A Simulation Study of Some Biasing Parameters for the Ridge Type Estimation of Poisson Regression

This article proposes several estimators for estimating the ridge parameter k based on Poisson ridge regression (RR) model. These estimators have been evaluated by means of Monte Carlo simulations. As performance criteria, we have calculated the mean squared error (MSE), the mean value, and the standard deviation of k. The first criterion is commonly used, while the other two have never been used when analyzing Poisson RR. However, these performance criteria are very informative because, if several estimators have an equal estimated MSE, then those with low average value and standard deviation of k should be preferred. Based on the simulated results, we may recommend some biasing parameters that may be useful for the practitioners in the field of health, social, and physical sciences.     

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.     

Evaluations of utility residential conservation programs in the Pacific Northwest: A critical review

Pacific Northwest utilities have sponsored the nation's earliest and most thorough residential conservation programs. The Northwest region also leads in the quality, quantity, and usefulness of utility-sponsored program evaluations. This article critically reviews the methods and findings of four major Northwest utility program evaluations. Recommendations for future evaluation and program management efforts also are discussed.     

Evaluating utility conservation programs: The BPA plan

This paper summarizes a detailed plan for evaluation of the residential energy conservation programs offered by the Bonneville Power Administration (BPA). The major purpose of the evaluation is to determine the energy savings attributable to the programs and to identify differences between program participants and nonparticipants. The focus of this paper is on the sampling issues associated with reliable estimates of energy savings. The methodology attempts to correct for the self-selection bias likely to occur in conservation program evaluation by using intentional nonrandom sampling. In addition, the paper suggests a variety of analytical approaches to analyzing the data with respect to energy savings.The proposed design, using four different groups to estimate program effects, is a comprehensive endeavor. This is appropriate for BPA because of the substantial investment it is making in conservation. However, the design is flexible and allows for use offewer groups to reduce evaluation costs.     

Tuning Parameter Selection and Various Good Fitting Characteristics for the Liu-Type Estimator in Linear Regression

Liu (2003 Liu , K. ( 2003 ). Using Liu-Type estimator to combat collinearity . Commun. Statist. Theor. Meth. 32 ( 5 ): 1009 – 1020 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) proposed the Liu-Type estimator (LTE) to combat the well-known multicollinearity problem in linear regression. In this article, various better fitting characteristics of the LTE than those of the ordinary ridge regression estimator (Hoerl and Kennard, 1970 Hoerl , A. E. , Kennard , R. W. ( 1970 ). Ridge regression: Biased estimation for non-orthogonal problems . Technometrics 12 : 55 – 67 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) are considered. In particular, we derived two methods to determine the parameter d for the LTE and find that the ridge parameter k could serve for regularization of an ill-conditioned design matrix, while the other parameter d could be used for tuning the fit quality. In addition, the coefficients of regression, coefficient of multiple determination, residual error variance, and generalized cross validation (GCV) of the prediction quality are very stable, and as the ridge parameter increases they eventually reach asymptotic levels, which produces robust regression models. Furthermore, a Monte Carlo evaluation of these features is also given to illustrate some of the theoretical results.     

An evaluation of ridge estimator in linear mixed models: an example from kidney failure data

This paper is concerned with the ridge estimation of fixed and random effects in the context of Henderson's mixed model equations in the linear mixed model. For this purpose, a penalized likelihood method is proposed. A linear combination of ridge estimator for fixed and random effects is compared to a linear combination of best linear unbiased estimator for fixed and random effects under the mean-square error (MSE) matrix criterion. Additionally, for choosing the biasing parameter, a method of MSE under the ridge estimator is given. A real data analysis is provided to illustrate the theoretical results and a simulation study is conducted to characterize the performance of ridge and best linear unbiased estimators approach in the linear mixed model.     

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.     

Performance of Kibria's Method for the Heteroscedastic Ridge Regression Model: Some Monte Carlo Evidence

It is common for a linear regression model that the error terms display some form of heteroscedasticity and at the same time, the regressors are also linearly correlated. Both of these problems have serious impact on the ordinary least squares (OLS) estimates. In the presence of heteroscedasticity, the OLS estimator becomes inefficient and the similar adverse impact can also be found on the ridge regression estimator that is alternatively used to cope with the problem of multicollinearity. In the available literature, the adaptive estimator has been established to be more efficient than the OLS estimator when there is heteroscedasticity of unknown form. The present article proposes the similar adaptation for the ridge regression setting with an attempt to have more efficient estimator. Our numerical results, based on the Monte Carlo simulations, provide very attractive performance of the proposed estimator in terms of efficiency. Three different existing methods have been used for the selection of biasing parameter. Moreover, three different distributions of the error term have been studied to evaluate the proposed estimator and these are normal, Student's t and F distribution.     

An Application of the Local Influence Diagnostics to Ridge Regression Under Elliptical Model

In this article, we study the effect of a minor perturbation on the ridge estimator considering the elliptical distribution for the errors. The necessary matrices for assessing the local influence under the perturbation of the explanatory variables and the scale matrix are derived. The Longley data is analyzed for illustration.     

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.