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.     

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.     

Seemingly Unrelated Ridge Regression in Semiparametric Models

This article is concerned with the problem of multicollinearity in the linear part of a seemingly unrelated semiparametric (SUS) model. It is also suspected that some additional non stochastic linear constraints hold on the whole parameter space. In the sequel, we propose semiparametric ridge and non ridge type estimators combining the restricted least squares methods in the model under study. For practical aspects, it is assumed that the covariance matrix of error terms is unknown and thus feasible estimators are proposed and their asymptotic distributional properties are derived. Also, necessary and sufficient conditions for the superiority of the ridge-type estimator over the non ridge type estimator for selecting the ridge parameter K are derived. Lastly, a Monte Carlo simulation study is conducted to estimate the parametric and nonparametric parts. In this regard, kernel smoothing and cross validation methods for estimating the nonparametric function are used.     

On the ridge regression estimator with sub-space restriction

In the linear regression model with elliptical errors, a shrinkage ridge estimator is proposed. In this regard, the restricted ridge regression estimator under sub-space restriction is improved by incorporating a general function which satisfies Taylor’s series expansion. Approximate quadratic risk function of the proposed shrinkage ridge estimator is evaluated in the elliptical regression model. A Monte Carlo simulation study and analysis based on a real data example are considered for performance analysis. It is evident from the numerical results that the shrinkage ridge estimator performs better than both unrestricted and restricted estimators in the multivariate t-regression model, for some specific cases.     

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.     

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.     

Preliminary testing of the Cobb–Douglas production function and related inferential issues

In this article, we consider the multiple regression model in the presence of multicollinearity and study the performance of the preliminary test estimator (PTE) both analytically and computationally, when it is a priori suspected that some constraints may hold on the vector parameter space. The performance of the PTE is further analyzed by comparing the risk of some well-known estimators of the ridge parameter through an extensive Monte Carlo simulation study under some bounded and or asymmetric loss functions. An application of the Cobb–Douglas production function is included and from these results as well as the simulation studies, it is clear that the bounded linear exponential loss function outperforms the other loss functions across all the proposed ridge parameters by comparing the risk values.     

Selection of the Ridge Parameter Using Mathematical Programming

Ridge regression solves multicollinearity problems by introducing a biasing parameter that is called ridge parameter; it shrinks the estimates and their standard errors in order to reach acceptable results. Selection of the ridge parameter was done using several subjective and objective techniques that are concerned with certain criteria. In this study, selection of the ridge parameter depends on other important statistical measures to reach a better value of the ridge parameter. The proposed ridge parameter selection technique depends on a mathematical programming model and the results are evaluated using a simulation study. The performance of the proposed method is good when the error variance is greater than or equal to one; the sample consists of 20 observations, the number of explanatory variables in the model is 2, and there is a very strong correlation between the two explanatory variables.     

Ridge regression methodology in partial linear models with correlated errors

In this article, a generalized restricted difference-based ridge estimator is defined for the vector parameter in a partial linear model when the errors are dependent. It is suspected that some additional linear constraints may hold on to the whole parameter space. The estimator is a generalization of the well-known restricted least-squares estimator and is confined to the (affine) subspace which is generated by the restrictions. The risk functions of the proposed estimators are derived under balanced loss function. Finally, the performance of the new estimators is evaluated by a simulated data set.     

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.     

A simulation study of deterministic ridge estimators

Some deterministic ridge rules are proposed and their finite sample properties are studied. Further, a simulation study is also conducted. Based on the simulation results, the proposed ridge estimators can improve the mean squared error over the least squares estimator, provided that the condition number of correlation matrices in the regression model is large, say at least 1,000.     

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.     

Nonparametric Shrinkage Estimation for Aalen's Additive Hazards Model

Aalen's nonparametric additive model in which the regression coefficients are assumed to be unspecified functions of time is a flexible alternative to Cox's proportional hazards model when the proportionality assumption is in doubt. In this paper, we incorporate a general linear hypothesis into the estimation of the time‐varying regression coefficients. We combine unrestricted least squares estimators and estimators that are restricted by the linear hypothesis and produce James‐Stein‐type shrinkage estimators of the regression coefficients. We develop the asymptotic joint distribution of such restricted and unrestricted estimators and use this to study the relative performance of the proposed estimators via their integrated asymptotic distributional risks. We conduct Monte Carlo simulations to examine the relative performance of the estimators in terms of their integrated mean square errors. We also compare the performance of the proposed estimators with a recently devised LASSO estimator as well as with ridge‐type estimators both via simulations and data on the survival of primary billiary cirhosis patients.     

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.     

Local influence in ridge semiparametric models

ABSTRACT

Ridge penalized least-squares estimators has been suggested as an alternative to the minimum penalized sum of squares estimates in the presence of collinearity among the explanatory variables in semiparametric regression models (SPRMs). This paper studies the local influence of minor perturbations on the ridge estimates in the SPRM. The diagnostics under the perturbation of ridge penalized sum of squares, response variable, explanatory variables and ridge parameter are considered. Some local influence diagnostics are given. A Monte Carlo simulation study and a real example are used to illustrate the proposed perturbations.     

On ridge parameter estimators under stochastic subspace hypothesis

This paper considers several estimators for estimating the restricted ridge parameter estimators. A simulation study has been conducted to compare the performance of these estimators. Based on the simulation study we found that, increasing the correlation between the independent variables has positive effect on the mean square error (MSE). However, increasing the value of ρ has negative effect on MSE. When the sample size increases, the MSE decreases even when the correlation between the independent variables is large. Two real life examples have been considered to illustrate the performance of the estimators.     

Fractional principal components regression: a general approach to biased estimators

Several biased estimators have been proposed as alternatives to the least squares estimator when multicollinearity is present in the multiple linear regression model. The ridge estimator and the principal components estimator are two techniques that have been proposed for such problems. In this paper the class of fractional principal component estimators is developed for the multiple linear regression model. This class contains many of the biased estimators commonly used to combat multicollinearity. In the fractional principal components framework, two new estimation techniques are introduced. The theoretical performances of the new estimators are evaluated and their small sample properties are compared via simulation with the ridge, generalized ridge and principal components estimators     

New modified moment estimators for the two-parameter weighted Lindley distribution

We propose a modification of the moment estimators for the two-parameter weighted Lindley distribution. The modification replaces the second sample moment (or equivalently the sample variance) by a certain sample average which is bounded on the unit interval for all values in the sample space. In this method, the estimates always exist uniquely over the entire parameter space and have consistency and asymptotic normality over the entire parameter space. The bias and mean squared error of the estimators are also examined by means of a Monte Carlo simulation study, and the empirical results show the small-sample superiority in addition to the desirable large sample properties. Monte Carlo simulation study showed that the proposed modified moment estimators have smaller biases and smaller mean-square errors than the existing moment estimators and are compared favourably with the maximum likelihood estimators in terms of bias and mean-square error. Three illustrative examples are finally presented.     

Two-stage Liu estimator in a simultaneous equations model

Two-stage least squares estimation in a simultaneous equations model has several desirable properties under the problem of multicollinearity. So, various kinds of improved estimation techniques can be developed to deal with the problem of multicollinearity. One of them is ridge regression estimation that can be applied at both stages and defined in Vinod and Ullah [Recent advances in regression methods. New York: Marcel Dekker; 1981]. We propose three different kinds of Liu estimators that are named by their implementation stages. Mean square errors are derived to compare the performances of the mentioned estimators and two different choices of the biasing parameter are offered. Moreover, a numerical example is given with a data analysis based on the Klein Model I and a Monte Carlo experiment is conducted.     

Empirical likelihood estimation for linear regression models with AR(p) error terms with numerical examples

Linear regression models are useful statistical tools to analyze data sets in different fields. There are several methods to estimate the parameters of a linear regression model. These methods usually perform under normally distributed and uncorrelated errors. If error terms are correlated the Conditional Maximum Likelihood (CML) estimation method under normality assumption is often used to estimate the parameters of interest. The CML estimation method is required a distributional assumption on error terms. However, in practice, such distributional assumptions on error terms may not be plausible. In this paper, we propose to estimate the parameters of a linear regression model with autoregressive error term using Empirical Likelihood (EL) method, which is a distribution free estimation method. A small simulation study is provided to evaluate the performance of the proposed estimation method over the CML method. The results of the simulation study show that the proposed estimators based on EL method are remarkably better than the estimators obtained from CML method in terms of mean squared errors (MSE) and bias in almost all the simulation configurations. These findings are also confirmed by the results of the numerical and real data examples.