Robust ridge and robust Liu estimator for regression based on the LTS estimator

In the multiple linear regression analysis, the ridge regression estimator and the Liu estimator are often used to address multicollinearity. Besides multicollinearity, outliers are also a problem in the multiple linear regression analysis. We propose new biased estimators based on the least trimmed squares (LTS) ridge estimator and the LTS Liu estimator in the case of the presence of both outliers and multicollinearity. For this purpose, a simulation study is conducted in order to see the difference between the robust ridge estimator and the robust Liu estimator in terms of their effectiveness; the mean square error. In our simulations, the behavior of the new biased estimators is examined for types of outliers: X-space outlier, Y-space outlier, and X-and Y-space outlier. The results for a number of different illustrative cases are presented. This paper also provides the results for the robust ridge regression and robust Liu estimators based on a real-life data set combining the problem of multicollinearity and outliers.     

THE EXAMINATION AND ANALYSIS OF RESIDUALS FOR SOME BIASED ESTIMATORS IN LINEAR REGRESSION

In the presence of collinearity certain biased estimation procedures like ridge regression, generalized inverse estimator, principal component regression, Liu estimator, or improved ridge and Liu estimators are used to improve the ordinary least squares (OLS) estimates in the linear regression model. In this paper new biased estimator (Liu estimator), almost unbiased (improved) Liu estimator and their residuals will be analyzed and compared with OLS residuals in terms of mean-squared error.     

A new modified Jackknifed estimator for the Poisson regression model

The Poisson regression is very popular in applied researches when analyzing the count data. However, multicollinearity problem arises for the Poisson regression model when the independent variables are highly intercorrelated. Shrinkage estimator is a commonly applied solution to the general problem caused by multicollinearity. Recently, the ridge regression (RR) estimators and some methods for estimating the ridge parameter k in the Poisson regression have been proposed. It has been found that some estimators are better than the commonly used maximum-likelihood (ML) estimator and some other RR estimators. In this study, the modified Jackknifed Poisson ridge regression (MJPR) estimator is proposed to remedy the multicollinearity. A simulation study and a real data example are provided to evaluate the performance of estimators. Both mean-squared error and the percentage relative error are considered as the performance criteria. The simulation study and the real data example results show that the proposed MJPR method outperforms the Poisson ridge regression, Jackknifed Poisson ridge regression and the ML in all of the different situations evaluated in this paper.     

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.     

Efficiency of two classes of stochastic restricted almost unbiased type principal component estimators in linear regression model

In this paper, we introduce two new classes of estimators called the stochastic restricted almost unbiased ridge-type principal component estimator (SRAURPCE) and the stochastic restricted almost unbiased Liu-type principal component estimator (SRAURPCE) to overcome the well-known multicollinearity problem in linear regression model. For the two cases when the restrictions are true and not true, necessary and sufficient conditions for the superiority of the proposed estimators are derived and compared, respectively. Furthermore, a Monte Carlo simulation study and a numerical example are given to illustrate the performance of the proposed estimators.     

r − k Class estimator in the linear regression model with correlated errors

Autocorrelation in errors and multicollinearity among the regressors are serious problems in regression analysis. The aim of this paper is to examine multicollinearity and autocorrelation problems concurrently and to compare the r ? k class estimator to the generalized least squares estimator, the principal components regression estimator and the ridge regression estimator by the scalar and matrix mean square error criteria in the linear regression model with correlated errors.     

A graphical evaluation of logistic ridge estimator in mixture experiments

In comparison to other experimental studies, multicollinearity appears frequently in mixture experiments, a special study area of response surface methodology, due to the constraints on the components composing the mixture. In the analysis of mixture experiments by using a special generalized linear model, logistic regression model, multicollinearity causes precision problems in the maximum-likelihood logistic regression estimate. Therefore, effects due to multicollinearity can be reduced to a certain extent by using alternative approaches. One of these approaches is to use biased estimators for the estimation of the coefficients. In this paper, we suggest the use of logistic ridge regression (RR) estimator in the cases where there is multicollinearity during the analysis of mixture experiments using logistic regression. Also, for the selection of the biasing parameter, we use fraction of design space plots for evaluating the effect of the logistic RR estimator with respect to the scaled mean squared error of prediction. The suggested graphical approaches are illustrated on the tumor incidence data set.     

On the euclidean distance between biased estimators

The purpose of this paper is to extend the results achieved by Hsuan (1981) to a wide class of biased estimators. It is shown that in the case of multicollinearity ridge and Iteration estimator can be made very close to the principal component estimator whereas the shrunken estimator does not have this property.     

Restricted minimum bias linear estimation in regression

In this article a class of restricted minimum bias linear estimators of the vector of unknown regression coefficients when multicollinearity among the columns of the design matrix exists, is obtained. The ordinary ridge regression, principal components and shrinkage estimators are members of this class. Moreover, our ap-proach can be used to improve, in some sense, certain classes of generalized ridge and shrinkage estimators of the vector of un-known parameters in linear models.     

On identification of transfer function models by biased regression methods

This paper investigates a biased regression approach to the preliminary estimation of the Box-Jenkins transfer function weights. Using statistical simulation to generate time series, 14 estimators (various OLS, ridge and principal components estimators) are compared in terms of MSE and standard error of the weight estimators. The estimators are investigated for different levels of multicollinearity, signal-to-noise ratio, number of independent variables, length of time series and number of lags included in the estimation. The results show that the ridge estimators nearly always give lower MSE than the OLS estimator, and in the computationally difficult cases give much lower MSE than the OLS estimator. The principal components estimators can give lower MSE than the OLS, but also higher values. All biased estimators nearly always give much lower estimated standard error than OLS when estimating the weights.     

The relative efficiency of the restricted estimators in linear regression models

This paper deals with the problem of multicollinearity in a multiple linear regression model with linear equality restrictions. The restricted two parameter estimator which was proposed in case of multicollinearity satisfies the restrictions. The performance of the restricted two parameter estimator over the restricted least squares (RLS) estimator and the ordinary least squares (OLS) estimator is examined under the mean square error (MSE) matrix criterion when the restrictions are correct and not correct. The necessary and sufficient conditions for the restricted ridge regression, restricted Liu and restricted shrunken estimators, which are the special cases of the restricted two parameter estimator, to have a smaller MSE matrix than the RLS and the OLS estimators are derived when the restrictions hold true and do not hold true. Theoretical results are illustrated with numerical examples based on Webster, Gunst and Mason data and Gorman and Toman data. We conduct a final demonstration of the performance of the estimators by running a Monte Carlo simulation which shows that when the variance of the error term and the correlation between the explanatory variables are large, the restricted two parameter estimator performs better than the RLS estimator and the OLS estimator under the configurations examined.     

Superiority of the r–k Class Estimator Over Some Estimators In A Linear Model

In regression analysis, to overcome the problem of multicollinearity, the r ? k class estimator is proposed as an alternative to the ordinary least squares estimator which is a general estimator including the ordinary ridge regression estimator, the principal components regression estimator and the ordinary least squares estimator. In this article, we derive the necessary and sufficient conditions for the superiority of the r ? k class estimator over each of these estimators under the Mahalanobis loss function by the average loss criterion. Then, we compare these estimators with each other using the same criterion. Also, we suggest to test to verify if these conditions are indeed satisfied. Finally, a numerical example and a Monte Carlo simulation are done to illustrate the theoretical results.     

A new biased estimator based on ridge estimation

In this paper we introduce a new biased estimator for the vector of parameters in a linear regression model and discuss its properties. We show that our new biased estimator is superior, in the mean square error(mse) sense, to the ordinary least squares (OLS) estimator, the ordinary ridge regression (ORR) estimator and the Liu estimator. We also compare the performance of our new biased estimator with two other special Liu-type estimators proposed in Liu (2003). We illustrate our findings with a numerical example based on the widely analysed dataset on Portland cement.     

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.     

On the Stochastic Restricted Almost Unbiased Estimators in Linear Regression Model

In this article, the stochastic restricted almost unbiased ridge regression estimator and stochastic restricted almost unbiased Liu estimator are proposed to overcome the well-known multicollinearity problem in linear regression model. The quadratic bias and mean square error matrix of the proposed estimators are derived and compared. Furthermore, a numerical example and a Monte Carlo simulation are given to illustrate some of the theoretical results.     

A Generalized Diagonal Ridge-type Estimator in Linear Regression

This article introduces a general class of biased estimator, namely a generalized diagonal ridge-type (GDR) estimator, for the linear regression model when multicollinearity occurs. The estimator represents different kinds of biased estimators when different parameters are obtained. Some properties of this estimator are discussed and an iterative procedure is provided for selecting the parameters. A Monte Carlo simulation study and an application show that the GDR estimator performs much better than the ordinary least squares (OLS) estimator under the mean square error (MSE) criterion when severe multicollinearity is present.     

Latent root regression: a biased regression methodology for use with collinear predictor variables

Many different biased regression techniques have been proposed for estimating parameters of a multiple linear regression model when the predictor variables are collinear. One particular alternative, latent root regression analysis, is a technique based on analyzing the latent roots and latent vectors of the correlation matrix of both the response and the predictor variables. It is the purpose of this paper to review the latent root regression estimator and to re-examine some of its properties and applications. It is shown that the latent root estimator is a member of a wider class of estimators for linear models     

The feasible generalized restricted ridge regression estimator

The presence of autocorrelation in errors and multicollinearity among the regressors have undesirable effects on the least-squares regression. There are a wide range of methods which are proposed to overcome the usefulness of the ordinary least-squares estimator or the generalized least-squares estimator, such as the Stein-rule, restricted least-squares or ridge estimator. Therefore, we introduce a new feasible generalized restricted ridge regression (FGRR) estimator to examine multicollinearity and autocorrelation problems simultaneously for the general linear regression model. We also derive some statistical properties of the FGRR estimator and comparisons have been conducted using matrix mean-square error. Moreover, a Monte Carlo simulation experiment is performed to investigate the performance of the proposed estimator over the others.     

Relative squared error prediction in the generalized linear regression model

In linear regression models, predictors based on least squares or on generalized least squares estimators are usually applied which, however, fail in case of multicollinearity. As an alternative biased estimators like ridge estimators, Kuks-Olman estimators, Bayes or minimax estimators are sometimes suggested. In our analysis the relative instead of the generally used absolute squared error enters the objective function. An explicit minimax solution is derived which, in an important special case, can be viewed as a predictor based on a Kuks-Olman estimator.     

A Jackknifed estimators for the negative binomial regression model

Shrinkage estimator is a commonly applied solution to the general problem caused by multicollinearity. Recently, the ridge regression (RR) estimators for estimating the ridge parameter k in the negative binomial (NB) regression have been proposed. The Jackknifed estimators are obtained to remedy the multicollinearity and reduce the bias. A simulation study is provided to evaluate the performance of estimators. Both mean squared error (MSE) and the percentage relative error (PRE) are considered as the performance criteria. The simulated result indicated that some of proposed Jackknifed estimators should be preferred to the ML method and ridge estimators to reduce MSE and bias.