Choice of the ridge factor from the correlation matrix determinant

Ridge regression is the alternative method to ordinary least squares, which is mostly applied when a multiple linear regression model presents a worrying degree of collinearity. A relevant topic in ridge regression is the selection of the ridge parameter, and different proposals have been presented in the scientific literature. Since the ridge estimator is biased, its estimation is normally based on the calculation of the mean square error (MSE) without considering (to the best of our knowledge) whether the proposed value for the ridge parameter really mitigates the collinearity. With this goal and different simulations, this paper proposes to estimate the ridge parameter from the determinant of the matrix of correlation of the data, which verifies that the variance inflation factor (VIF) is lower than the traditionally established threshold. The possible relation between the VIF and the determinant of the matrix of correlation is also analysed. Finally, the contribution is illustrated with three real examples.     

Collinearity diagnostic applied in ridge estimation through the variance inflation factor

The variance inflation factor (VIF) is used to detect the presence of linear relationships between two or more independent variables (i.e. collinearity) in the multiple linear regression model. However, the traditionally used VIF definitions encounter some problems when extended to the case of the ridge estimation (RE). This paper presents an extension of the VIF in RE by providing two alternative VIF expressions that overcome these problems in the general case. Some characteristics of these expressions are also presented and compared with the traditional expression. The results are illustrated with an economic example in the case of three independent variables and with a Monte Carlo simulation for the general case.     

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.     

Confidence Interval for Shrinkage Parameters in Ridge Regression

In ridge regression, the estimation of ridge parameter k is an important problem. There are several methods available in the literature to do this job some what efficiently. However, no attempts were made to suggest a confidence interval for the ridge parameter using the knwoledge from the data. In this article, we propose a data dependent confidence interval for the ridge parameter k. The method of obtaining the confidence interval is illustrated with the help of a data set. A simulation study indicates that the empirical coverage probability of the suggested confidence intervals are quite high.     

Cox regression analysis in presence of collinearity: an application to assessment of health risks associated with occupational radiation exposure

This paper considers the analysis of time to event data in the presence of collinearity between covariates. In linear and logistic regression models, the ridge regression estimator has been applied as an alternative to the maximum likelihood estimator in the presence of collinearity. The advantage of the ridge regression estimator over the usual maximum likelihood estimator is that the former often has a smaller total mean square error and is thus more precise. In this paper, we generalized this approach for addressing collinearity to the Cox proportional hazards model. Simulation studies were conducted to evaluate the performance of the ridge regression estimator. Our approach was motivated by an occupational radiation study conducted at Oak Ridge National Laboratory to evaluate health risks associated with occupational radiation exposure in which the exposure tends to be correlated with possible confounders such as years of exposure and attained age. We applied the proposed methods to this study to evaluate the association of radiation exposure with all-cause mortality.     

Using Liu-Type Estimator to Combat Collinearity

Abstract

Linear regression model and least squares method are widely used in many fields of natural and social sciences. In the presence of collinearity, the least squares estimator is unstable and often gives misleading information. Ridge regression is the most common method to overcome this problem. We find that when there exists severe collinearity, the shrinkage parameter selected by existing methods for ridge regression may not fully address the ill conditioning problem. To solve this problem, we propose a new two-parameter estimator. We show using both theoretic results and simulation that our new estimator has two advantages over ridge regression. First, our estimator has less mean squared error (MSE). Second, our estimator can fully address the ill conditioning problem. A numerical example from literature is used to illustrate the results.     

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.     

Tests of regression coefficients under ridge regression models

This paper investigates two “non-exact” t-type tests, t( k₂) and t(k₂), of the individual coefficients of a linear regression model, based on two ordinary ridge estimators. The reported results are built on a simulation study covering 84 different models. For models with large standard errors, the ridge-based t-tests have correct levels with considerable gain in powers over those of the least squares t-test, t(0). For models with small standard errors, t(k₁) is found to be liberal and is not safe to use while, t(k₂) is found to slightly exceed the nominal level in few cases. When tie two ridge tests art: not winners, the results indicate that they don't loose much against t(0).     

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.     

Developing Ridge Parameters for SUR Model

This paper proposes a number of procedures for developing new biased estimators of the seemingly unrelated regression (SUR) parameters, when the explanatory variables are affected by multicollinearity. Several ridge parameters are proposed and then compared in terms of the trace mean squared error (TMSE) and (PR) criteria. The PR criterion is the proportion of replication (out of 1,000) for which the SUR version of the generalized least squares (SGLS) estimator has a smaller TMSE than others. The study was performed using Monte Carlo simulations where the number of equations in the system, the number of observations, the correlation among equations, and the correlation between explanatory variables have been varied. For each model, we performed 1,000 replications. Our results show that under certain conditions some of the proposed SUR ridge parameters, (R _Sgeom , R _Skmed , R _Sqarith , and R _Sqmax ), performed well when compared, in terms of TMSE and PR criteria, with other proposed and popular existing ridge parameters. In large samples and when the collinearity between the explanatory variables is not high, the unbiased SUR estimator (SGLS), performed better than the other ridge parameters.     

Comparisons of the <Emphasis Type="Italic">r</Emphasis> − <Emphasis Type="Italic">k</Emphasis> class estimator to the ordinary least squares estimator under the Pitman’s closeness criterion

In the presence of multicollinearity, the r − k class estimator is proposed as an alternative to the ordinary least squares (OLS) estimator which is a general estimator including the ordinary ridge regression (ORR), the principal components regression (PCR) and the OLS estimators. Comparison of competing estimators of a parameter in the sense of mean square error (MSE) criterion is of central interest. An alternative criterion to the MSE criterion is the Pitman’s (1937) closeness (PC) criterion. In this paper, we compare the r − k class estimator to the OLS estimator in terms of PC criterion so that we can get the comparison of the ORR estimator to the OLS estimator under the PC criterion which was done by Mason et al. (1990) and also the comparison of the PCR estimator to the OLS estimator by means of the PC criterion which was done by Lin and Wei (2002).     

On Ridge Parameters in Logistic Regression

This article applies and investigates a number of logistic ridge regression (RR) parameters that are estimable by using the maximum likelihood (ML) method. By conducting an extensive Monte Carlo study, the performances of ML and logistic RR are investigated in the presence of multicollinearity and under different conditions. The simulation study evaluates a number of methods of estimating the RR parameter k that has recently been developed for use in linear regression analysis. The results from the simulation study show that there is at least one RR estimator that has a lower mean squared error (MSE) than the ML method for all the different evaluated situations.     

Ridge estimation in regression problems with autocorrelated errors: A monte carlo study

This paper presents the results of a Monte Carlo study of OLS and GLS based adaptive ridge estimators for regression problems in which the independent variables are collinear and the errors are autocorrelated. It studies the effects of degree of collinearity, magnitude of error variance, orientation of the parameter vector and serial correlation of the independent variables on the mean squared error performance of these estimators. Results suggest that such estimators produce greatly improved performance in favorable portions of the parameter space. The GLS based methods are best when the independent variables are also serially correlated.     

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.     

Positive-rule stein-type almost unbiased ridge estimator in linear regression model

Abstract

In this article, when it is suspected that regression coefficients may be restricted to a subspace, we discuss the parameter estimation of regression coefficients in a multiple regression model. Then, in order to improve the preliminary test almost ridge estimator, we study the positive-rule Stein-type almost unbiased ridge estimator based on the positive-rule stein-type shrinkage estimator and almost unbiased ridge estimator. After that, quadratic bias and quadratic risk values of the new estimator are derived and compared with some relative estimators. And we also discuss the option of parameter k. Finally, we perform a real data example and a Monte Carlo study to illustrate theoretical results.     

Simultaneous Confidence Bands for Linear Regression with Covariates Constrained in Intervals

Abstract. The focus of this article is on simultaneous confidence bands over a rectangular covariate region for a linear regression model with k>1 covariates, for which only conservative or approximate confidence bands are available in the statistical literature stretching back to Working & Hotelling (J. Amer. Statist. Assoc. 24 , 1929; 73–85). Formulas of simultaneous confidence levels of the hyperbolic and constant width bands are provided. These involve only a k‐dimensional integral; it is unlikely that the simultaneous confidence levels can be expressed as an integral of less than k‐dimension. These formulas allow the construction for the first time of exact hyperbolic and constant width confidence bands for at least a small k(>1) by using numerical quadrature. Comparison between the hyperbolic and constant width bands is then addressed under both the average width and minimum volume confidence set criteria. It is observed that the constant width band can be drastically less efficient than the hyperbolic band when k>1. Finally it is pointed out how the methods given in this article can be applied to more general regression models such as fixed‐effect or random‐effect generalized linear regression models.     

Unbiased ridge estimation with prior information and ridge trace

A procedure is illustrated to incorporate prior information in the ridge regression model. Unbiased ridge estimators with prior information are defined and a robust estimate of the ridge parameter k is proposed.     

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.     

Stability of the inverse correlation matrix. Partial ridge regression

In this paper we evaluate the stability of the inverse of a correlation matrix by studying the derivatives of each of its entries with respect to each entry of the correlation matrix. From them we deduce the derivatives of the squared length of the inverse matrix, the variance inflation factors (VIF), and the regression coefficients. To illustrate the procedure, we use a correlation matrix that has already been analyzed by Hoerl and Kennard (1970, Technometrics 12, 69–82), and, by looking at the derivatives of the squared length of the regression vector, we show that the addition of a constant to some of the diagonal entries of the matrix is sufficient for obtaining satisfying estimates of the regression coefficients. This ‘partial ridge regression’ is carried out on the previous matrix and modifies only the coefficients which are perturbed by the collinearity.     

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.