Preliminary Test Estimator Research Articles

Generalized preliminary test stochastic restricted estimator in the linear regression model

ABSTRACTIn this paper, we propose three generalized estimators, namely, generalized unrestricted estimator (GURE), generalized stochastic restricted estimator (GSRE), and generalized preliminary test stochastic restricted estimator (GPTSRE). The GURE can be used to represent the ridge estimator, almost unbiased ridge estimator (AURE), Liu estimator, and almost unbiased Liu estimator. When stochastic restrictions are available in addition to the sample information, the GSRE can be used to represent stochastic mixed ridge estimator, stochastic restricted Liu estimator, stochastic restricted almost unbiased ridge estimator, and stochastic restricted almost unbiased Liu estimator. The GPTSRE can be used to represent the preliminary test estimators based on mixed estimator. Using the GPTSRE, the properties of three other preliminary test estimators, namely preliminary test stochastic mixed ridge estimator, preliminary test stochastic restricted almost unbiased Liu estimator, and preliminary test stochastic restricted almost unbiased ridge estimator can also be discussed. The mean square error matrix criterion is used to obtain the superiority conditions to compare the estimators based on GPTSRE with some biased estimators for the two cases for which the stochastic restrictions are correct, and are not correct. Finally, a numerical example and a Monte Carlo simulation study are done to illustrate the theoretical findings of the proposed estimators.

The generalized preliminary test estimator when different sets of stochastic restrictions are available

Arumairajan and Wijekoon (Commun Stat—Theor Methods, in press, 2014) proposed a generalized preliminary test stochastic restricted estimator (GPTSRE) to represent the preliminary test estimators when stochastic restrictions are available in addition to sample model. The aim of this paper is to define the GPTSRE when two different sets of competing stochastic restrictions are available. Moreover the conditions for superiority of GPTSRE based on one stochastic restriction over the other are derived with respect to mean square error (MSE) matrix criterion. Furthermore the estimator GPTSRE is theoretically compared with almost unbiased ridge estimator and almost unbiased Liu estimator in the MSE matrix sense. Finally a Monte Carlo simulation study and numerical example are done to illustrate the theoretical findings.

Preliminary Test Estimation and Shrinkage Preliminary Test Estimation in Normal and Negative Exponential Distribution Using Linex Loss Function

In statistical estimation procedure prior information regarding the unknown value of parameter is utilizing and it may result in a decrease of sampling variability of the estimator or it may save sample size which is desirable in many estimation procedures. The commonly used approaches in statistical inference which utilize prior information are Bayesian approach, preliminary test procedure and shrinkage estimation. The paper proposes preliminary test estimator and shrinkage preliminary test estimator for the variance in normal distribution and studies its property under Linex loss function. The paper also proposes and suggests shrinkage preliminary test estimator for the variance in negative exponential distribution and studies its property under Linex loss function.

More on the Preliminary Test Stochastic Restricted Liu Estimator in Linear Regression Model

In this paper we compare recently developed preliminary test estimator called Preliminary Test Stochastic Restricted Liu Estimator (PTSRLE) with Ordinary Least Square Estimator (OLSE) and Mixed Estimator (ME) in the Mean Square Error Matrix (MSEM) sense for the two cases in which the stochastic restrictions are correct and not correct. Finally a numerical example and a Monte Carlo simulation study are done to illustrate the theoretical findings.

On the Construction of Preliminary Test Estimator Based on Record Values for the Burr XII Model

In this paper some different sorts of estimators are proposed based on record breaking observations in the Burr type XII model. We define Bayes as well as empirical Bayes preliminary test estimators in the same fashion as in the ordinary preliminary test estimator using relevant combinations of uniformly minimum variance unbiased (UMVU) and Bayes estimators. Exact and asymptotic bias and mean square error (MSE) expressions for the proposed estimators are derived under two different conditions of knowing the shape parameters. We compare the MSEs and obtain the confidence interval for the parameter of interest in which the preliminary test type estimators outperform the UMVU, Bayes and empirical Bayes estimators. An application of the ordinary preliminary test estimator is also considered. We conclude this approach by a useful discussion for practical purposes and a summary.

Comparisons of variance estimators in a misspecified linear model with elliptically contoured errors

In a misspecified linear regression model with elliptically contoured errors, the exact biases and risks of least squares, restricted least squares, preliminary test and Stein-type estimators of the error variance are derived. Also, we compare the risk performances of the underlying estimators and give the dominance pictures for them. It is shown that the risk of preliminary test estimator attains the smallest value if the critical value equals one. Moreover, we give a bootstrap procedure for estimating the risks of proposed estimators in order to overcome the difficulty of computing the exact risks when the sample size becomes larger.

Preliminary Test Estimators Induced by Three Large Sample Tests for Stochastic Constraints in a Regression Model with Multivariate Student-t Error

In this article, we consider the preliminary test approach to the estimation of the regression parameter in a multiple regression model with multivariate Student-t distribution. The preliminary test estimators (PTE) based on the Wald (W), Likelihood Ratio (LR), and Lagrangian Multiplier (LM) tests are given under the suspicion of stochastic constraints occurring. The bias, mean square error matrix (MSEM), and weighted mean square error (WMSE) of the proposed estimators are derived and compared. The conditions of superiority of the proposed estimators are obtained. Finally, we conclude that the optimum choice of the level of significance becomes the traditional choice by using the W test.

Preliminary test and Bayes Estimation of a Location Parameter Under Blinex Loss

In this article, the preliminary test estimator is considered under the BLINEX loss function. The problem under consideration is the estimation of the location parameter from a normal distribution. The risk under the null hypothesis for the preliminary test estimator, the exact risk function for restricted maximum likelihood and approximated risk function for the unrestricted maximum likelihood estimator, are derived under BLINEX loss and the different risk structures are compared to one another both analytically and computationally. As a motivation on the use of BLINEX rather than LINEX, the risk for the preliminary test estimator under BLINEX loss is compared to the risk of the preliminary test estimator under LINEX loss and it is shown that the LINEX expected loss is higher than BLINEX expected loss. Furthermore, two feasible Bayes estimators are derived under BLINEX loss, and a feasible Bayes preliminary test estimator is defined and compared to the classical preliminary test estimator.

Improved estimators of the distribution function based on lower record values

In this paper, we define different types of estimators for the distribution function, namely preliminary test (PT), shrinkage PT (SPT), Stein type (S), and Thompson shrinkage (TS) estimators based on lower record observations and their inter record times. Their asymptotic distributional bias and mean square error are explicitly derived. The superiority conditions of the new proposed estimators over the existing estimator of distribution function are also obtained. It is shown that about the neighborhood of the null hypothesis \(F_0\), the PTE is superior to the SE in the sense of having higher asymptotic relative efficiency. Further for the reasonable values of \(\alpha \) the newly proposed SPT estimator uniformly dominates the non-parametric maximum like likelihood estimators in the literatures. A table is also given to be more specifics along the exhibited theoretical results for practical sake. Some graphical representations are given as proofs of our assertions. A simulation study is also carried out for some life time distribution, to examine the accuracy of the proposed estimators with a limited sample size. The results show that combination of the parametric and nonparametric estimators will give more efficient estimators. This study is finally concluded by applying the theoretic results to a real data set.

Improved confidence intervals for the scale parameter of Burr XII model based on record values

In this paper some different sorts of confidence intervals are considered for the scale parameter of the Burr type XII distribution based on the upper record values. In this regard, the coverage probability is adopted as a measure of improvement when the endpoints are the same for all types of confidence intervals. Proposed confidence intervals are based on the preliminary test estimator, Thompson shrinkage estimator and Bayes estimator with conjugate prior information. It is nicely demonstrated that the confidence intervals based on the above methodologies are superior to the equal tail confidence interval on specific intervals. Subsequently, to construct a uniformly dominant confidence interval, the result of Kubokawa (Ann Stat 22(1):290---299, 1994) is extended for dependent observations by making use of the information that exists in a covariate record value.

Improved preliminary test and Stein-rule Liu estimators for the ill-conditioned elliptical linear regression model

Recently, Liu (1993) estimator draws an important attention to estimate the regression parameters for an ill-conditioned linear regression model when the vector of errors is distributed according to the law belonging to the class of elliptically contoured distributions (ECDs). This paper proposed some improved Liu type estimators, namely, the unrestricted Liu estimator (ULE), restricted Liu estimator (RLE), preliminary test Liu estimator (PTLE), shrinkage Liu estimator (SLE) and positive rule Liu estimator (PRLE) for estimating the regression parameters @b. The performance of the proposed estimators is compared based on the quadratic bias and risk functions under both null and alternative hypotheses, which specify certain restrictions on the regression parameters. The conditions of superiority of the proposed estimators for parameter d and non-centrality parameter @D are given.

Regression model with elliptically contoured errors

For the regression model y=X β+ε where the errors follow the elliptically contoured distribution, we consider the least squares, restricted least squares, preliminary test, Stein-type shrinkage and positive-rule shrinkage estimators for the regression parameters, β. We compare the quadratic risks of the estimators to determine the relative dominance properties of the five estimators.

Shrinkage Ridge Estimators in Linear Regression

The problem of estimation of the regression coefficients in a multiple regression model (MRM) is considered under multicollinearity situation. Further it is suspected that the regression coefficients may be restricted to a subspace. In this approach, 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 elliptically contoured distributions (ECD). Accordingly, their exact risk expressions in addition to biases are derived. The estimators under study are compared 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.

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), 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.

Singular Ridge Regression With Stochastic Constraints

This article considers the estimation of the restricted ridge regression parameter in singular models. The problem is commenced with considering elliptically contoured equality constrained and then followed by proposing the preliminary test estimator. Along with proposing some important properties of this estimator, a real example satisfying the elliptical assumption is also given to bring the problem into a noticeable issue.

Robust Analysis of High Throughput Screening (HTS) Assay Data

Quantitative high throughput screening (qHTS) assays use cells or tissues to screen thousands of compounds in a short period of time. Data generated from qHTS assays are then evaluated using nonlinear regression models, such as the Hill model, and decisions regarding toxicity are made using the estimates of the parameters of the model. For any given compound, the variability in the observed response may either be constant across dose groups (homoscedasticity) or vary with dose (heteroscedasticity). Since thousands of compounds are simultaneously evaluated in a qHTS assay, it is not practically feasible for an investigator to perform residual analysis to determine the variance structure before performing statistical inferences on each compound. Since it is well known that the variance structure plays an important role in the analysis of linear and nonlinear regression models, it is therefore important to have practically useful and easy to interpret methodology that is robust to the variance structure. Furthermore, given the number of chemicals that are investigated in the qHTS assay, outliers and influential observations are not uncommon. In this article, we describe preliminary test estimation (PTE)-based methodology that is robust to the variance structure as well as any potential outliers and influential observations. Performance of the proposed methodology is evaluated in terms of false discovery rate (FDR) and power using a simulation study mimicking a real qHTS data. Of the two methods currently in use, our simulations studies suggest that one is extremely conservative with very small power in comparison to the proposed PTE-based method whereas the other method is very liberal. In contrast, the proposed PTE-based methodology achieves a better control of FDR while maintaining good power. The proposed methodology is illustrated using a dataset obtained from the National Toxicology Program (NTP). Additional information, simulation results, data, and computer code are available online as supplementary materials.

Preliminary test and estimation in some multifactor diffusion processes

We consider the general inference problem of the drift parameters matrices of m independent multivariate diffusion processes. In particular, we develop preliminary test and Stein-type estimation theory for m-parameters matrices which are suspected to satisfy some constraints. By using asymptotic distributional quadratic risk criterion, we analyze the relative performance of the pretest and shrinkage estimators. Further, we generalize some inference problems recently studied for a single multifactor Vasicek model. Finally, we carry out some simulation studies whose results corroborate our theoretical findings for observation periods of small and moderate lengths of time.

Improvement of the Preliminary Test Estimator When Stochastic Restrictions are Available in Linear Regression Model

Ridge type estimators are used to estimate regression parameters in a multiple linear regression model when multicolinearity exists among predictor variables. When different estimators are available, preliminary test estimation procedure is adopted to select a suitable estimator. In this paper, two ridge estimators, the Stochastic Restricted Liu Estimator and Liu Estimator are combined to define a new preliminary test estimator, namely the Preliminary Test Stochastic Restricted Liu Estimator (PTSRLE). The stochastic properties of the proposed estimator are derived, and the performance of PTSRLE is compared with SRLE in the sense of mean square error matrix (MSEM) and scalar mean square error (SMSE) for the two cases in which the stochastic restrictions are correct and not correct. Moreover the SMSE of PTSRLE based on Wald (WA), Likelihood Ratio (LR) and Lagrangian Multiplier (LM) tests are derived, and the performance of PTSRLE is compared using WA, LR and LM tests as a function of the shrinkage parameter d with respect to the SMSE. Finally a numerical example is given to illustrate some of the theoretical findings.

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 and Phi-divergence Measures in Pooling Binomial Data

Two independent random samples are drawn from two Binomial populations with parameters theta1 and theta2 respectively. Ahmed (1991) considered a preliminary test estimator based on maximum likelihood estimator for estimating theta1 when it is suspected that theta1=theta2. In this paper we combine minimum phi-divergence estimators as well as phi-divergence test statistics in order to define a preliminary phi-divergence test estimators. These new estimators are compared with the classical estimator as well as the pooled estimator.