A. K. Md. Ehsanes Saleh

Performance of some new preliminary test ridge regression estimators and their properties

The problem of estimation of the regression coefficients in a multiple regression model is considered under multicollinearity situation when it is suspected that the regression coefficients may be restricted to a subspace. We present the estimators of the regression coefficients combining the idea of preliminary test and ridge regression methodology. Accordingly, we consider three estimators, namely, the unrestricted ridge regression estimator (URRE), the restricted ridge regression estimator (RRRE), and finally, the preliminary test ridge regression estimator (PTRRE). The biases, variancematrices and mean square errors (mse) of the estimators are derived and compared with the usual estimators. Regions of optimality of the estimators are determined by studying the mse criterion. The conditions of superiority of the estimators over the traditional estimators as in Saleh and Han (1990) and Ali and Saleh (1991) have also been discussed.

Estimation of noncentrality parameters

We propose some estimators of noncentrality parameters which improve upon usual unbiased estimators under quadratic loss. The distributions we consider are the noncentral chi-square and the noncentral F. However, we give more general results for the family of elliptically contoured distributions and propose a robust dominating estimator.

Nonparametric estimation of location parameter after a preliminary test on regression in the multivariate case

For a simple multivariate regression model, nonparametric estimation of the (vector of) intercept following a preliminary test on the regression vector is considered. Along with the asymptotic distribution of these estimators, their asymptotic bias and dispersion matrices are studied and allied efficiency results are presented.

EMPIRICAL BAYES SUBSET ESTIMATION IN REGRESSION MODELS

The paper considers estimation of β γ in the general regression model Y = X j / J j + e ^e ~ N(0, σ^Ι)^ where it is suspected that β^ = 0. Empirical Bayes estimators of β^ are proposed which shrink the unrestricted least squares estimator β j to the restricted least squares estimator β γ under the hypothesis HQ: β2 = 0. The empirical Bayes estimators serve as a compromise between β γ and βρ and lean more towards /3j if HQ is true, and towards β γ otherwise. Also these estimators when slightly modified, enjoy both Bayesian and frequentesl risk superiority over the preliminary test estimators. As an application, one may consider factorial experiments where the primary interest is to estimate the main effects, and one shrinks the unrestricted least squares estimators of the main effects towards the restricted least squares estimators under the hypothesis that higher order interactions are not significant. AMS 1980 Subject Classification: 62J07, 62C12, 62C20.

Optimum critical value for pre-test estimator

ABSTRACT We propose a preliminary test least squares estimator (PTLSE) based on a fixed critical value for the preliminary test (PT). We compare the performance of the proposed estimator with that of the Brook (1976) and Han and Bancroft (1968) criterion. Table and graphs of relative efficiencies are presented to support the view of using fixed critical value for the PT. It is observed that the proposed or Brooks method are conservative for fixed q, whereas that of Han and Bancroft is flexible. If the researchers are concern about the minimum guaranteed efficiency, they might select our or Brooks method. However, if they are willing to accept higher size of test and want to have higher minimum guaranteed efficiency, they should select the Han and Bancroft method.

Estimation strategies for the intercept vector in a simple linear multivariate normal regression model

Abstract For a simple multivariate regression model, the problem of estimating the intercept vector is considered when it is apriori suspected that the slope may be restricted to a subspace. Four estimation strategies have been developed for the intercept parameter. In this situation, the estimates based on a preliminary test as well as on the Stein-rule are desirable. Exact bias and risks of all of these estimators are derived and their efficiencies relative to classical estimators are studied under quadratic loss function. An optimum rule for the preliminary test estimator is discussed. It is shown that the shrinkage estimator dominates the classical one, whereas none of the preliminary test and shrinkage estimator dominate each other. It is found that shrinkage estimator dominates the preliminary test estimator except in a range around the restriction. Further, for large values of α, the level of statistical significance, shrinkage estimator dominates the preliminary test estimator uniformly.

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, shrinkag...

On some ridge regression estimators: a nonparametric approach

This paper considers the R-estimation of the parameters of a multiple regression model when the design matrix is ill-conditioned. Accordingly, we introduce the ridge regression (RR) modification to the usual R-estimators and consider five RR R-estimators when it is suspected that the regression parameters may belong to a linear subspace of the parameter space. The regions of optimality of the proposed estimators are determined based on the quadratic risks. Asymptotic relative efficiency tables and risk graphs are provided for the numerical and graphical comparisons of the five estimators.

Improved Estimation in a Contingency Table: Independence Structure

Abstract Estimation of the cell probabilities in a two-way contingency table is considered when it is plausible that the table might have an independence structure relating to the two traits. In a classical nonparametric setup, the unrestricted maximum likelihood estimators of the cell probabilities are the corresponding sample proportions; under the assumption of independence, the restricted estimators are the product of the respective row and column sample proportions. The latter estimators behave better than the former when independence actually holds, but a different picture may emerge for possible departure from the assumed independence structure; the restricted estimators may be heavily biased, inefficient, and even inconsistent. For this reason, a preliminary test on independence based on the classical contingency chi-squared statistic may be conveniently incorporated in the formulation of a preliminary test estimator of the matrix of cell probabilities. Since, typically, we have a multiparameter e...

Rank tests and regression rank score tests in measurement error models

The rank and regression rank score tests of linear hypothesis in the linear regression model are modified for measurement error models. The modified tests are still distribution free. Some tests of linear subhypotheses are invariant to the nuisance parameter, others are based on the aligned ranks using the R-estimators. The asymptotic relative efficiencies of tests with respect to tests in models without measurement errors are evaluated. The simulation study illustrates the powers of the tests.