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.     

Shrinkage and pretest estimators for longitudinal data analysis under partially linear models

In this paper, we develop marginal analysis methods for longitudinal data under partially linear models. We employ the pretest and shrinkage estimation procedures to estimate the mean response parameters as well as the association parameters, which may be subject to certain restrictions. We provide the analytic expressions for the asymptotic biases and risks of the proposed estimators, and investigate their relative performance to the unrestricted semiparametric least-squares estimator (USLSE). We show that if the dimension of association parameters exceeds two, the risk of the shrinkage estimators is strictly less than that of the USLSE in most of the parameter space. On the other hand, the risk of the pretest estimator depends on the validity of the restrictions of association parameters. A simulation study is conducted to evaluate the performance of the proposed estimators relative to that of the USLSE. A real data example is applied to illustrate the practical usefulness of the proposed estimation procedures.     

Model selection and post estimation based on a pretest for logistic regression models

ABSTRACT

This article addresses the problem of parameter estimation of the logistic regression model under subspace information via linear shrinkage, pretest, and shrinkage pretest estimators along with the traditional unrestricted maximum likelihood estimator and restricted estimator. We developed an asymptotic theory for the linear shrinkage and pretest estimators and compared their relative performance using the notion of asymptotic distributional bias and asymptotic quadratic risk. The analytical results demonstrated that the proposed estimation strategies outperformed the classical estimation strategies in a meaningful parameter space. Detailed Monte-Carlo simulation studies were conducted for different combinations and the performance of each estimation method was evaluated in terms of simulated relative efficiency. The results of the simulation study were in strong agreement with the asymptotic analytical findings. Two real-data examples are also given to appraise the performance of the estimators.     

Merging data from multiple sources: pretest and shrinkage perspectives

In this article, we have developed asymptotic theory for the simultaneous estimation of the k means of arbitrary populations under the common mean hypothesis and further assuming that corresponding population variances are unknown and unequal. The unrestricted estimator, the Graybill-Deal-type restricted estimator, the preliminary test, and the Stein-type shrinkage estimators are suggested. A large sample test statistic is also proposed as a pretest for testing the common mean hypothesis. Under the sequence of local alternatives and squared error loss, we have compared the asymptotic properties of the estimators by means of asymptotic distributional quadratic bias and risk. Comprehensive Monte-Carlo simulation experiments were conducted to study the relative risk performance of the estimators with reference to the unrestricted estimator in finite samples. Two real-data examples are also furnished to illustrate the application of the suggested estimation strategies.     

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.     

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.     

Improved pretest nonparametric estimation in a multivariate regression model

Shrinkage pretest nonparametric estimation of the location parameter vector in a multivariate regression model is considered when nonsample information (NSI) about the regression parameters is available. By using the quadratic risk criterion, the dominance of the pretest estimators over the usual estimators has been investigated. We demonstrate analytically and computationally that the proposed improved pretest estimator establishes a wider dominance range for the parameter under consideration than that of the usual pretest estimator in which it is superior over the unrestricted estimator.     

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.     

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, 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.     

Shrinkage and Penalty Estimators of a Poisson Regression Model

In this paper we propose Stein‐type shrinkage estimators for the parameter vector of a Poisson regression model when it is suspected that some of the parameters may be restricted to a subspace. 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. Furthermore, we consider three different penalty estimators: the LASSO, adaptive LASSO, and SCAD estimators and compare their relative performance with that of the shrinkage estimators. Monte Carlo simulation studies reveal that the shrinkage strategy compares favorably to the use of penalty estimators, in terms of relative mean squared error, when the number of inactive predictors in the model is moderate to large. The shrinkage and penalty strategies are applied to two real data sets to illustrate the usefulness of the procedures in practice.     

Meta-analysis,pretest, and shrinkage estimation of kurtosis parameters

An asymptotic theory for the improved estimation of kurtosis parameter vector is developed for multi-sample case using uncertain prior information (UPI) that several kurtosis parameters are the same. Meta-analysis is performed to obtain pooled estimator, as it is a statistical methodology for pooling quantitative evidence. Pooled estimator is a good choice when assumption of homogeneity holds but it becomes inconsistent as assumption violates, therefore pretest and Stein-type shrinkage estimators are proposed as they combine sample and nonsample information in a superior way. Asymptotic properties of suggested estimators are discussed and their risk comparisons are also mentioned.     

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.     

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.     

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.     

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 Estimation of the Memory Parameter in Stationary Gaussian Processes

The correct and efficient estimation of memory parameters in a stationary Gaussian processes is an important issue, since otherwise, forecasts based on the resulting time series would be misleading. On the other hand, if the memory parameters are suspected to fall in a smaller subspace through some hypothesis restrictions, it becomes a hard decision whether to use estimators based on the restricted spaces or to use unrestricted estimators over the full parameter space. In this article, we propose James-Stein-type estimators of the memory parameters of a stationary Gaussian times series process, which can efficiently incorporate the hypothetical restrictions. We show theoretically that the proposed estimators are more efficient than the usual unrestricted maximum likelihood estimators over the entire parameter space.     

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.     

On the ridge regression estimator with sub-space restriction

In the linear regression model with elliptical errors, a shrinkage ridge estimator is proposed. In this regard, the restricted ridge regression estimator under sub-space restriction is improved by incorporating a general function which satisfies Taylor’s series expansion. Approximate quadratic risk function of the proposed shrinkage ridge estimator is evaluated in the elliptical regression model. A Monte Carlo simulation study and analysis based on a real data example are considered for performance analysis. It is evident from the numerical results that the shrinkage ridge estimator performs better than both unrestricted and restricted estimators in the multivariate t-regression model, for some specific cases.     

Relative performance of stein-rule and preliminary test estimators in linear models least squares theory

In the classical (univariare) linear model, bearing the plausibility of a subset of the regression parameters being close to a pivot, shrinkage least squares estimation of the complementary subset is considered. Based on the usual James-Stein rule, shrinkage least squares estimators are constructed, and under an asymptotic setup (allowing the shrinkage parameters to be 'close to ' the pivot), the relative performance of such estimators and the prcliminary test estimators is studied. In this context, the normality of the errors is also avoided under the same asymptotic setup. None of the shrinkage and preliminary test estimators may dominate the other (in the light of the asymptotic distributional risk criterion, as has been developed here), though each of them fares well relative to the classical least squeres estimator. The chice of the shrinkage factor is also examined properly.