Some finite sample properties of generalized ridge regression estimators

In this paper we study the mean square error properties of the generalized ridge estimator. We obtain the exact and the approximate bias and the mean square error of the operational generalized ridge estimator in terms of G( ) functions. We show, among other things, that the operational generalized ridge estimator does not dominate the ordinary least squares estimator up to a certain order of approximation. Finally, we note that the iterative procedures to obtain coverging ridge estimators should be used with caution.     

Bayes minimax ridge regression estimators

The problem of estimating of the vector β of the linear regression model y = Aβ + ? with ? ～ N_p(0, σ²I_p) under quadratic loss function is considered when common variance σ² is unknown. We first find a class of minimax estimators for this problem which extends a class given by Maruyama and Strawderman (2005 Maruyama, Y., and W. E. Strawderman. 2005. A new class of generalized Bayes minimax ridge regression estimators. Annals of Statistics 33:1753–70.[Crossref], [Web of Science ®] , [Google Scholar]) and using these estimators, we obtain a large class of (proper and generalized) Bayes minimax estimators and show that the result of Maruyama and Strawderman (2005 Maruyama, Y., and W. E. Strawderman. 2005. A new class of generalized Bayes minimax ridge regression estimators. Annals of Statistics 33:1753–70.[Crossref], [Web of Science ®] , [Google Scholar]) is a special case of our result. We also show that under certain conditions, these generalized Bayes minimax estimators have greater numerical stability (i.e., smaller condition number) than the least-squares estimator.     

On small sample properties of the almost unbiased generalized ridge estimator

The purpose of this paper is to examine small sample properties of the operational almost unbiased generalized ridge estimator (E) . The exact first two moments of theAUGRE are derived. It is shown that although the reduction of the bias of the AUGRE is substantial, the AUGRE is rather inefficient than the generalized ridge estimator without the bias correction in a wide range of a noncen-trality parameter in terms of the mean square error.     

Some properties of generalized ridge estimators

Exact expressions, in the form of infinite series expansions, are given for the first and second moments of two well known generalized ridge estimators. These series expansions are then evaluated using recursive formulas and computations are verified using approximations. Results are presented for the relative mean square error and bias of these estimators as well as their relative efficiency with respect to least squares.     

On the small sample properties of norm-restricted maximum likelihood estimators for logistic regression models

This paper develops alternatives to maximum likelihood estimators (MLE) for logistic regression models and compares the mean squared error (MSE) of the estimators. The MLE for the vector of underlying success probabilities has low MSE only when the true probabilities are extreme (i.e., near 0 or 1). Extreme probabilities correspond to logistic regression parameter vectors which are large in norm. A competing “restricted” MLE and an empirical version of it are suggested as estimators with better performance than the MLE for central probabilities. An approximate EM-algorithm for estimating the restriction is described. As in the case of normal theory ridge estimators, the proposed estimators are shown to be formally derivable by Bayes and empirical Bayes arguments. The small sample operating characteristics of the proposed estimators are compared to the MLE via a simulation study; both the estimation of individual probabilities and of logistic parameters are considered.     

A class of generalized ridge estimators

Presence of collinearity among the explanatory variables results in larger standard errors of parameters estimated. When multicollinearity is present among the explanatory variables, the ordinary least-square (OLS) estimators tend to be unstable due to larger variance of the estimators of the regression coefficients. As alternatives to OLS estimators few ridge estimators are available in the literature. This article presents some of the popular ridge estimators and attempts to provide (i) a generalized class of ridge estimators and (ii) a modified ridge estimator. The performance of the proposed estimators is investigated with the help of Monte Carlo simulation technique. Simulation results indicate that the suggested estimators perform better than the ordinary least-square (OLS) estimators and other estimators considered in this article.     

On small sample properties of the generalized signed rank and generalized sign tests

The generalized signed rank (GSR) and generalized sign (GS) tests were recently proposed for matched pair studies with censored observations (Woolson and Lechenbruch, 1980). The results provided in that paper were asymptotic, and no indicatin of small sample behavior was given. In this paper we report on simulation studied of these statistics for a variety of distributions. We find that the GSR is more powerful than the GS, and that censoring does not affect power greatly. In the original paper, we assumed each member of the pair has the same censoring time. We consider a variant of this in which each member of the pair has a censoring time chosen from a uniform distribution, and the minimum of these times is selected as the censoring time for the pair. It is found that the power of the test is slightly reduced because the number of doubly censored pairs is increased.     

Characterization of minimax linear estimators in linear regression

Two characterization theorems of the minimax linear estimator (Mile) are proven for the case, where the regression parameter varies only in an arbitrary ellipsoid. Furthermore, the existence, uniqueness and admissibility of Mile are shown. The explicit determination of Mile is carried out for a special case.     

A note on generalized ridge estimators

It is shown that a necessary and sufficient condition derived by Farebrother (1984)for a generalized ridge estimator to dominate the ordinary least-squares estimator with respect to the mean-square-error-matrix criterion in the linear regression model admits a similar interpretation as the well known criterion of Toro-Viz-carrondo and Wallace (1968)for the dominance of a restricted least-squares estimator over the ordinary least-squares estimator. Two other properties of the generalized ridge estimators, referring to the concept of admissibility, are also pointed out.     

Small sample properties of ridge estimators with normal and non-normal disturbances

In this paper we consider five well known and widely used ridge estimators when the convenient assumption of normality of the disturbances is abandoned and report on a Monte Carlo study of their small sample properties. The Monte Carlo experiment is applied to four different data sets with artificially varied degrees of multicollinearity, while the disturbances follow normal, lognormal, uniform and Laplace distributions with small and large variances. The results show that the best estimates are obtained for all ridge estimators when the disturbances follow the lognormal distribution. Also, none of the examined ridge estimators shows a consistent behavior under the different settings considered.     

On small sample properties of the mixed regression predictor under misspecification

This paper relaxes the Mittelhammer's (1981) assumption that the value of the true variance is known in the mixed regression model and examines the small sample, properties of the feasible mixed regression predictor under misspecification. The paper shows that the feasible mixed regression predictor is not always superior to the ordinary least squares predictor in terms of the weak mean square error when there exist omitted variables in the model. Further it shows that misspecificstion works favorably for the ordinary least squares predictor.     

Operational ridge regression estimators under the prediction goal

In this paper we study the Mean Square Error and Conditional Mean Forecasting of Operational Ordinary Ridge Estimator. We use the G( ) functions to provide both the exact and the approximate bias and Mean Square Error of ordinary ridge estimator (ORE), We show, among other things, that ORE dominates OLS up to a certain order of approximation under the conditional mean forecasting sense.     

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.     

Minicommunication computation of moments of generalized ridge estimators

This paper deals with a formal identification of outliers in regression based on tests of hypotheses. The hypothesis is not the standard one but is based on performance criteria that relates to the coefficient estimation and predictive capabilities of the model. The cri-teria include the trace of the mean square error matrix on the coefficients and integrated mean square error of prediction. Both the mean shift outlier model and the variance in-flation model are discussed.     

Some ridge regression estimators for the zero-inflated Poisson model

The zero-inflated Poisson regression model is commonly used when analyzing economic data that come in the form of non-negative integers since it accounts for excess zeros and overdispersion of the dependent variable. However, a problem often encountered when analyzing economic data that has not been addressed for this model is multicollinearity. This paper proposes ridge regression (RR) estimators and some methods for estimating the ridge parameter k for a non-negative model. A simulation study has been conducted to compare the performance of the estimators. Both mean squared error and mean absolute error are considered as the performance criteria. The simulation study shows that some estimators are better than the commonly used maximum-likelihood estimator and some other RR estimators. Based on the simulation study and an empirical application, some useful estimators are recommended for practitioners.     

Generalized ridge regression estimators under the LINEX loss function

In this paper, we examine the risk performance of the generalized ridge regression (GRR) and feasible GRR estimators when the LINEX loss function is used. A sufficient condition for the GRR estimator to dominate the OLS estimator is shown, and the risk functions of the feasible GRR estimator and the OLS estimator are numerically compared.     

The exact properties of the lawless and wang operational ridge regression estimator in a misspecified regression equation

The exact properties of the Lawless and Wang Operational Ridge Regression estimator are derived in the context of a misspecified regression equation.     

Improved ridge estimators in a linear regression model

In this paper, the notion of the improved ridge estimator (IRE) is put forward in the linear regression model y=X β+e. The problem arises if augmenting the equation 0=c′α+ε instead of 0=C α+? to the model. Three special IREs are considered and studied under the mean-squared error criterion and the prediction error sum of squares criterion. The simulations demonstrate that the proposed estimators are effective and recommendable, especially when multicollinearity is severe.