Pre-test Estimator Research Articles

Conflict Between Pretest Estimators Induced by Three Large Sample Tests Under a Regression Model with Student t-error

We consider the preliminary test approach to the estimation of regression parameters, in a multiple-regression model with non-normal error. Here, we consider the multivariate t-distribution as the error distribution. The preliminary test estimators (PTEs) based on the Wald, likelihood ratio (LR) and Lagrangian multiplier (LM) tests are given. Their bias, mean-square error matrix M and weighted mean-square error (WMSE) functions are derived and compared. We show that there is conflict between the three estimators with respect to their bias, M and WMSE because of certain inequalities that exist inherently among the test statistics and because of approximating them by the same asymptotic distribution. In the neighbourhood of the null hypothesis the PTE based on the LM test has the smallest WMSE followed by the LR-based estimator, and the estimator based on the Wald test is the worst. However, the PTE based on the Wald test performs the best followed by the LR-based estimator when the parameter moves away from the subspace of the restriction and the LM-based estimator is the worst. It has been shown that in the choice of the smallest level of significance to yield the best estimator in the sense of the highest minimum guaranteed efficiency the Wald-based PTE dominates the other two estimators. Further, the optimum choice of the level of significance becomes the traditional choice by using the Wald test.

A Monte Carlo Study of Robustness of Pretest and Shrinkage Estimators in Pooling Coefficients of Variation

Pooling data, when justified, is advantageous for estimating the true parameter. In this paper the problem of estimating the coefficient of variation is considered when it is a priori suspected that two coefficients of variation are the same. Various estimators based on pretest and shrinkage rules are considered. A comparison through the Simulated Mean Squared Error (SMSE) criterion is carried out among various proposed estimators of the target coefficient of variation. The relative simulated efficiencies of the restricted, shrinkage restricted and shrinkage pretest estimators are studied. It is found that the proposed estimators are quite robust when the sample sizes are not too large. The result of Monte Carlo study indicates that the proposed shrinkage pretest estimator is efficient than the usual estimator in a wider range.

Value of cerebrospinal fluid cytochemical examination for the diagnosis of congenital toxoplasmosis at birth in France.

Because of routine screening and treatment of pregnant women for Toxoplasma infection in France, most neonates born to mothers who seroconverted during pregnancy are either not infected or asymptomatic. Early diagnosis relies mainly on radiologic, ophthalmologic and biologic tests. Cerebrospinal fluid (CSF) cytochemical evaluation is one of several tests performed in parallel to increase the overall sensitivity of the diagnostic evaluation. Our goal was to assess the value of cytochemical examination and to confirm whether using a portion of available CSF for this analysis is legitimate. The individual performance of each of the two cytochemical tests and their combined value when used in parallel were assessed. These findings were then compared with the anti-Toxoplasma IgM and IgA serum titers and the clinical, ophthalmologic and radiologic findings at birth. CSF cytochemical analysis was possible in only 52% of the 233 children in the study. Our results in 112 children indicated poor sensitivity estimates. There was no significant change in the posttest probability of infection compared with the pretest estimation of risk in cases of a negative finding. After a mean follow-up of 80 months there was no evidence that CSF cytochemistry helped predict the risk of sequelae. In our setting cytochemical examination did not significantly contribute to the diagnosis of congenital infection at birth. Because of the limited quantity of CSF available, we suggest the use of other methods with higher yield.

The exact distribution and density functions of a pre-test estimator of the error variance in a linear regression model with proxy variables

We consider a linear regression model when some independent variables are unobservable, but proxy variables are available instead of them. We derive the distribution and density functions of a pre-test estimator of the error variance after a pre-test for the null hypothesis that the coefficients for the unobservable variables are zeros. Based on the density function, we show that when the critical value of the pre-test is unity, the coverage probability in the interval estimation of the error variance is maximum.

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.

Bayesian estimation of the linear regression model with an uncertain interval constraint on coefficients

This article considers Bayesian inference in the interval constrained normal linear regression model. Whereas much of the previous literature has concentrated on the case where the prior constraint is correctly specified, our framework explicitly allows for the possibility of an invalid constraint. We adopt a non-informative prior and uncertainty concerning the interval restriction is represented by two prior odds ratios. The sampling theoretic risk of the resulting Bayesian interval pre-test estimator is derived, illustrated and explored.

Effects on inference of pretesting the exogeneity of a regressor

This paper investigates, via simulation, the performance of the inference of a pretest estimator based on a Hausman's test of exogeneity. I find that an inference based on conventional pretest procedure leads to a substantial size distortion. A bootstrap-based inferential procedure is proposed to take into account the pretesting effect on the small sample distribution of the pretest estimator in independently (possibly non-identically) distributed samples. My simulation shows that the bootstrap-based inference improves substantially over the conventional procedure.

The exact density and distribution functions of the inequality constrained and pre-test estimators

We consider a bivariate normal linear regression model with an inequality restriction imposed on one of the regression coefficients. The exact analytical expressions for the density and distribution functions of the inequality constrained and pre-test estimators are derived and numerically evaluated. The implications of using the inequality constrained and pre-test estimators in confidence interval construction are also discussed and explored.

The exact risk performance of a pre-test estimator in a heteroskedastic linear regression model under the balanced loss function

We examine the risk of a pre-test estimator for regression coefficients after a pre-test for homoskedasticity under the Balanced Loss Function (BLF). We show analytically that the two stage Aitken estimator is dominated by the pre-test estimator with the critical value of unity, even if the BLF is used. We also show numerically that both the two stage Aitken estimator and the pre-test estimator can be dominated by the ordinary least squares estimator when “goodness of fit” is regarded as more important than precision of estimation.

Monte Carlo Results on Pure and Pretest Estimators of an Error Component Model with Autocorrelated Disturbances

A Monte Carlo study is used to compare the finite sample relative efficiency of a number of pure and pre-test estimators for an error component model with first-order autocorrelated remainder disturbances.

Optimality of the Stein‐type Estimator for Variance

This paper considers the optimality of the Stein-type estimator of the disturbance variance among the class of pre-test estimators after the pre-test for a linear hypothesis on coefficients. The Stein-type estimator is a member in this class with a critical value, say cs. It is known that the Stein-type estimator is the best in the subclass which consists of the pre- test estimators with smaller critical values than cs. It is analytically shown that an extension of the optimality to a larger class is not possible.

The density function and the MSE dominance of the pre-test estimator in a heteroscedastic linear regression model with omitted variables

In this paper, we consider a heteroscedastic linear regression model with omitted variables. We derive the density function of the pre-test estimator consisting of the two-stage Aitken estimator (2SAE) and the ordinary least squares estimator (OLSE) after the pre-test for homoscedasticity. We also derive the first two moments based on the density function and show the sufficient condition for the pre-test estimator to dominate the 2SAE in terms of the MSE. Our numerical evaluations show that when this sufficient condition does not hold and when the magnitude of the specification error is large, the pre-test estimator can be dominated by the 2SAE, and further, the 2SAE can be dominated by the OLSE.

Pre-test estimation and design in the linear model

As a robust method against model deviation we consider a pre-test estimation function. To optimize a continuous design for this problem we give an asymptotic risk matrix for the quadratic loss. The risk will then be given by an isotonic criterion function of the asymptotic risk matrix. As an optimization criterion we look for a design that minimizes the maximal risk in the deviation model under the restriction that the risk in the original model does not exceed a given bound. This optimization problem will be solved for the polynomial regression, the deviation consisting in one additional regression function and the criterion function being the determinant.

Estimating the error variance after a pre-test for an inequality restriction on the coefficients

Estimation of the regression error variance after a preliminary test of an inequality constraint on the coefficient vector is considered. We derive the exact finite sample risk of several inequality restricted and pre-test estimators of σ 2. These estimators are associated with the maximum likelihood, least squares and minimum mean squared error component estimators. Optimal critical values for the pre-test according to a mini-max regret criterion are numerically calculated. Furthermore, we examine the robustness of the optimal choice of critical values and the risk properties of the estimators of σ 2 to model mis-specification through the exclusion of relevant regressors.

The exact risks of some pre-test and stein-type regression estimators umder balanced loss

In regression analysis we are often interested in using an estimator which is “precise” and which simultaneously provides a model with “good fit”, In this paper we consider the risk properties of several estimators of the regression coefficient vector "trader “balanced” loss, This loss function (Zellner, 1994) reflects both of the described attributes. Under a particular form of balanced loss, we derive the predictive risk of the pre-test estimator which results after a test for exact linear restrictions on the coefficient vector. The corresponding risks of Stein-rule and positive-part Stein-rale estimators are also established. The risks based on loss functions which allow only for estimation precision, or only for goodness of fit, are special cases of our results, and we draw appropriate comparisons, In particular, we show that some of the well-known results under (quadratic) precision-only loss are not robust to our generalization of the loss function

On a simultaneous equations pre-test estimator

This paper investigates the finite-sample properties of a pre-test estimator for the linear regression model when endogenous regressors render the ordinary least squares estimator inconsistent. A Wu-Hausman exogeneity test is applied to determine whether two-stage least squares is appropriate. Both analytic and nonparametric approximations to the pre-test risk and bias are discussed.

Shrinkage estimation in nonlinear regression The Box-Cox transformation

The risk superiority of the Stein-rule estimator over the maximum likelihood (ML) estimator is known in the context of the normal linear regression model. We present a positive-part Stein-like estimator that dominates the ML and pretest estimators under quadratic loss, weighted by the information matrix, in nonlinear regression. The risk properties of ML, constrained ML, pretest, and shrinkage estimators for the Box-Cox model are discussed. In a Monte Carlo experiment, the shrinkage estimator dominates the ML estimator under the unweighted quadratic loss. However, the ML estimator may have lower risk than the shrinkage estimator under other weighted quadratic loss functions in small samples.

Improved estimators of energy models

It has been suggested that Stein-rule estimators of econometric models can be used to improve inferences in those models by combining sample information with uncertain non-sample information in the estimation process. Pretest estimators are often used for model specification, even though it is well known that they possess poor sampling properties. The Stein-rule contains the same elements used by the classical pretest estimator but performs much better than the pretest estimator in repeated sampling. Based on Monte Carlo experiments the Stein-rule estimator is shown to offer significant quadratic risk improvements over maximum likelihood, restricted maximum likelihood, and pretest estimators of energy models.

On risk comparisons of some estimators in a linear regression model under inagaki’S loss function and nonnormal error terms

In this paper we consider the risk performances of some estimators for both location and scale parameters in a linear regression model under Inagaki’s loss function We prove that the pre-test estimator for location parameter is dominated by the Stein-rule estimator under Inagaki’s loss function when the distribution of error terms is expressed by the scale mixture of normal distribution and the variance of error terms is unknown.. It is an extension of the results in Nagata (1983) to our situation Also we perform numerical calculations to draw the shapes of the risks.

Risk behavior of a pre-test estimator for normal variance with the Stein-type estimator

In this paper, we examine the risk behavior of a pre-test estimator for normal variance with the Stein-type estimator. The one-sided pre-test is conducted for the null hypothesis that the population variance is equal to a specific value, and the Stein-type estimator is used if the null hypothesis is rejected. A sufficient condition for the pre-test estimator to dominate the Stein-type estimator is shown.