Stochastic Linear Restrictions Research Articles

A Stochastic Restricted Two-Parameter Estimator in Linear Regression Model

Özkale and Kaciranlar (2007) 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.

A new ridge-type estimator in stochastic restricted linear regression

In this paper, we propose a new ridge-type estimator called the weighted mixed ridge estimator by unifying the sample and prior information in linear model with additional stochastic linear restrictions. The new estimator is a generalization of the weighted mixed estimator [B. Schaffrin and H. Toutenburg, Weighted mixed regression, Zeitschrift fur Angewandte Mathematik und Mechanik 70 (1990), pp. 735–738] and ordinary ridge estimator (ORE) [A.E. Hoerl and R.W. Kennard, Ridge regression: Biased estimation for non-orthogonal problems, Technometrics 12 (1970), pp. 55–67]. The performances of this new estimator against the weighted mixed estimator, ORE and the mixed ridge estimator [Y.L. Li and H. Yang, A new stochastic mixed ridge estimator in linear regression, Stat. Pap. (2008) (in press, DOI 10.1007/s00362-008-0169-5)] are examined in terms of the mean squared error matrix sense. Finally, a numerical example and a Monte Carlo simulation are also given to show the theoretical results.

A new stochastic restricted biased estimator under heteroscedastic or correlated error

In this paper, under the linear regression model with heteroscedastic and/or correlated errors when the stochastic linear restrictions on the parameter vector are assumed to be held, a generalization of the ordinary mixed estimator (GOME), ordinary ridge regression estimator (GORR) and Generalized least squares estimator (GLSE) is proposed. The performance of this new estimator against GOME, GORR, GLS and the stochastic restricted Liu estimator (SRLE) [Yang and Xu, Statist. Papers 50 (2007) 639–647] are examined in terms of matrix mean square error criterion. A numerical example is considered to illustrate the theoretical results.

LIU-TYPE ESTIMATION FOR THE LINEAR MODEL WITH LINEAR RESTRICTIONS

For the multicollinearity in restricted linear regression,there are some problems still left open in current popular ordinary restricted least square estimation.By introducing stochastic linear restrictions,a restricted Liu-type estimation is proposed,which generalizes the ordinary restricted least square estimation and restricted ridge estimation.Then the selection method of the scalars k and d is disscussed and is generalized to the estimation of restricted linear model and canonical form.

A stochastic restricted ridge regression estimator

Groß [J. Groß, Restricted ridge estimation, Statistics & Probability Letters 65 (2003) 57–64] proposed a restricted ridge regression estimator when exact restrictions are assumed to hold. When there are stochastic linear restrictions on the parameter vector, we introduce a new estimator by combining ideas underlying the mixed and the ridge regression estimators under the assumption that the errors are not independent and identically distributed. Apart from [J. Groß, Restricted ridge estimation, Statistics & Probability Letters 65 (2003) 57–64], we call this new estimator as the stochastic restricted ridge regression (SRRR) estimator. The performance of the SRRR estimator over the mixed estimator in respect of the variance and the mean square error matrices is examined. We also illustrate our findings with a numerical example. The shrinkage generalized least squares (GLS) and the stochastic restricted shrinkage GLS estimators are proposed.

A new stochastic mixed ridge estimator in linear regression model

This paper is concerned with the parameter estimation in linear regression model with additional stochastic linear restrictions. To overcome the multicollinearity problem, a new stochastic mixed ridge estimator is proposed and its efficiency is discussed. Necessary and sufficient conditions for the superiority of the stochastic mixed ridge estimator over the ridge estimator and the mixed estimator in the mean squared error matrix sense are derived for the two cases in which the parametric restrictions are correct and are not correct. Finally, a numerical example is also given to show the theoretical results.

Consistent estimation of regression coefficients in ultrastructural measurement error model using stochastic prior information

The problem of consistent estimation of regression coefficients in a multivariate linear ultrastructural measurement error model is considered in this article when some additional information on regression coefficients is available a priori. Such additional information is expressible in the form of stochastic linear restrictions. Utilizing stochastic restrictions given a priori, some methodologies are presented to obtain the consistent estimators of regression coefficients under two types of additional information separately, viz., covariance matrix of measurement errors and reliability matrix associated with explanatory variables. The measurement errors are assumed to be not necessarily normally distributed. The asymptotic properties of the proposed estimators are derived and analyzed analytically as well as numerically through a Monte Carlo simulation experiment.

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.

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.

Mixed linear regression with equi-cross-correlated errors

This paper investigates an one-step procedure for the General Least Squares Estimation with an estimated covariance matrix (GLSE) in the Linear Regression Model with stochastic linear restrictions on the regression coefficients. The covariance matrix of the errors is assumed to be known up to the correlations between the errors of the observations and the errors of the restrictions. It is assumed that all of these correlations are equal. After transforming the model into its canonical form we show, that the GLSE coincides with the Ordinary Least Squares Estimation (OLS) of the complete model, with OLS after deleting one appropriate observation or with the OLS after replacing two appropriate observations by their arithmetic average. Furthermore, we calculate the relative efficiencies of these estimators with respect to the BLUE using the generalized variance criterion. The resulting estimator may be better than the OLS-estimator, what depends essentially on the diagonal elements of the hat matrix. An example demonstrating this fact is given. Additionally, a new test is proposed for the independence of the disturbances.

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.

Optimal estimation methods under weakened linear restrictions in regression

This paper deals with the linear regression model under additional stochastic linear restrictions when the dispersion matrix of the restriction disturbances is unknown and the traditional mixed model approach is not applicable. The concept of weak unbiasedness is introduced. Since the optimal weakly unbiased estimators, already mentioned in the previous sentence, are not operational, some replacement strategies for the unknown quantities are discussed.

Comparing Stochastically Restricted Estimators in a Linear Regression Model

AbstractThe minimum dispersion linear unbiased estimators of the vector of parameters in a linear regression model are compared when the parameters of the model are subject to stochastic linear restrictions with different dispersion matrices of the disturbances involved in them.

General instrumental variables estimation under stochastic linear restrictions

Abstract This letter extends the Theil-Goldberger ‘mixed’ regression estimator, for models subject to stochastic linear restrictions, to the case of stochastic regressors. A general instrumental variables ‘mixed’ estimator is discussed. The asymptotic distribution of the estimator is obtained, and an asymptotic test of the compatibility of the sample and prior information is presented.