to illustrate the usefulness of shrinkage and LASSO estimators.     

Oracle model selection for correlated data via residuals

This paper concerns model selection for autoregressive time series when the observations are contaminated with trend. We propose an adaptive least absolute shrinkage and selection operator (LASSO) type model selection method, in which the trend is estimated by B-splines, the detrended residuals are calculated, and then the residuals are used as if they were observations to optimize an adaptive LASSO type objective function. The oracle properties of such an adaptive LASSO model selection procedure are established; that is, the proposed method can identify the true model with probability approaching one as the sample size increases, and the asymptotic properties of estimators are not affected by the replacement of observations with detrended residuals. The intensive simulation studies of several constrained and unconstrained autoregressive models also confirm the theoretical results. The method is illustrated by two time series data sets, the annual U.S. tobacco production and annual tree ring width measurements.     

Une condition nécessaire d'admissibilité et ses conséquences sur les estimateurs à rétrécisseur de la moyenne d'un vecteur normal

Considering exponential families of distributions, we estimate parameters which are not the natural parameters. We prove that the admissible estimators of these parameters are limits of Bayes estimators and can be expressed through a given functional form. An important particular case of this model pertains to the estimation of the mean of a multidimensional normal distribution when the variance is known up to a multiplicative factor. We deduce from the main result a necessry condition for the admissibility of matricial shrinkage estimators.     

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.     

Construction of improved estimators of multinomial proportions

The improved large sample estimation theory for the probabilities of multi¬nomial distribution is developed under uncertain prior information (UPI) that the true proportion is a known quantity. Several estimators based on pretest and the Stein-type shrinkage rules are constructed. The expressions for the bias and risk of the proposed estimators are derived and compared with the maximum likelihood (ml) estimators. It is demonstrated that the shrinkage estimators are superior to the ml estimators. It is also shown that none of the preliminary test and shrinkage estimators dominate each other, though they perform y/ell relative to the ml estimators. The relative dominance picture of the estimators is presented. A simulation study is carried out to assess the performance of the estimators numerically in small samples.     

A lower bound for the risk of classes of shrinkage estimators ina general multivariate estimation problem and some deduced estimators

Given a general statistical model and an arbitrary quadratic loss, we propose a lower bound for the associated risk of a class of shrinkage estimators. With respect to the considered class of shrinkage estimators, this bound is optimal.In the particular case of the estimation of the location parameter of an ellipti-cally symmetric distribution, this bound can be used to find the relative improvement brought by a given estimator and the remaining possible improvement, using a Monte-Carlo method. We deduce from these results a new type of shrinkage estimators whose risk can be as close as one wants of the lower bound near a chosen pole and yet remain bounded. Some of them are good alternatives to the positive-part James-Stein estimator.     

Shrinkage and LASSO strategies in high-dimensional heteroscedastic models

ABSTRACT

In this paper, we consider the estimation problem of the parameter vector in the linear regression model with heteroscedastic errors. First, under heteroscedastic errors, we study the performance of shrinkage-type estimators and their performance as compared to theunrestricted and restricted least squares estimators. In order to accommodate the heteroscedastic structure, we generalize an identity which is useful in deriving the risk function. Thanks to the established identity, we prove that shrinkage estimators dominate the unrestricted estimator. Finally, we explore the performance of high-dimensional heteroscedastic regression estimator as compared to classical LASSO and shrinkage estimators.     

Estimation of the intercept parameter for linear regression model with uncertain non-sample prior information

This paper considers alternative estimators of the intercept parameter of the linear regression model with normal error when uncertain non-sample prior information about the value of the slope parameter is available. The maximum likelihood, restricted, preliminary test and shrinkage estimators are considered. Based on their quadratic biases and mean square errors the relative performances of the estimators are investigated. Both analytical and graphical comparisons are explored. None of the estimators is found to be uniformly dominating the others. However, if the non-sample prior information regarding the value of the slope is not too far from its true value, the shrinkage estimator of the intercept parameter dominates the rest of the estimators.     

Bayes estimators in the presence of a guess value

The paper deals with the problem of parameter estimation in the presence of a guess value and attempts to justify the use of Bayes estimators as an alternative to ordinary shrinkage estimators. Finally, certain Bayes estimators of exponential parameters are obtained under type II censoring, and these are compared with the corresponding MLEs and ordinary shrinkage estimators using a Monte Carlo study.     

Shrinkage Estimators of Intercept Parameters of Two Simple Regression Models with Suspected Equal Slopes

Estimators of the intercept parameter of a simple linear regression model involves the slope estimator. In this article, we consider the estimation of the intercept parameters of two linear regression models with normal errors, when it is a priori suspected that the two regression lines are parallel, but in doubt. We also introduce a coefficient of distrust as a measure of degree of lack of trust on the uncertain prior information regarding the equality of two slopes. Three different estimators of the intercept parameters are defined by using the sample data, the non sample uncertain prior information, an appropriate test statistic, and the coefficient of distrust. The relative performances of the unrestricted, shrinkage restricted and shrinkage preliminary test estimators are investigated based on the analyses of the bias and risk functions under quadratic loss. If the prior information is precise and the coefficient of distrust is small, the shrinkage preliminary test estimator overperforms the other estimators. An example based on a medical study is used to illustrate the method.     

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.     

A sufficient condition for the MSE dominance of the positive-part shrinkage estimator when each individual regression coefficient is estimated in a misspecified linear regression model

In this paper, assuming that there exist omitted explanatory variables in the specified model, we derive the exact formula for the mean squared error (MSE) of a general family of shrinkage estimators for each individual regression coefficient. It is shown analytically that when our concern is to estimate each individual regression coefficient, the positive-part shrinkage estimators have smaller MSE than the original shrinkage estimators under some conditions even when the relevant regressors are omitted. Also, by numerical evaluations, we showed the effects of our theorem for several specific cases. It is shown that the positive-part shrinkage estimators have smaller MSE than the original shrinkage estimators for wide region of parameter space even when there exist omitted variables in the specified model.     

Efficient estimation for time series following generalized linear models

In this paper, we consider James–Stein shrinkage and pretest estimation methods for time series following generalized linear models when it is conjectured that some of the regression parameters may be restricted to a subspace. Efficient estimation strategies are developed when there are many covariates in the model and some of them are not statistically significant. Statistical properties of the pretest and shrinkage estimation methods including asymptotic distributional bias and risk are developed. We investigate the relative performances of shrinkage and pretest estimators with respect to the unrestricted maximum partial likelihood estimator (MPLE). We show that the shrinkage estimators have a lower relative mean squared error as compared to the unrestricted MPLE when the number of significant covariates exceeds two. Monte Carlo simulation experiments were conducted for different combinations of inactive covariates and the performance of each estimator was evaluated in terms of its mean squared error. The practical benefits of the proposed methods are illustrated using two real data sets.