Preliminary Test Estimator Research Articles

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.

Preliminary test estimators and phi-divergence measures in generalized linear models with binary data

We consider the problem of estimation of the parameters in Generalized Linear Models (GLM) with binary data when it is suspected that the parameter vector obeys some exact linear restrictions which are linearly independent with some degree of uncertainty. Based on minimum ϕ -divergence estimation ( M ϕ E ) , we consider some estimators for the parameters of the GLM: Unrestricted M ϕ E , restricted M ϕ E , Preliminary M ϕ E , Shrinkage M ϕ E , Shrinkage preliminary M ϕ E , James–Stein M ϕ E , Positive-part of Stein-Rule M ϕ E and Modified preliminary M ϕ E . Asymptotic bias as well as risk with a quadratic loss function are studied under contiguous alternative hypotheses. Some discussion about dominance among the estimators studied is presented. Finally, a simulation study is carried out.

Stein-type improvement under stochastic constraints: Use of multivariate Student-t model in regression

Recently, many researchers have considered the use of heavy-tailed models for processing multiplicative economic and business data for validity of robustness. As a reliable justification, fat-tailed models contain outliers and extreme values reasonably well. In this paper, we assume in the multiple regression model, that the error vector follows multivariate Student-t distribution as a viable alternative to the multivariate normal and obtain unrestricted and restricted estimators under the suspicion of stochastic constraints occurring. Also the preliminary test, Stein-type shrinkage and positive-rule shrinkage estimators are derived when the variable term in the restriction is assumed to follow multivariate Student-t distribution. The conditions of superiority of the proposed estimators are provided under weighted quadratic loss function.

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.

On some pre-test and Stein-rule phi-divergence test estimators in the independence model of categorical data

For the model of independence in a two way contingency table, shrinkage estimators based on minimum φ -divergence estimators and φ -divergence statistics are considered. These estimators are based on the James–Stein-type rule and incorporate the idea of preliminary test estimator. The asymptotic bias and risk are obtained under contiguous alternative hypotheses, and on the basis of them a comparison study is carried out.

Preliminary phi-divergence test estimators for linear restrictions in a logistic regression model

The problem of estimation of the parameters in a logistic regression model is considered under multicollinearity situation when it is suspected that the parameter of the logistic regression model may be restricted to a subspace. We study the properties of the preliminary test based on the minimum ϕ -divergence estimator as well as in the ϕ -divergence test statistic. The minimum ϕ -divergence estimator is a natural extension of the maximum likelihood estimator and the ϕ -divergence test statistics is a family of the test statistics for testing the hypothesis that the regression coefficients may be restricted to a subspace.

Theory of Preliminary Test and Stein-Type Estimation with Applications by A. K. E. Saleh

Theory of Preliminary Test and Stein-Type Estimation with Applications by A. K. E. Saleh

Observed Data Based Estimator and Both Observed Data and Prior Information Based Estimator Under Asymmetric Losses

SYNOPTIC ABSTRACTIn the Bayesian approach to statistical analyses we incorporate prior information about the parameter of the model with observed data. This prior information is in the form of a prior distribution of the parameter. If the prior information is available as a constant value of the parameter rather than its prior distribution, the Bayesian approach cannot be pursued. However, there are estimation methods that incorporate such prior information with the observed data. The expectation is that the incorporation of such additional information in the estimation process would result in a better estimator than that based on the observed data alone. In some cases this may be true, but in many other cases the risk of worse consequences cannot be ruled out. This paper studies the performance of the observed data based unrestricted estimator (UE), and both observed data and prior information based preliminary test estimator (PTE) of the univariate normal mean under the linex loss function. The risk functions of both UE and PTE are derived. The moment generating function (MGF) of PTE is derived which turns out to be a component of the risk function. From the MGF the first two moments of PTE are obtained and found to be identical to those obtained using different approaches in Khan and Saleh (2001) and Zellner (1986). Under the linex loss criterion the performance of the PTE is compared with that of UE. It is revealed that if the uncertain non-sample prior information about the value of the mean is not too far from its true value, PTE outperforms UE.

Testing and Merging Information for Effect Size Estimation

A large-sample test for testing the equality of two effect sizes is presented. The null and non-null distributions of the proposed test statistic are derived. Further, the problem of estimating the effect size is considered when it is a priori suspected that two effect sizes may be close to each other. The combined data from all the samples leads to more efficient estimator of the effect size. We propose a basis for optimally combining estimation problems when there is uncertainty concerning the appropriate statistical model-estimator to use in representing the sampling process. The objective here is to produce natural adaptive estimators with some good statistical properties. In the context of two bivariate statistical models, the expressions for the asymptotic mean squared error of the proposed estimators are derived and compared with the parallel expressions for the benchmark estimators. We demonstrate that the suggested preliminary test estimator has superior asymptotic mean squared error performance relative to the benchmark and pooled estimators. A simulation study and application of the methodology to real data are presented.

Shrinkage estimation of the slope parameters of two parallel regression lines under uncertain prior information

The estimation of the slope parameter of two linear regression models with normal errors are considered, when it is suspected that the two lines are parallel. The uncertain prior information about the equality of slopes is presented by a null hypothesis and a coefficient of distrust on the null hypothesis is introduced. The unrestricted estimator (UE) based on the sample responses and shrinkage restricted estimator (SRE) as well as shrinkage preliminary test estimator (SPTE) based on the sample responses and prior information are defined. The relative performances of the UE, SRE and SPTE are investigated based on the analysis of the bias, quadratic bias and quadratic risk functions. An example based on a health study data is used to illustrate the method. The SPTE dominates other two estimators if the coefficient of distrust is not far from 0 and the difference between the population slopes is small.

Improving the Estimation of Eigenvectors Under Quadratic Loss

Improved estimation of eigenvectors of a covariance matrix is considered under uncertain prior information (UP!) regarding the parameter vector. Like statistical models underlying the statistical inferences to be made, the prior information will be susceptible to uncertainty and the practitioners may be reluctant to impose the additional information regarding parameters in the estimation process. A very large gain in precision may be achieved by judiciously exploiting the information about the parameters which in practice will be available in any realistic problem. Several estimators based on preliminary test 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 usual estimators. We demonstrate how the classical large sample theory of the conventional estimator can be extended to shrinkage and preliminary test estimators for the eigenvector of a covariance matrix. It is established that shrinkage estimators are asymptotically superior to the usual sample estimators. For illustration purposes, the method is applied to three data sets.

Quasi-Empirical Bayes Methods of Estimation in Arma (p, q) Models with Vague Prior Information on MA(q)1

This paper investigates the asymptotic properties of various estimators of autocorrelations in an ARMA{p,q) model when vague non-sampling information on the moving average part is available. In particular, we discuss the usual MLE {called unrestricted estimator), restricted MLE (based on vague information), preliminary test estimator, shrinkage estimator and the positive rule estimator of the autocorrelations. It is shown that near the prior information on the MA-parameters, the restricted , preliminary test and shrinkage estimators of the autoregressive parameters perform better than the unrestricted estimator, while their superiority changes as MA-parameters divert from the prior informations. The analysis is based on the asymptotic properties of the estimators under contiguous alternatives.

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.

On the Size Corrected Tests in Improved Estimation

In this paper we propose shrinkage preliminary test estimator (SPTE) of the coefficient vector in the multiple linear regression model based on the size corrected Wald ( W), likelihood ratio ( LR) and Lagrangian multiplier ( LM) tests. The correction factors used are those obt,ained from degrees of freedom corrections to the estimate of the error variance and those obtained from the second­order Edgeworth approximations to the exact distributions of the test statistics. The bias and weighted mean squared error (WMSE) fun ctions of the estimators are derived. With respect to WMSE, the relative efficiencies of the SPTEs relative to the maximum likelihood estimator are calculated. This study shows that the amount of conflict can be substantial when the three t ests are based on the same asymptotic chi­square critical value. The conflict among the SPTEs is due to the asymptotic tests not having the correct significance level. The Edgeworth size corrected W, LR and LM tests reduce the conflict remarkably.

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.

Preliminary Phi-divergence test estimator for multinomial probabilities

The four minimum Phi-divegence estimators of multinomial probabilities under a model constraint, namely, unrestricted estimator, restricted estimator, preliminary test estimator, shrinkage estimator, and the positive-rule shrinkage estimator, have been studied. The asymptotic distribution of these estimators are given under Pittman alternatives, together with the asymptotic bias and the asymptotic risk under the quadratic loss. Comparisons within these four estimators using their asymptotic function risks are made, and several conclusions are obtained

Estimating Fisher information in normal population with prior information

This paper is contemplated to propose a class of shrunken estimators which is further used in constructing a class of preliminary test estimators for amount of information in complete samples from normal population when some ‘apriori’ or guessed value of amount of information is available and analyses their characteristics. The proposed classes of shrunken estimators and preliminary test estimators are compared with the usual unbiased estimator and MMSE estimator. Eventually, empirical study is carried out to demonstrate the performance of some shrunken estimators and preliminary test estimators of the proposed classes over the MMSE estimator. The suggested class of estimators is found to give gratifying results. Subsequently, the usage of proposed classes of estimators in estimating the precision of sample mean has been exclusively discussed followed by an example.

Estimation of the mean vector of a multivariate normal distribution: subspace hypothesis

This paper considers the estimation of the mean vector θ of a p-variate normal distribution with unknown covariance matrix Σ when it is suspected that for a p × r known matrix B the hypothesis θ = B η , η ∈ R r may hold. We consider empirical Bayes estimators which includes (i) the unrestricted unbiased (UE) estimator, namely, the sample mean vector (ii) the restricted estimator (RE) which is obtained when the hypothesis θ = B η holds (iii) the preliminary test estimator (PTE), (iv) the James–Stein estimator (JSE), and (v) the positive-rule Stein estimator (PRSE). The biases and the risks under the squared loss function are evaluated for all the five estimators and compared. The numerical computations show that PRSE is the best among all the five estimators even when the hypothesis θ = B η is true.

Improved estimation of regression parameters in measurement error models

The problem of simultaneous estimation of the regression parameters in a multiple regression model with measurement errors is considered when it is suspected that the regression parameter vector may be the null-vector with some degree of uncertainty. In this regard, we propose two sets of four estimators, namely, (i) the unrestricted estimator, (ii) the preliminary test estimator, (iii) the Stein-type estimator and (iv) the postive-rule Stein-type estimator. In an asymptotic setup, properties of these estimators are studied based on asymptotic distributional bias, MSE matrices, and risks under a quadratic loss function. In addition to the asymptotic dominance of the Stein-type estimators, the paper contains discussion of dominating confidence sets based on the Stein-type estimation. Asymptotic analysis is considered based on a sequence of local alternatives to obtain the desired results.

Comparisons of improved risk estimators of the multivariate mean vector

The estimation of the mean vector of a multivariate normal distribution, under the uncertain prior information (UPI) that component means are equal but unknown, is considered. The positive part of Stein-Rule (PSE) and improved preliminary test (IPE) estimators are proposed. It is demonstrated analytically as well as computationally that the positive part of Stein-Rule estimator is superior to the usual Stein-Rule estimator (SE). Furthermore, it is shown that the proposed improved pretest estimator dominates the traditional preliminary test estimator (PE) regardless the correctness of the nonsample information. The relative dominance of the proposed estimators are presented analytically as well as graphically. Percentage improvements of the proposed estimators over the unrestricted estimator (UE) are computed. It is shown that for p ⩾ 3 , SE or PSE is the best to use while for p ⩽ 2 , UE is preferable.