Modified Restricted Almost Unbiased Liu Estimator in Linear Regression Model

In this article we introduce a modified restricted almost unbiased Liu estimator in linear regression model which satisfies the linear restrictions. The mean squared error matrix (MSEM) of the proposed estimator is derived and compared with the corresponding competitors in literature. Finally, a numerical example and a Monte Carlo simulation are given to illustrate some of the theoretical 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.     

Linearized Restricted Ridge Regression Estimator in Linear Regression

This article primarily aims to put forward the linearized restricted ridge regression (LRRR) estimator in linear regression models. Two types of LRRR estimators are investigated under the PRESS criterion and the optimal LRRR estimators and the optimal restricted generalized ridge regression estimator are obtained. We apply the results to the Hald data and finally make a simulation study by using the method of McDonald and Galarneau.     

Two Stochastic Restricted Principal Components Regression Estimator in Linear Regression

In this article, we propose two stochastic restricted principal components regression estimator by combining the approach followed in obtaining the ordinary mixed estimator and the principal components regression estimator in linear regression model. The performance of the two new estimators in terms of matrix MSE criterion is studied. We also give an example and a Monte Carlo simulation to show the theoretical results.     

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.     

A New Two-Parameter Estimator in Linear Regression

This article is concerned with the parameter estimation in linear regression model. To overcome the multicollinearity problem, a new two-parameter estimator is proposed. This new estimator is a general estimator which includes the ordinary least squares (OLS) estimator, the ridge regression (RR) estimator, and the Liu estimator as special cases. Necessary and sufficient conditions for the superiority of the new estimator over the OLS, RR, Liu estimators, and the two-parameter estimator proposed by Ozkale and Kaciranlar (2007 Ozkale , M. R. , Kaciranlar , S. ( 2007 ). The restricted and unrestricted two-parameter estimators . Commun. Statist. Theor. Meth. 36 : 2707 – 2725 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) in the mean squared error matrix (MSEM) sense are derived. Furthermore, we obtain the estimators of the biasing parameters and give a numerical example to illustrate some of the theoretical results.     

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 Stochastic Restricted Ridge Regression Estimator

In this article, we introduce a new stochastic restricted estimator for the unknown vector parameter in the linear regression model when stochastic linear restrictions on the parameters hold. We show that the new estimator is a generalization of the ordinary mixed estimator (OME), Liu estimator (LE), ordinary ridge estimator (ORR), (k-d) class estimator, stochastic restricted Liu estimator (SRLE), and stochastic restricted ridge estimator (SRRE). Performance of the new estimator in comparison to other estimators in terms of the mean squares error matrix (MMSE) is examined. Numerical example from literature have been given to illustrate the results.     

A Stochastic Restricted Two-Parameter Estimator in Linear Regression Model

Özkale and Kaciranlar (2007 Özakle , M. R. , Kaciranlar , S. ( 2007 ). The restricted and unrestricted two-parameter estimators . Commun. Statist. Theor. Meth. 36 : 2707 – 2725 . [Google Scholar]) proposed a two-parameter estimator (TPE) for the unknown parameter vector in linear regression when exact restrictions are assumed to hold. In this article, under the assumption that the errors are not independent and identically distributed, we introduce a new estimator by combining the ideas underlying the mixed estimator (ME) and the two-parameter estimator when stochastic linear restrictions are assumed to hold. The new estimator is called the stochastic restricted two-parameter estimator (SRTPE) and necessary and sufficient conditions for the superiority of the SRTPE over the ME and TPE are derived by the mean squared error matrix (MSEM) criterion. Furthermore, selection of the biasing parameters is discussed and a numerical example is given to illustrate some of the theoretical results.     

On Second Order Admissibilities in Two-Parameter Logistic Regression Model

In two-parameter family of distribution, conditions for a modified maximum likelihood estimator to be second-order admissible are given. Applying these results to two-parameter logistic regression model, it is shown that the maximum likelihood estimator is always second-order inadmissible and the Rao-Blackwellized minimum logit chi-squared estimator is second-order admissible if and only if the number of the doses is greater than or equal to 6.     

Linearized Ridge Regression Estimator in Linear Regression

In this article, we aim to study the linearized ridge regression (LRR) estimator in a linear regression model motivated by the work of Liu (1993). The LRR estimator and the two types of generalized Liu estimators are investigated under the PRESS criterion. The method of obtaining the optimal generalized ridge regression (GRR) estimator is derived from the optimal LRR estimator. We apply the Hald data as a numerical example and then make a simulation study to show the main results. It is concluded that the idea of transforming the GRR estimator as a complicated function of the biasing parameters to a linearized version should be paid more attention in the future.     

Improvement of the Liu Estimator in Weighted Mixed Regression

ABSTRACT

The maximum likelihood approach to the proportional hazards model is considered. The purpose is to find a general approach to the analysis of the proportional hazards model, whether the baseline distribution is absolutely continuous, discrete, or a mixture. The advantage is that ties are treated without pain, while the performance for continuous data is almost the same as Cox's partial likelihood. The potential disadvantage with many nuisance parameters is taken care of by profiling them out for risk sets containing only one failure.     

On the performance of some new Liu parameters for the gamma regression model

The maximum likelihood (ML) method is used to estimate the unknown Gamma regression (GR) coefficients. In the presence of multicollinearity, the variance of the ML method becomes overstated and the inference based on the ML method may not be trustworthy. To combat multicollinearity, the Liu estimator has been used. In this estimator, estimation of the Liu parameter d is an important problem. A few estimation methods are available in the literature for estimating such a parameter. This study has considered some of these methods and also proposed some new methods for estimation of the d. The Monte Carlo simulation study has been conducted to assess the performance of the proposed methods where the mean squared error (MSE) is considered as a performance criterion. Based on the Monte Carlo simulation and application results, it is shown that the Liu estimator is always superior to the ML and recommendation about which best Liu parameter should be used in the Liu estimator for the GR model is given.     

An improved and efficient biased estimation technique in logistic regression model

Abstract

In this article, we propose a new improved and efficient biased estimation method which is a modified restricted Liu-type estimator satisfying some sub-space linear restrictions in the binary logistic regression model. We study the properties of the new estimator under the mean squared error matrix criterion and our results show that under certain conditions the new estimator is superior to some other estimators. Moreover, a Monte Carlo simulation study is conducted to show the performance of the new estimator in the simulated mean squared error and predictive median squared errors sense. Finally, a real application is considered.     

Linearized Ridge Regression Estimator Under the Mean Squared Error Criterion in a Linear Regression Model

This article mainly aims to study the superiority of the notion of linearized ridge regression estimator (LRRE) under the mean squared error criterion in a linear regression model. Firstly, we derive uniform lower bound of MSE for the class of the generalized shrinkage estimator (GSE), based on which it is shown that the optimal LRRE is the best estimator in the class of GSE's. Secondly, we propose the notion of the almost unbiased completeness and show that LRRE possesses such a property. Thirdly, the simulation study is given, from which it indicates that the LRRE performs desirably. Finally, the main results are applied to the well known Hald data.     

On the predictive performance of the almost unbiased Liu estimator

ABSTRACT

Regression models are usually used in forecasting (predicting) unknown values of the response variable y. This article considers the predictive performance of the almost unbiased Liu estimator compared to the ordinary least-squares estimator, principal component regression estimator, and Liu estimator. Finally, we present a numerical example to explain the theoretical results and we obtain a region where the almost unbiased Liu estimator is uniformly superior to the ordinary least-squares estimator, principal component regression estimator, and Liu estimator.     

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.     

New Ridge Regression Estimator in Semiparametric Regression Models

In the context of ridge regression, the estimation of shrinkage parameter plays an important role in analyzing data. Many efforts have been put to develop the computation of risk function in different full-parametric ridge regression approaches using eigenvalues and then bringing an efficient estimator of shrinkage parameter based on them. In this respect, the estimation of shrinkage parameter is neglected for semiparametric regression model. Not restricted, but the main focus of this approach is to develop necessary tools for computing the risk function of regression coefficient based on the eigenvalues of design matrix in semiparametric regression. For this purpose the differencing methodology is applied. We also propose a new estimator for shrinkage parameter which is of harmonic type mean of ridge estimators. It is shown that this estimator performs better than all the existing ones for the regression coefficient. For our proposal, a Monte Carlo simulation study and a real dataset analysis related to housing attributes are conducted to illustrate the efficiency of shrinkage estimators based on the minimum risk and mean squared error criteria.     

r-d Class Estimator Under Misspecification

Omission of some relevant explanatory variables and multicollinearity in regression models are very serious problems in applied works. There are some papers examining the multicollinearity and misspecification which is due to omission of some relevant explanatory variables, concurrently. To remedy the problem of multicollinearity, Kaç?ranlar and Sakall?o?lu (2001 Kaç?ranlar, S., Sakall?o?lu, S. (2001). Combining the Liu estimator and the principal component regression estimator. Commun. Stat. Theory Methods. 30:2699–2705.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) proposed the r-d class estimator that includes the ordinary least squares, principal components regression, and Liu estimators as special cases. The aim of this paper is to examine the performance of the r-d class estimator in misspecificied linear models.