Robust two parameter ridge M-estimator for linear regression

The problem of multicollinearity and outliers in the data set produce undesirable effects on the ordinary least squares estimator. Therefore, robust two parameter ridge estimation based on M-estimator (ME) is introduced to deal with multicollinearity and outliers in the y-direction. The proposed estimator outperforms ME, two parameter ridge estimator and robust ridge M-estimator according to mean square error criterion. Moreover, a numerical example and a Monte Carlo simulation experiment are presented.     

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.     

A robust Liu regression estimator

The least-squares regression estimator can be very sensitive in the presence of multicollinearity and outliers in the data. We introduce a new robust estimator based on the MM estimator. By considering weights, also the resulting MM-Liu estimator is highly robust, but also the estimation of the biasing parameter is robustified. Also for high-dimensional data, a robust Liu-type estimator is introduced, based on the Partial Robust M-estimator. Simulation experiments and a real dataset show the advantages over the standard estimators and other robustness proposals.     

Robust Linearized Ridge M-estimator for Linear Regression Model

In the multiple linear regression, multicollinearity and outliers are commonly occurring problems. They produce undesirable effects on the ordinary least squares estimator. Many alternative parameter estimation methods are available in the literature which deals with these problems independently. In practice, it may happen that the multicollinearity and outliers occur simultaneously. In this article, we present a new estimator called as Linearized Ridge M-estimator which combats the problem of simultaneous occurrence of multicollinearity and outliers. A real data example and a simulation study is carried out to illustrate the performance of the proposed estimator.     

Robust Liu-type estimator for regression based on M-estimator

The problem of multicollinearity and outliers in the dataset can strongly distort ordinary least-square estimates and lead to unreliable results. We propose a new Robust Liu-type M-estimator to cope with this combined problem of multicollinearity and outliers in the y-direction. Our new estimator has advantages over two-parameter Liu-type estimator, Ridge-type M-estimator, and M-estimator. Furthermore, we give a numerical example and a simulation study to illustrate some of the theoretical results.     

Robust Liu estimator for regression based on an M-estimator

Consider the regression model y = beta 0 1 + Xbeta + epsilon. Recently, the Liu estimator, which is an alternative biased estimator beta L (d) = (X'X + I) -1 (X'X + dI)beta OLS , where 0<d<1 is a parameter, has been proposed to overcome multicollinearity . The advantage of beta L (d) over the ridge estimator beta R (k) is that beta L (d) is a linear function of d. Therefore, it is easier to choose d than to choose k in the ridge estimator. However, beta L (d) is obtained by shrinking the ordinary least squares (OLS) estimator using the matrix (X'X + I) -1 (X'X + dI) so that the presence of outliers in the y direction may affect the beta L (d) estimator. To cope with this combined problem of multicollinearity and outliers, we propose an alternative class of Liu-type M-estimators (LM-estimators) obtained by shrinking an M-estimator beta M , instead of the OLS estimator using the matrix (X'X + I) -1 (X'X + dI).     

Restricted Two Parameter Ridge Estimator

In 2005 Lipovetsky and Conklin proposed an estimator, the two parameter ridge estimator (TRE), as an alternative to the ordinary least squares estimator (OLSE) and the ordinary ridge estimator (RE) in the presence of multicollinearity, and in 2006 Lipovetsky improved the two parameter model. In this paper, we introduce two new estimators, one of which is the modified two parameter ridge estimator (MTRE) defined by following Swindel's paper of 1976. The other one is the restricted two parameter ridge estimator (RTRE) which is derived by setting additional linear restrictions on the parameter vectors. This estimator is a generalization of the restricted least squares estimator (RLSE) and includes the restricted ridge estimator (RRE) proposed by Groß in 2003. A numerical example is provided and a simulation study is conducted for the comparisons of the RTRE with the OLSE, RLSE, RE, RRE and TRE.     

Improved robust ridge M-estimation

It is developed that non-sample prior information about regression vector-parameter, usually in the form of constraints, improves the risk performance of the ordinary least squares estimator (OLSE) when it is shrunken. However, in practice, it may happen that both multicollinearity and outliers exist simultaneously in the data. In such a situation, the use of robust ridge estimator is suggested to overcome the undesirable effects of the OLSE. In this article, some prior information in the form of constraints is employed to improve the performance of this estimator in the multiple regression model. In this regard, shrinkage ridge robust estimators are defined. Advantages of the proposed estimators over the usual robust ridge estimator are also investigated using Monte-Carlo simulation as well as a real data example.     

ROBUST RIDGE REGRESSION BASED ON AN M-ESTIMATOR

Consider the linear regression model y =β₀1 +Xβ+ in the usual notation. It is argued that the class of ordinary ridge estimators obtained by shrinking the least squares estimator by the matrix (X¹X + kI)^-1X'X is sensitive to outliers in the ^variable. To overcome this problem, we propose a new class of ridge-type M-estimators, obtained by shrinking an M-estimator (instead of the least squares estimator) by the same matrix. Since the optimal value of the ridge parameter k is unknown, we suggest a procedure for choosing it adaptively. In a reasonably large scale simulation study with a particular M-estimator, we found that if the conditions are such that the M-estimator is more efficient than the least squares estimator then the corresponding ridge-type M-estimator proposed here is better, in terms of a Mean Squared Error criteria, than the ordinary ridge estimator with k chosen suitably. An example illustrates that the estimators proposed here are less sensitive to outliers in the y-variable than ordinary ridge estimators.     

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.     

MORE ON THE PRE-TEST ESTIMATOR IN RIDGE REGRESSION

ABSTRACT

The problem of estimation of the regression coefficients in a multiple regression model is considered under a multicollinearity situation when it is suspected that the regression coefficients may be restricted to a subspace. The objective of this paper is to compare the usual preliminary test estimator and the preliminary test ridge regression estimator in the sense of the dispersion matrix of one dominating that of the other. In particular we proved two results giving necessary and sufficient conditions for the superiority of the preliminary test ridge regression estimator over the preliminary test estimator associated with the δ = 0 (or Δ = 0) and δ ≠ 0 (or Δ ≠ 0).     

Influence measures in affine combination type regression

The detection of outliers and influential observations has received a great deal of attention in the statistical literature in the context of least-squares (LS) regression. However, the explanatory variables can be correlated with each other and alternatives to LS come out to address outliers/influential observations and multicollinearity, simultaneously. This paper proposes new influence measures based on the affine combination type regression for the detection of influential observations in the linear regression model when multicollinearity exists. Approximate influence measures are also proposed for the affine combination type regression. Since the affine combination type regression includes the ridge, the Liu and the shrunken regressions as special cases, influence measures under the ridge, the Liu and the shrunken regressions are also examined to see the possible effect that multicollinearity can have on the influence of an observation. The Longley data set is given illustrating the influence measures in affine combination type regression and also in ridge, Liu and shrunken regressions so that the performance of different biased regressions on detecting and assessing the influential observations is examined.     

Liu-Type Logistic Estimator

It is known that multicollinearity inflates the variance of the maximum likelihood estimator in logistic regression. Especially, if the primary interest is in the coefficients, the impact of collinearity can be very serious. To deal with collinearity, a ridge estimator was proposed by Schaefer et al. The primary interest of this article is to introduce a Liu-type estimator that had a smaller total mean squared error (MSE) than the Schaefer's ridge estimator under certain conditions. Simulation studies were conducted that evaluated the performance of this estimator. Furthermore, the proposed estimator was applied to a real-life dataset.     

An Iterative Weighted Least Squares Algorithm and Simulation Study for Censored Data M-Estimates

A popular linear regression estimator for censored data is the one proposed by Buckley and James (1979). However, this estimator is not robust to outliers, which is not surprising since it is a modified version of the uncensored data least squares estimator. Lai and Ying (1994) have proposed an M-estimator for censored data that is a generalization of the Buckley- James estimator. In this paper we discuss a weighted least squares algorithm for computing these M-estimates and compare the performance of two Huber M-estimators with the Buckley-James estimator in a simulation study. We find that the Huber M-estimators perform more robustly for a broad range of censoring and error distributions.     

A two-parameter estimator in the negative binomial regression model

In this article, a two-parameter estimator is proposed to combat multicollinearity in the negative binomial regression model. The proposed two-parameter estimator is a general estimator which includes the maximum likelihood (ML) estimator, the ridge estimator (RE) and the Liu estimator as special cases. Some properties on the asymptotic mean-squared error (MSE) are derived and necessary and sufficient conditions for the superiority of the two-parameter estimator over the ML estimator and sufficient conditions for the superiority of the two-parameter estimator over the RE and the Liu estimator in the asymptotic mean-squared error (MSE) matrix sense are obtained. Furthermore, several methods and three rules for choosing appropriate shrinkage parameters are proposed. Finally, a Monte Carlo simulation study is given to illustrate some of the theoretical results.     

Performance of ridge estimator in inverse Gaussian regression model

The presence of multicollinearity among the explanatory variables has undesirable effects on the maximum likelihood estimator (MLE). Ridge estimator (RE) is a widely used estimator in overcoming this issue. The RE enjoys the advantage that its mean squared error (MSE) is less than that of MLE. The inverse Gaussian regression (IGR) model is a well-known model in the application when the response variable positively skewed. The purpose of this paper is to derive the RE of the IGR under multicollinearity problem. In addition, the performance of this estimator is investigated under numerous methods for estimating the ridge parameter. Monte Carlo simulation results indicate that the suggested estimator performs better than the MLE estimator in terms of MSE. Furthermore, a real chemometrics dataset application is utilized and the results demonstrate the excellent performance of the suggested estimator when the multicollinearity is present in IGR model.     

Restricted ridge estimator in the logistic regression model

It is known that when the multicollinearity exists in the logistic regression model, variance of maximum likelihood estimator is unstable. As a remedy, Schaefer et al. presented a ridge estimator in the logistic regression model. Making use of the ridge estimator, when some linear restrictions are also present, we introduce a restricted ridge estimator in the logistic regression model. Statistical properties of this newly defined estimator will be studied and comparisons are done in the simulation study in the sense of mean squared error criterion. A real-data example and a simulation study are introduced to discuss the performance of this estimator.     

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 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.     

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.