On ridge parameter estimators under stochastic subspace hypothesis

This paper considers several estimators for estimating the restricted ridge parameter estimators. A simulation study has been conducted to compare the performance of these estimators. Based on the simulation study we found that, increasing the correlation between the independent variables has positive effect on the mean square error (MSE). However, increasing the value of ρ has negative effect on MSE. When the sample size increases, the MSE decreases even when the correlation between the independent variables is large. Two real life examples have been considered to illustrate the performance of the estimators.     

On the inadmssibelity of ridge estimator in a linear model

The necessary and sufficient conditions for the inadmissibility of the ridge regression is discussed under two different criteria, namely, average loss and Pitman nearness. Although the two criteria are very different, same conclusions are obtained. The loss functions considered in this article are th likelihood loss function and the Mahalanobis loss function. The two loss functions are motivated from the point of view of classification of two normal populations. Under the Mahalanobis loss it is demonstrated that the ridge regression is always inadmissible as long as the errors are assumed to be symmetrically distributed about the origin.     

A new estimator combining the ridge regression and the restricted least squares methods of estimation

It is well-known in the literature on multicollinearity that one of the major consequences of multicollinearity on the ordinary least squares estimator is that the estimator produces large sampling variances, which in turn might inappropriately lead to exclusion of otherwise significant coefficients from the model. To circumvent this problem, two accepted estimation procedures which are often suggested are the restricted least squares method and the ridge regression method. While the former leads to a reduction in the sampling variance of the estimator, the later ensures a smaller mean square error value for the estimator. In this paper we have proposed a new estimator which is based on a criterion that combines the ideas underlying these two estimators. The standard properties of this new estimator have been studied in the paper. It has also been shown that this estimator is superior to both the restricted least squares as well as the ordinary ridge regression estimators by the criterion of mean sauare error of the estimator of the regression coefficients when the restrictions are indeed correct. The conditions for superiority of this estimator over the other two have also been derived for the situation when the restrictions are not correct.     

Two kinds of restricted modified estimators in linear regression model

In this paper, we introduce two kinds of new restricted estimators called restricted modified Liu estimator and restricted modified ridge estimator based on prior information for the vector of parameters in a linear regression model with linear restrictions. Furthermore, the performance of the proposed estimators in mean squares error matrix sense is derived and compared. Finally, a numerical example and a Monte Carlo simulation are given to illustrate some of the theoretical results.     

The approximation of an unbiased test with a small level for the bioequivalence problem

In the bioequivalence problem. Brown. Hwang and Munk (1997) constructed an unbiased level a test and other tests which are uniformly more powerful than the two one-sided tests procedures when a iscomparatively larger. In this paper, for a small level, an unbiased test is shown to be approxirnately constructeQ lor tnis prooiem oy using tneir Metnog. ine numerical construction is also given.     

On the ridge regression estimator with sub-space restriction

In the linear regression model with elliptical errors, a shrinkage ridge estimator is proposed. In this regard, the restricted ridge regression estimator under sub-space restriction is improved by incorporating a general function which satisfies Taylor’s series expansion. Approximate quadratic risk function of the proposed shrinkage ridge estimator is evaluated in the elliptical regression model. A Monte Carlo simulation study and analysis based on a real data example are considered for performance analysis. It is evident from the numerical results that the shrinkage ridge estimator performs better than both unrestricted and restricted estimators in the multivariate t-regression model, for some specific cases.     

A note on the moments of stochastic shrinkage parameters in ridge regression

The purpose of this note is to gain insight on the performance of two well known operational Ridge Regression estimators by deriving the moments of their stochastic shrinkage parameters. We also show that, under certain conditions, one of them has bounded moments.     

Fractional principal components regression: a general approach to biased estimators

Several biased estimators have been proposed as alternatives to the least squares estimator when multicollinearity is present in the multiple linear regression model. The ridge estimator and the principal components estimator are two techniques that have been proposed for such problems. In this paper the class of fractional principal component estimators is developed for the multiple linear regression model. This class contains many of the biased estimators commonly used to combat multicollinearity. In the fractional principal components framework, two new estimation techniques are introduced. The theoretical performances of the new estimators are evaluated and their small sample properties are compared via simulation with the ridge, generalized ridge and principal components estimators     

On the minimum mean squared error estimators in a regression model

In this paper we analyze the properties of two estimators oroposed by Farebrother (1975) for linear regression models.     

Moments of the trace of a noncentral wishart matrix

Several articles have appeared on the moments of the trace of a noncentral Wishart matrix. Partial results are given in these papers but the representations of these partial results are often too cumbersome for any practical use. In this paper general results are given in compact form which are readily usable in practical problems.