Comment on a generalized stochastic restricted ridge regression estimator

In this note, we make some comments about the paper of Alheety and Kibria (2014 Alheety, M.I., Kibria, B.M.G. (2014). A generalized stochastic restricted ridge regression estimator. Commun. Stat. Theor. Meth. 43:4415–4427.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and correct the wrongly proved Theorems in that paper.     

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.     

Ridge estimator with correlated errors and two-stage ridge estimator under inequality restrictions

Liew (1976a Liew, C.K. (1976a). A two-stage least-squares estimation with inequality restrictions on parameters. Rev. Econ. Stat. LVIII(2):234–238.[Crossref], [Web of Science ®] , [Google Scholar]) introduced generalized inequality constrained least squares (GICLS) estimator and inequality constrained two-stage and three-stage least squares estimators by reducing primal–dual relation to problem of Dantzig and Cottle (1967 Dantzig, G.B., Cottle, R.W. (1967). Positive (semi-) definite matrices and mathematical programming. In: Abadie, J., ed. Nonlinear Programming (pp. 55–73). Amsterdam: North Holland Publishing Co. [Google Scholar]), Cottle and Dantzig (1974 Cottle, R.W., Dantzig, G.B. (1974). Complementary pivot of mathematical programming. In: Dantzig, G.B., Eaves, B.C., eds. Studies in OptimizationVol. 10. Washington: Mathematical Association of America. [Google Scholar]) and solving with Lemke (1962 Lemke, C.E. (1962). A method of solution for quadratic programs. Manage. Sci. 8(4):442–453.[Crossref], [Web of Science ®] , [Google Scholar]) algorithm. The purpose of this article is to present inequality constrained ridge regression (ICRR) estimator with correlated errors and inequality constrained two-stage and three-stage ridge regression estimators in the presence of multicollinearity. Untruncated variance–covariance matrix and mean square error are derived for the ICRR estimator with correlated errors, and its superiority over the GICLS estimator is examined via Monte Carlo simulation.     

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.     

On preliminary test almost unbiased two-parameter estimator in linear regression model with student's t errors

In this paper, the preliminary test approach to the estimation of the linear regression model with student's t errors is considered. The preliminary test almost unbiased two-parameter estimator is proposed, when it is suspected that the regression parameter may be restricted to a constraint. The quadratic biases and quadratic risks of the proposed estimators are derived and compared under both null and alternative hypotheses. The conditions of superiority of the proposed estimators for departure parameter and biasing parameters k and d are derived, respectively. Furthermore, a real data example and a Monte Carlo simulation study are provided to illustrate some of the theoretical results.     

Comments on “On the weighted mixed Liu-type estimator under unbiased stochastic restrictions”

We make some comments about the paper of Yildiz (2017 Yildiz, N. 2017. On the weighted mixed Liu-type estimator under unbiased stochastic restrictions. Communications in Statistics Simulation and Computation 46 (9):7238–48. do?:10.1080/03610918.2016.1235189.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and correct the theorems in that paper.     

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.     

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 modified Liu-type estimator with an intercept term under mixture experiments

We consider ridge regression with an intercept term under mixture experiments. We propose a new estimator which is shown to be a modified version of the Liu-type estimator. The so-called compound covariate estimator is applied to modify the Liu-type estimator. We then derive a formula of the total mean squared error (TMSE) of the proposed estimator. It is shown that the new estimator improves upon existing estimators in terms of the TMSE, and the performance of the new estimator is invariant under the change of the intercept term. We demonstrate the new estimator using a real dataset on mixture experiments.     

A jackknifed difference-based ridge estimator in the partial linear model with correlated errors

Tabakan and Akdeniz [Difference-based ridge estimator of parameters in partial linear model. Statist Pap. 2010;51(2):357–368] proposed a difference-based ridge estimator (DBRE) in the partial linear model. In this paper, a new estimator is introduced by jackknifing the DBRE that Tabakan and Akdeniz presented. We investigate the performance of this new estimator over the DBRE and difference-based estimator introduced by Yatchew [An elementary estimator of the partial linear model. Econom Lett. 1997;57:135–143] in terms of mean-squared error and mean-squared error matrix and a numerical example is provided to demonstrate the performance of the estimators.     

Recursive improvement of estimates in a Gauss-Markov model with linear restrictions

The minimum-dispersion linear unbiased estimator of a set of estimable functions in a general Gauss-Markov model with double linear restrictions is considered. The attention is focused on developing a recursive formula in which an initial estimator, obtained from the unrestricted model, is corrected with respect to the restrictions successively incorporated into the model. The established formula generalizes known results developed for the simple Gauss-Markov model.     

On a preliminary test estimator for one of the parameters of the log-zero-poisson distribution

A discrete distribution called the log-zero-Poisson distribution has been recommended by Katti (c.f. Biometrics 1970) as an alternate to the negative binomial and other distributions usually called "contagious" distributions.A major problem in the use of this and all other contagious distributions has been the difficulty of obtaining the maximum likelihood esti-mates. A custom-made ad hoc estimator, λ, has been proposed for the parameter λ of this distribution in Katti and Khedr (1980). In this paper, its efficiency relative to Fisher information is studied, only to discover that λ can be 30 times better than the maximum likelihood estimate in some parts of the parameter space and much weaker in other parts.A preliminary test is recommended to choose between the estimates, and the efficiency of the procedure is tabulated. As it is to be expected, the resultant estimator equals the better of the two estimators with some error at the values of the parameters where the two estimators are equivalent.     

A non stochastic ridge regression estimator and comparison with the James-Stein estimator

Abstract

This article presents a non-stochastic version of the Generalized Ridge Regression estimator that arises from a discussion of the properties of a Generalized Ridge Regression estimator whose shrinkage parameters are found to be close to their upper bounds. The resulting estimator takes the form of a shrinkage estimator that is superior to both the Ordinary Least Squares estimator and the James-Stein estimator under certain conditions. A numerical study is provided to investigate the range of signal to noise ratio under which the new estimator dominates the James-Stein estimator with respect to the prediction mean square error.     

Asymptotic properties of maximum quasi-likelihood estimator in quasi-likelihood non linear models with stochastic regression

In this paper, we establish the asymptotic properties of maximum quasi-likelihood estimator (MQLE) in quasi-likelihood non linear models (QLNMs) with stochastic regression under some mild regular conditions. We also investigate the existence, strong consistency, and asymptotic normality of MQLE in QLNMs with stochastic regression.     

Principal components regression estimator and a test for the restrictions

In this article, we introduce restricted principal components regression (RPCR) estimator by combining the approaches followed in obtaining the restricted least squares estimator and the principal components regression estimator. The performance of the RPCR estimator with respect to the matrix and the generalized mean square error are examined. We also suggest a testing procedure for linear restrictions in principal components regression by using singly and doubly non-central F distribution.     

On the weighted mixed Liu-type estimator under unbiased stochastic restrictions

In this article, we introduce the weighted mixed Liu-type estimator (WMLTE) based on the weighted mixed and Liu-type estimator (LTE) in linear regression model. We will also present necessary and sufficient conditions for superiority of the weighted mixed Liu-type estimator over the weighted mixed estimator (WME) and Liu type estimator (LTE) in terms of mean square error matrix (MSEM) criterion. Finally, a numerical example and a Monte Carlo simulation is also given to show the theoretical results.     

Finite sample properties of an HPT estimator when each individual regression coefficient is estimated in a misspecified linear regression model

ABSTRACT

In this paper, assuming that there exist omitted variables in the specified model, we analytically derive the exact formula for the mean squared error (MSE) of a heterogeneous pre-test (HPT) estimator whose components are the ordinary least squares (OLS) and feasible ridge regression (FRR) estimators. Since we cannot examine the MSE performance analytically, we execute numerical evaluations to investigate small sample properties of the HPT estimator, and compare the MSE performance of the HPT estimator with those of the FRR estimator and the usual OLS estimator. Our numerical results show that (1) the HPT estimator is more efficient when the model misspecification is severe; (2) the HPT estimator with the optimal critical value obtained under the correctly specified model can be safely used even when there exist omitted variables in the specified model.     

Variable selection in linear regression based on ridge estimator

In this article, we consider the problem of variable selection in linear regression when multicollinearity is present in the data. It is well known that in the presence of multicollinearity, performance of least square (LS) estimator of regression parameters is not satisfactory. Consequently, subset selection methods, such as Mallow's Cp, which are based on LS estimates lead to selection of inadequate subsets. To overcome the problem of multicollinearity in subset selection, a new subset selection algorithm based on the ridge estimator is proposed. It is shown that the new algorithm is a better alternative to Mallow's Cp when the data exhibit multicollinearity.     

Performance of ridge estimator in inverse Gaussian regression model

The presence of multicollinearity among the explanatory variables has undesirable effects on the maximum likelihood estimator (MLE). Ridge estimator (RE) is a widely used estimator in overcoming this issue. The RE enjoys the advantage that its mean squared error (MSE) is less than that of MLE. The inverse Gaussian regression (IGR) model is a well-known model in the application when the response variable positively skewed. The purpose of this paper is to derive the RE of the IGR under multicollinearity problem. In addition, the performance of this estimator is investigated under numerous methods for estimating the ridge parameter. Monte Carlo simulation results indicate that the suggested estimator performs better than the MLE estimator in terms of MSE. Furthermore, a real chemometrics dataset application is utilized and the results demonstrate the excellent performance of the suggested estimator when the multicollinearity is present in IGR model.     

MSE performance and minimax regret significance points for a HPT estimator under a multivariate t error distribution

In this article, assuming that the error terms follow a multivariate t distribution,we derive the exact formulae forthe moments of the heterogeneous preliminary test (HPT) estimator proposed by Xu (2012b Xu, H. (2012b). MSE performance and minimax regret significance points for a HPT estimator when each individual regression coefficient is estimated. Commun. Stat. Theory Methods 42:2152–2164.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). We also execute the numerical evaluation to investigate the mean squared error (MSE) performance of the HPT estimator and compare it with those of the feasible ridge regression (FRR) estimator and the usual ordinary least squared (OLS) estimator. Further, we derive the optimal critical values of the preliminary F test for the HPT estimator, using the minimax regret function proposed by Sawa and Hiromatsu (1973 Sawa, T., Hiromatsu, T. (1973). Minimax regret significance points for a preliminary test in regression analysis. Econometrica 41:1093–1101.[Crossref], [Web of Science ®] , [Google Scholar]). Our results show that (1) the optimal significance level (α*) increases as the degrees of freedom of multivariate t distribution (ν₀) increases; (2) when ν₀ ? 10, the value of α* is close to that in the normal error case.