On the equality of usual and amemiya's partially generalized least squares estimator

Equivalent conditions are derived for the equality of GLSE (generalized least squares estimator) and partially GLSE (PGLSE), the latter introduced by Amemiya (1983). By adopting a more general approach the ordinary least squares estimator (OLSE) can shown to be a special PGLSE. Furthcrmore, linearly restricted estimators proposed by Balestra (1983) are investigated in this context. To facilitate the comparison of estimators extensive use of oblique and orthogonal projectors is made.     

A new estimator of covariance matrix

The problem of estimating a covariance matrix is considered in this paper. Using the so-called partial Iwasawa coordinates of the covariance matrix, a new improved estimator dominating the James-Stein estimator is proposed. The results of a simulation study verifies that the new estimator provides a substantial improvement in risk under Stein's loss.     

A new heteroskedasticity-consistent covariance matrix estimator and inference under heteroskedasticity

To solve the heteroscedastic problem in linear regression, many different heteroskedasticity-consistent covariance matrix estimators have been proposed, including HC0 estimator and its variants, such as HC1, HC2, HC3, HC4, HC5 and HC4m. Each variant of the HC0 estimator aims at correcting the tendency of underestimating the true variances. In this paper, a new variant of HC0 estimator, HC5m, which is a combination of HC5 and HC4m, is proposed. Both the numerical analysis and the empirical analysis show that the quasi-t inference based on HC5m is typically more reliable than inferences based on other covariance matrix estimators, regardless of the existence of high leverage points.     

Mean square error and efficiency of the least squares estimator over interval constraints

We derive the mean square error of an interval constrained least squares estimator (INCLS) for a regression model. We then show that the INCLS estimator dominates, in mean square error, the unconstrained least squares estimator provided the regression residuais are normally distri'iiuted and Ynat Yrie imposed coii-

straint is satisfied or nearly satisfied.     

Robust alternatives of the least squares estimator

The purpose of this paper consists in deriving estimators which are less sensitive than the least squares estimator, when the assumption that the expectation vector lies in a certain linear subspace is violated. The obtained robust estimators are convex combinations of the least squares estimator and of the random vector Y.     

Improving on the sample covariance matrix for a complex elliptically contoured distribution

In this paper the problem of estimating the scale matrix in a complex elliptically contoured distribution (complex ECD) is addressed. An extended Haff–Stein identity for this model is derived. It is shown that the minimax estimators of the covariance matrix obtained under the complex normal model remain robust under the complex ECD model when the Stein loss function is employed.     

A study on the least squares estimator of multivariate isotonic regression function

Consider the problem of pointwise estimation of f in a multivariate isotonic regression model Z=f(X₁,…,X_d)+ϵ, where Z is the response variable, f is an unknown nonparametric regression function, which is isotonic with respect to each component, and ϵ is the error term. In this article, we investigate the behavior of the least squares estimator of f. We generalize the greatest convex minorant characterization of isotonic regression estimator for the multivariate case and use it to establish the asymptotic distribution of properly normalized version of the estimator. Moreover, we test whether the multivariate isotonic regression function at a fixed point is larger (or smaller) than a specified value or not based on this estimator, and the consistency of the test is established. The practicability of the estimator and the test are shown on simulated and real data as well.     

An error-free generalized matrix inversion and linear least squares method based on bordering

The paper develops a method from which algorithms can be constructed to numerically compute error-free (free from computer roundoff error) generalized inverses and solutions to linear least squares problems having rational entries. A multiple modulus system is used to avoid error accumulation that is inherent in the floating-point number system. Some properties of finite fields of characteristic p, GF(p), are used in conjunction with a bordering method for matrix inversion to find nonsingular minors of a matrix over the field of rational numbers.     

A new heteroskedasticity-consistent covariance matrix estimator for the linear regression model

The assumption that all random errors in the linear regression model share the same variance (homoskedasticity) is often violated in practice. The ordinary least squares estimator of the vector of regression parameters remains unbiased, consistent and asymptotically normal under unequal error variances. Many practitioners then choose to base their inferences on such an estimator. The usual practice is to couple it with an asymptotically valid estimation of its covariance matrix, and then carry out hypothesis tests that are valid under heteroskedasticity of unknown form. We use numerical integration methods to compute the exact null distributions of some quasi-t test statistics, and propose a new covariance matrix estimator. The numerical results favor testing inference based on the estimator we propose.     

Adaptive ridge estimator in a linear regression model with spherically symmetric error under constraint

Abstract

We consider adaptive ridge regression estimators in the general linear model with homogeneous spherically symmetric errors. A restriction on the parameter of regression is considered. We assume that all components are non negative (i.e. on the positive orthant). For this setting, we produce under general quadratic loss such estimators whose risk function dominates that of the least squares provided the number of regressors in the least fore.     

Consistency of the structured total least squares estimator in a multivariate errors-in-variables model

The structured total least squares estimator, defined via a constrained optimization problem, is a generalization of the total least squares estimator when the data matrix and the applied correction satisfy given structural constraints. In the paper, an affine structure with additional assumptions is considered. In particular, Toeplitz and Hankel structured, noise free and unstructured blocks are allowed simultaneously in the augmented data matrix. An equivalent optimization problem is derived that has as decision variables only the estimated parameters. The cost function of the equivalent problem is used to prove consistency of the structured total least squares estimator. The results for the general affine structured multivariate model are illustrated by examples of special models. Modification of the results for block-Hankel/Toeplitz structures is also given. As a by-product of the analysis of the cost function, an iterative algorithm for the computation of the structured total least squares estimator is proposed.     

An extension of a rank criterion for the least squares estimator to be the best linear unbiased estimator

Among criteria for the least squares estimator in a linear model (y, Xβ, V) to be simultaneously the best linear unbiased estimator, one convenient for applications is that of Anderson (1971, 1972). His result, however, has been developed under assumptions of full column rank for X and nonsingularity for V. Subsequently, this result has been extended by Styan (1973) to the case when the restriction on X is removed. In this note, it is shown that also the restriction on V can be relaxed and, consequently, that Anderson's criterion is applicable to the general linear model without any rank assumptions at all.     

Consistency of the least median of squares estimator in nonlinear regression

Recent research by Sakata and White (1995) presents the consistency and asymptotic normality of S-estimators in nonlinear regression. It is well known from research in linear regression that it is important to use a consistent high breakdown estimator as an initial estimate when computing an S-estimate. This paper presents the proof of the weak consistency of the least median of squares estimator in a nonlinear regression setting, thus suggesting that it is a reasonable choice for the starting value for computing S-estimates in nonlinear regression.     

On generalized least squares estimation of the weibull distribution

A simple estimation procedure, based on the generalized least squares method, for the parameters of the Weibull distribution is described and investigated. Through a simulation study, this estimation technique is compared with maximum likelihood estimation, ordinary least squares estimation, and Menon's estimation procedure; this comparison is based on observed relative efficiencies (that is, the ratio of the Cramer-Rao lower bound to the observed mean squared error). Simulation results are presented for samples of size 25. Among the estimators considered in this simulation study, the generalized least squares estimator was found to be the "best" estimator for the shape parameter and a close competitor to the maximum likelihood estimator of the scale parameter.     

Small sample behavior of a robust heteroskedasticity consistent covariance matrix estimator

In heteroskedastic regression models, the least squares (OLS) covariance matrix estimator is inconsistent and inference is not reliable. To deal with inconsistency one can estimate the regression coefficients by OLS, and then implement a heteroskedasticity consistent covariance matrix (HCCM) estimator. Unfortunately the HCCM estimator is biased. The bias is reduced by implementing a robust regression, and by using the robust residuals to compute the HCCM estimator (RHCCM). A Monte-Carlo study analyzes the behavior of RHCCM and of other HCCM estimators, in the presence of systematic and random heteroskedasticity, and of outliers in the explanatory variables.     

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.     

A note on the correlation betweenS 2 and the least squares estimator in the linear regression model

In the following we consider the correlation betweenS ² and the least squares estimator in the linear regression model. We are interested in situations where these two statistics are uncorrelated though the errors are correlated. Conditions are developed without normality assumption, only assuming finite fourth moments of the error distributions. Support by Deutsche Forschungsgemeinschaft Grant No. Tr 253/1-2 is gratefully acknowledged.     

A note on the correlation betweenS 2 and the least squares estimator in the linear regression model

The online version of the original article can be found at     

Estimation of parameters in a generalized GMANOVA model based on an outer product analogy and least squares

This paper investigates estimation of parameters in a combination of the multivariate linear model and growth curve model, called a generalized GMANOVA model. Making analogy between the outer product of data vectors and covariance yields an approach to directly do least squares to covariance. An outer product least squares estimator of covariance (COPLS estimator) is obtained and its distribution is presented if a normal assumption is imposed on the error matrix. Based on the COPLS estimator, two-stage generalized least squares estimators of the regression coefficients are derived. In addition, asymptotic normalities of these estimators are investigated. Simulation studies have shown that the COPLS estimator and two-stage GLS estimators are alternative competitors with more efficiency in the sense of sample mean, standard deviations and mean of the variance estimates to the existing ML estimator in finite samples. An example of application is also illustrated.