Generalized preliminary test stochastic restricted estimator in the linear regression model

ABSTRACT

In this paper, we propose three generalized estimators, namely, generalized unrestricted estimator (GURE), generalized stochastic restricted estimator (GSRE), and generalized preliminary test stochastic restricted estimator (GPTSRE). The GURE can be used to represent the ridge estimator, almost unbiased ridge estimator (AURE), Liu estimator, and almost unbiased Liu estimator. When stochastic restrictions are available in addition to the sample information, the GSRE can be used to represent stochastic mixed ridge estimator, stochastic restricted Liu estimator, stochastic restricted almost unbiased ridge estimator, and stochastic restricted almost unbiased Liu estimator. The GPTSRE can be used to represent the preliminary test estimators based on mixed estimator. Using the GPTSRE, the properties of three other preliminary test estimators, namely preliminary test stochastic mixed ridge estimator, preliminary test stochastic restricted almost unbiased Liu estimator, and preliminary test stochastic restricted almost unbiased ridge estimator can also be discussed. The mean square error matrix criterion is used to obtain the superiority conditions to compare the estimators based on GPTSRE with some biased estimators for the two cases for which the stochastic restrictions are correct, and are not correct. Finally, a numerical example and a Monte Carlo simulation study are done to illustrate the theoretical findings of the proposed estimators.     

A weighted stochastic restricted ridge estimator in partially linear model

In this article, we consider the estimation of a partially linear model when stochastic linear restrictions on the parameter components are assumed to hold. Based on the weighted mixed estimator, profile least-squares method, and ridge method, a weighted stochastic restricted ridge estimator of the parametric component is introduced. The properties of the new estimator are also discussed. Finally, a simulation study is given to show the performance of the new estimator.     

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.     

Local Influence Analysis for The Ridge Regression Under Stochastic Linear Restrictions

In this article, we assess the local influence for the ridge regression of linear models with stochastic linear restrictions in the spirit of Cook by using the log-likelihood of the stochastic restricted ridge regression estimator. The diagnostics under the perturbations of constant variance, responses and individual explanatory variables are derived. We also assess the local influence of the stochastic restricted ridge regression estimator under the approach suggested by Billor and Loynes. At the end, a numerical example on the Longley data is given to illustrate the theoretic results.     

Estimation in Singular Linear Models with Stochastic Linear Restrictions and Linear Equality Restrictions

This article is concerned with the parameter estimation in a singular linear regression model with stochastic linear restrictions and linear equality restrictions simultaneously. A new estimator is introduced and it is proved that the proposed estimator is superior to the least squares estimator and singular mixed estimator in the mean squared error sense under certain conditions.     

A stochastic restricted k–d class estimator

In this paper, we introduce a stochastic restricted k–d class estimator for the vector of parameters in a linear model when additional linear restrictions on the parameter vector are assumed to hold. The stochastic restricted k–d class estimator is a generalization of the ordinary mixed estimator and the k–d class estimator. We show that our new biased estimator is superior in the mean squared error matrix sense to the k–d class estimator [S. Sakall?o?lu and S. Kaçiranlar, A new biased estimator based on ridge estimation, Statist. Papers 49 (2008), pp. 669–689] and the stochastic restricted Liu estimator [H. Yang and J.W. Xu, An alternative stochastic restricted Liu estimator in linear regression, Statist. Papers 50 (2009), pp. 639–647]. Finally, a numerical example is given to show the theoretical results.     

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.     

Econometric Reviews

This paper considers the problem of estimating a nonlinear statistical model subject to stochastic linear constraints among unknown parameters. These constraints represent prior information which originates from a previous estimation of the same model using an alternative database. One feature of this specification allows for the disign matrix of stochastic linear restrictions to be estimated. The mixed regression technique and the maximum likelihood approach are used to derive the estimator for both the model coefficients and the unknown elements of this design matrix. The proposed estimator whose asymptotic properties are studied, contains as a special case the conventional mixed regression estimator based on a fixed design matrix. A new test of compatibility between prior and sample information is also introduced. Thesuggested estimator is tested empirically with both simulated and actual marketing data.     

Generalized mixed estimator for nonlinear models: a maximum likelihood approach

This paper considers the problem of estimating a nonlinear statistical model subject to stochastic linear constraints among unknown parameters. These constraints represent prior information which originates from a previous estimation of the same model using an alternative database. One feature of this specification allows for the disign matrix of stochastic linear restrictions to be estimated. The mixed regression technique and the maximum likelihood approach are used to derive the estimator for both the model coefficients and the unknown elements of this design matrix. The proposed estimator whose asymptotic properties are studied, contains as a special case the conventional mixed regression estimator based on a fixed design matrix. A new test of compatibility between prior and sample information is also introduced. Thesuggested estimator is tested empirically with both simulated and actual marketing data.     

Estimation in a linear regression model with stochastic linear restrictions: a new two-parameter-weighted mixed estimator

The present paper considers the weighted mixed regression estimation of the coefficient vector in a linear regression model with stochastic linear restrictions binding the regression coefficients. We introduce a new two-parameter-weighted mixed estimator (TPWME) by unifying the weighted mixed estimator of Schaffrin and Toutenburg [1] and the two-parameter estimator (TPE) of Özkale and Kaç?ranlar [2]. This new estimator is a general estimator which includes the weighted mixed estimator, the TPE and the restricted two-parameter estimator (RTPE) proposed by Özkale and Kaç?ranlar [2] as special cases. Furthermore, we compare the TPWME with the weighted mixed estimator and the TPE with respect to the matrix mean square error criterion. A numerical example and a Monte Carlo simulation experiment are presented by using different estimators of the biasing parameters to illustrate some of the theoretical results.     

Ridge Estimation under the Stochastic Restriction

In linear programming and modeling of an economic system, there may occur some linear stochastic artificial or unnatural manners, which may need serious attentions. These stochastic unusual uncertainty, say stochastic constraints, definitely cause some changes in the estimators under work and their behaviors. In this approach, we are basically concerned with the problem of multicollinearity, when it is suspected that the parameter space may be restricted to some stochastic restrictions. We develop the estimation strategy form unbiasedness to some improved biased adjustment. In this regard, we study the performance of shrinkage estimators under the assumption of elliptically contoured errors and derive the region of optimality of each one. Lastly, a numerical example is taken to determine the adequate ridge parameter for each given estimator.     

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.     

Estimation in Singular Linear Models with Stochastic Linear Restrictions

This article generalizes the ordinary mixed estimator (OME) in theory, and obtains the estimator of the unknown regression parameters in singular linear models with stochastic linear restrictions: singular mixed estimator (SME). We also give some properties of SME obtained in this article, and prove that it is superior to unrestricted least squared estimator (LSE) in singular linear models in the sense of the covariance matrix and generalized mean square error (GMSE). After that, we also have a discussion about the two-stage estimator of SME. The result we give in this article could be regarded as generalizations of both OME and unrestricted LSE at the same time.     

Semiparametric Ridge Regression Approach in Partially Linear Models

In this article, we introduce a semiparametric ridge regression estimator for the vector-parameter in a partial linear model. It is also assumed that some additional artificial linear restrictions are imposed to the whole parameter space and the errors are dependent. This 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. Asymptotic distributional bias and risk are also derived and the comparison result is then given.     

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.     

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.     

Performance analysis of the preliminary test estimator with series of stochastic restrictions

In this paper, the problem of estimation of the regression coefficients in a multiple regression model is considered under the multicollinearity situation when there are series of stochastic linear restrictions available on the regression parameter vector. We have considered the preliminary test ridge regression estimators (PTRREs) based on the Wald, likelihood ratio, and lagrangian multiplier tests. Tables for the maximum and minimum guaranteed efficiency of the PTRREs are obtained, which allow us to determine the optimum choice of the level of significance corresponding to the optimum estimator. Some numerical results support the findings.     

Restricted Ridge Estimators of the Parameters in Semiparametric Regression Model

In this article, we introduce a ridge estimator for the vector of parameters β in a semiparametric model when additional linear restrictions on the parameter vector are assumed to hold. We also obtain the semiparametric restricted ridge estimator for the parametric component in the semiparametric regression model. The ideas in this article are illustrated with a data set consisting of housing prices and through a comparison of the performances of the proposed and related estimators via a Monte Carlo simulation.     

An alternative stochastic restricted Liu estimator in linear regression

In this paper, we introduce an alternative stochastic restricted Liu estimator for the vector of parameters in a linear regression model when additional stochastic linear restrictions on the parameter vector are assumed to hold. The new estimator is a generalization of the ordinary mixed estimator (OME) (Durbin in J Am Stat Assoc 48:799–808, 1953; Theil and Goldberger in Int Econ Rev 2:65–78, 1961; Theil in J Am Stat Assoc 58:401–414, 1963) and Liu estimator proposed by Liu (Commun Stat Theory Methods 22:393–402, 1993). Necessary and sufficient conditions for the superiority of the new stochastic restricted Liu estimator over the OME, the Liu estimator and the estimator proposed by Hubert and Wijekoon (Stat Pap 47:471–479, 2006) in the mean squared error matrix (MSEM) sense are derived. Furthermore, a numerical example based on the widely analysed dataset on Portland cement (Woods et al. in Ind Eng Chem 24:1207–1241, 1932) and a Monte Carlo evaluation of the estimators are also given to illustrate some of the theoretical results.     

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.