Estimation of Slope for Linear Regression Model with Uncertain Prior Information and Student-t Error

This article considers estimation of the slope parameter of the linear regression model with Student-t errors in the presence of uncertain prior information on the value of the unknown slope. Incorporating uncertain non sample prior information with the sample data the unrestricted, restricted, preliminary test, and shrinkage estimators are defined. The performances of the estimators are compared based on the criteria of unbiasedness and mean squared errors. Both analytical and graphical methods are explored. Although none of the estimators is uniformly superior to the others, if the non sample information is close to its true value, the shrinkage estimator over performs the rest of the 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.     

Pooling means under uncertain prior information with application to discrete distributions

The problem of pooling means is considered based on two samples in presence of the uncertain prior information that these samples are taken from possibly identical populations. Two discrete models, Poisson and binomial are considered in particular. Three estimators, i.e. the unrestricted estimator, shrinkage restricted estimator and estimators based on preliminary test are proposed. Their asymptotic mean squared errors are derived and compared. It is demonstrated via asymptotic results that the range of the parameter space in which shrinkage preliminary test estimator dominates the unrestricted estimator is wider than that of the usual preliminary test estimator. A Monte Carlo study for Poisson model is presented to compare the performance of the estimators for small samples.     

On the comparison of the pre-test and shrinkage estimators for the univariate normal mean

The estimation of the mean of an univariate normal population with unknown variance is considered when uncertain non-sample prior information is available. Alternative estimators are defined to incorporate both the sample as well as the non-sample information in the estimation process. Some of the important statistical properties of the restricted, preliminary test, and shrinkage estimators are investigated. The performances of the estimators are compared based on the criteria of unbiasedness and mean square error in order to search for a ‘best’ estimator. Both analytical and graphical methods are explored. There is no superior estimator that uniformly dominates the others. However, if the non-sample information regarding the value of the mean is close to its true value, the shrinkage estimator over performs the rest of the estimators. Received: June 19, 1999; revised version: March 23, 2000     

On the estimation of reliabilty function in a Weibull lifetime distribution

The estimation of the reliability function of the Weibull lifetime model is considered in the presence of uncertain prior information (not in the form of prior distribution) on the parameter of interest. This information is assumed to be available in some sort of a realistic conjecture. In this article, we focus on how to combine sample and non-sample information together in order to achieve improved estimation performance. Three classes of point estimatiors, namely, the unrestricted estimator, the shrinkage estimator and shrinkage preliminary test estimator (SPTE) are proposed. Their asymptotic biases and mean-squared errors are derived and compared. The relative dominance picture of the estimators is presented. Interestingly, the proposed SPTE dominates the unrestricted estimator in a range that is wider than that of the usual preliminary test estimator. A small-scale simulation experiment is used to examine the small sample properties of the proposed estimators. Our simulation investigations have provided strong evidence that corroborates with asymptotic theory. The suggested estimation methods are applied to a published data set to illustrate the performance of the estimators in a real-life situation.     

An Improved Estimation in Regression Parameter Matrix in Multivariate Regression Model

We consider shrinkage and preliminary test estimation strategies for the matrix of regression parameters in multivariate multiple regression model in the presence of a natural linear constraint. We suggest a shrinkage and preliminary test estimation strategies for the parameter matrix. The goal of this article is to critically examine the relative performances of these estimators in the direction of the subspace and candidate subspace restricted type estimators. Our analytical and numerical results show that the proposed shrinkage and preliminary test estimators perform better than the benchmark estimator under candidate subspace and beyond. The methods are also applied on a real data set for illustrative purposes.     

Estimating and testing effect size from an arbitrary population

In this article, we develop inference tools for an effect size parameter in a paired experiment. A class of estimators is defined that includes natural, shrinkage and shrinkage preliminary test estimators. The shrinkage and preliminary test methods incorporate uncertain prior information on the parameter. This information may be available in the form of a realistic guess on the basis of the experimenter’s knowledge and experience, which can be incorporated into the estimation process to increase the efficiency of the estimator. Asymptotic properties of the proposed estimators are investigated both analytically and computationally. A simulation study is also conducted to assess the performance of the estimators for moderate and large samples. For illustration purposes, the method is applied to a data set.     

Improved estimation of the mean vector for student-t model

Improved James-Stein type estimation of the mean vector μ of a multovaroate Student-t population of dimension p with ν degrees of freedom is considered. In addition to the sample data, uncertain prior information on the value of the mean vector, in the form of a null hypothesis, is used for the estiamtion. The usual maximum liklihood estimator((mle) of μ is obtained and a test statistic for testing H₀:μ=μ₀ is derived. Based on the mle of μ and the tes statistic the preliminary test estimator (PTE), Stein-type shrinkage estimator (SE) and positive-rule shrinkage esiimator (PRSE) are defined. The bias and the quadratic risk of the estimators are evaiuated. The relative performances of the estimators are mvestigated by analyzing the risks under different condltlons It is observed that the FRSE dommates over he other three estimators, regardless of the vaiidity of the null hypothesis and the value ν.     

Combining poisson means

The problem of estimating the Poisson mean is considered based on the two samples in the presence of uncertain prior information (not in the form of distribution) that two independent random samples taken from two possibly identical Poisson populations. The parameter of interest is λ₁ from population I. Three estimators, i.e. the unrestricted estimator, restricted estimator and preliminary test estimator are proposed. Their asymptotic mean squared errors are derived and compared; parameter regions have been found for which restricted and preliminary test estimators are always asymptotically more efficient than the classical estimator. The relative dominance picture of the estimators is presented. Maximum and minimum asymptotic efficiencies of the estimators relative to the classical estimator are tabulated. A max-min rule for the size of the preliminary test is also discussed. A Monte Carlo study is presented to compare the performance of the estimator with that of Kale and Bancroft (1967).     

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.     

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.     

Comparison of preliminary test estimators based on generalized order statistics from proportional hazard family using Pitman measure of closeness

In this article, based on generalized order statistics from a family of proportional hazard rate model, we use a statistical test to generate a class of preliminary test estimators and shrinkage preliminary test estimators for the proportionality parameter. These estimators are compared under Pitman measure of closeness (PMC) as well as MSE criteria. Although the PMC suffers from non transitivity, in the first class of estimators, it has the transitivity property and we obtain the Pitman-closest estimator. Analytical and graphical methods are used to show the range of parameter in which preliminary test and shrinkage preliminary test estimators perform better than their competitor estimators. Results reveal that when the prior information is not too far from its real value, the proposed estimators are superior based on both mentioned criteria.     

Empirical bayes shrinkage estimation of reliability in the exponential distribution

In this paper we propose two empirical Bayes shrinkage estimators for the reliability of the exponential distribution and study their properties. Under the uniform prior distribution and the inverted gamma prior distribution these estimators are developed and compared with a preliminary test estimator and with a shrinkage testimator in terms of mean squared error. The proposed empirical Bayes shrinkage estimator under the inverted gamma prior distribution is shown to be preferable to the preliminary test estimator and the shrinkage testimator when the prior value of mean life is clsoe to the true mean life.     

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.     

Two kinds of restricted modified estimators in linear regression model

In this paper, we introduce two kinds of new restricted estimators called restricted modified Liu estimator and restricted modified ridge estimator based on prior information for the vector of parameters in a linear regression model with linear restrictions. Furthermore, the performance of the proposed estimators in mean squares error matrix sense is derived and compared. Finally, a numerical example and a Monte Carlo simulation are given to illustrate some of the theoretical results.     

On the estimation of the mean vector of a multivariate normal distribution under symmetry

The problem of estimation of the mean vector of a multivariate normal distribution with unknown covariance matrix, under uncertain prior information (UPI) that the component mean vectors are equal, is considered. The shrinkage preliminary test maximum likelihood estimator (SPTMLE) for the parameter vector is proposed. The risk and covariance matrix of the proposed estimato are derived and parameter range in which SPTMLE dominates the usual preliminary test maximum likelihood estimator (PTMLE) is investigated. It is shown that the proposed estimator provides a wider range than the usual premilinary test estimator in which it dominates the classical estimator. Further, the SPTMLE has more appropriate size for the preliminary test than the PTMLE.     

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.     

Pooling multivariate data

In case it is doubtful whether two sets of data have the same mean vector, four estimation strategies have been developed for the target mean vector. In this situation, the estimates based on a preliminary test as well as on Stein-rule are advantageous. Two measures of relative efficiency are considered; one based on thequadratic loss function, and the other on the determinant of the mean square error matrix. A max-min rule for the size of the preliminary test of significance is presented. It is demonstrated that the shrinkage estimator dominates the classical estimator, whereas none of the shrinkage estimator and the preliminary test estimator dominate each other. The range in the parameter space where preliminary test estimator dominates shrinkage is investigated analytically and computationally. It is found that the shrinkage estimator outperform the preliminary test estimator except in a region around the null hypothesis. Moreover, for large values of a, the level of statistical significance, shrinkage estimator dominates the preliminary test estimator uniformly. The relative dominance of the estimators is presented.     

On a Comparison of Several Competing Estimates of a Univariate Normal Mean by the Multiple Criteria Decision-Making Method

In this article we consider the problem of estimation of the mean of a univariate normal population with an unknown variance when uncertain nonsample prior information about the mean is available. We compare four estimators of the mean, including pretest and shrinkage estimators. The performances of the estimators are compared based on the multiple criteria decision making (MCDM) procedure in order to find the best estimator.     

Using improved estimation strategies to combat multicollinearity

In this approach, some generalized ridge estimators are defined based on shrinkage foundation. Completely under the suspicion that some sub-space restrictions may occur, we present the estimators of the regression coefficients combining the idea of preliminary test estimator and Stein-rule estimator with the ridge regression methodology for normal models. Their exact risk expressions in addition to biases are derived and the regions of optimality of the estimators are exactly determined along with some numerical analysis. In this regard, the ridge parameter is determined in different disciplines.