Shrinkage testimators for the shape parameter of weibull distributin under type ii censoring

In this paper we propose some shrinkage testimators for the shape parameter of the Weibull distribution when censored samples are available and study their properties. Comparison of the testimators with Singh and Bhatkulikar (1978) and with the usual estimator, interms of mean squared error are made. It is shown that the proposed testimators     

Pooling reliability functions

In this article large sample pooling procedures for reliability functions of an exponential life testing model is considered. Asymptotic properties of shrinkage estimation procedure subsequent to preliminary tests are developed. It is shown that the proposed estimator possesses substantially snakker asymptotic mean squared error than the usual estimator in a region of the lparameter space. Relative efficiencies of the purposed estimators to the usual estimators are obtained and recommendations of the level of the preliminary tests are provided. Relative dominance picture of the estimators is presented. It is shown that the proposed estimator provides a wider dominance range over usual estimator than the usual preliminary test estimator. More importantly, the size of the preliminary test is meaningful. Simulation studies is also carried out to appraise the performance of the estimators when samples are small.     

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.     

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.     

Shrinkage Testimators for the Shape Parameter of Pareto Distribution Using LINEX Loss Function

In this article, shrinkage testimators for the shape parameter of a Pareto distribution are considered, when its prior guess value is available. The choices of shrinkage factor are also suggested. The proposed testimators are compared with the minimum risk estimator among the class of unbiased estimators with the LINEX loss function.     

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.     

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.     

MSE Performance and Minimax Regret Significance Points for a HPT Estimator when each Individual Regression Coefficient is Estimated

In this article, we consider a heterogeneous preliminary test (HPT) estimator whose components are the OLS and feasible ridge regression (FRR) estimators, and derive the exact formulae for the moments of the HPT estimator using mathematical method. Since we cannot examine the MSE of the HPT estimator analytically, we execute the numerical evaluation to investigate the MSE performance of the HPT estimator, and compare the MSE performance of the HPT estimator with those of the FRR estimator and the usual OLS estimator. Furthermore, using the minimax regret criterion proposed by Sawa and Hiromatsu (1973 Sawa , T. , Hiromatsu , T. ( 1973 ). Minimax regret significance points for a preliminary test in regression analysis . Econometrica 41 : 1093 – 1101 .[Crossref], [Web of Science ®] , [Google Scholar]), we derive the optimal critical points of the preliminary F test. Our results show that the optimal significance points are greater than 19% and the optimal signicance points decrease as the denominator degrees of freedom of the preliminary F test statistic increases.     

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.     

Sometimes pool t estimation – via shrinkage

The performance of an estimator can be improved by incorporating some additional information(s) available besides the sample information. If two censored samples are available from the same exponential distribution, it is advantageous to pool the two samples for estimating the mean life. Further, incorporating guess information facilitates accuracy borrowing by shrinkage to a guess point or interval. Both the views have been taken into consideration in the present study. The present paper proposes an estimator for the mean life time of a two parameter exponential distribution, using conditional and/or guess information on it, when the two guarantees are equal but unknown. The bias, mean square error and relative efficiency of the proposed estimator have been studied. Some theoretical results have been derived. It is observed that the proposed testimator dominates the conventional estimator in certain range of life ratio, guess life ratio and shrinkage factor. Further, it is claimed that it always fares better than the preliminary test estimator for mean life proposed by Gupta and Singh (Microelectron. Reliab., 1985, 25, 881–887).     

OPTIMAL CRITICAL VALUES FOR A PRELIMINARY TEST IN POOLING MEANS

Given two random samples from normal populations with the same known variance, the experimenter wishes to estimate the mean of the fist population. Whether to pool the two samples or not is made to depend on the result of a preliminary normal test. The bias and mean square error of the pre-test estimator are presented. Based on two different criteria, a minimax regret and a minimum expected regret ones, the optimal critical values for the pre-test are given. The minimax regret value is unique (about 1.37), while the alternative values vary depending on the mean and precision included in the prior distribution.     

Preliminary Test Regression Estimator When two Prior Values of Population Mean (μx) of an Auxiliary Variable (x) are Available

A regression estimator using two prior values of population mean (μ_x) of an auxiliary variable (x) is proposed after a preliminary test of closeness of these prior values to the true valueμ_x. The proposed preliminary test regression estimator has been found to be more efficient in general than the usual regression estimator when prior values are used in place of μ_xwithout preliminary test of significance. The efficiency of the proposed estimator over the usual regression estimator has also been computed for different values of Δ₀, Δ₁, n, and ρ, which showed considerable gain in precision.     

On estimating the common mean in two normal distributions after a preliminary test for equality of variances

Given two random samples of equal size from two normal distributions with common mean but possibly different variances, we examine the sampling performance of the pre-test estimator for the common mean after a preliminary test for equality of variances. It is shown that when the alternative in the pretest is one-sided, the Graybill-Deal estimator is dominated by the pre-test estimator if the critical value is chosen appropriately. It is also shown that all estimators, the grand mean, the Graybill-Deal estimator and the pre-test estimator, are admissible when the alternative in the pre-test is two-sided. The optimal critical values in the two-sided pre-test are sought based on the minimax regret and the minimum average risk criteria, and it is shown that the Graybill-Deal estimator is most preferable under the minimum average risk criterion when the alternative in the pre-test is two-sided.     

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.     

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     

Some improved estimators for a measures of dispersion of an inverse gaussian distribution

In this paper some improved estimators for the measure of dispersion of an inverse Gaussian distribution have been obtained. If some guessed value of λ is available in the form of a point esitmate λ₀ the shrikage technique has been applied and an estimator has been proposed which has smaller mean squared error than the usual estimator. Since the shrinkage estimator has better performance if the guessed value is in the vicinity of the true value, a shrinkage testimator has also been proposed and compared with the usual estimator.     

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.     

Bayesian shrinkage estimators for a measure of dispersion of an inverse gaussian distribution

The Bayesian shrinkage estimation for a measure of dispersion with known mean is studied for the inverse Gaussian distribution. An optimum choice of the shrinkage factor and the properties of the proposed Bayesian shrinkage estimators are being studied. It is shown that these estimators have smaller risk than the usual estimator of the reciprocal measure of dispersion.     

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.     

Preliminary test estimators in double sampling with two auxiliary variables

An attempt has been mads to suggest some estimators for population mean in double sampling with two auxiliary variables., alternative to the usual regression estimator. When the experimenter has partial Information about the mean of the auxiliary variable or variables, preliminary test estimators can be used. The bias, mean square error, relative efficiency and optimum allocation of sample sizes are obtained for the suggested estimators.