A NOTE ON THE PROPERTIES OF STEIN-RULE AND INEQUALITY RESTRICTED ESTIMATORS WHEN THE REGRESSION MODEL IS OVER-FITTED

This paper considers the effect of an erroneous inclusion of regressors on the risk propertiesof the Stein-rule,positive-part Stein-rule and inequality restricted and pre-test estimators in a linearregression model.The two Stein-rule estimators are considered when extraneous information is availablein the form of a set of multiple equality constraints on the coefficients,while the inequality estimatorsare considered under the case of a single inequality constraint.It is shown that the inclusion ofwrong regressors has only minimal effect on the properties of the Stein-rule and positive-part Stein-ruleestimators,and no effect at all on the inequality restricted and pre-test estimators when there is asingle inequality constraint.

On the Stochastic Restricted Modified Almost Unbiased Liu Estimator in Linear Regression Model

In this article,we propose a new biased estimator,namely stochastic restricted modified almost unbiased Liu estimator by combining modified almost unbiased Liu estimator(MAULE)and mixed estimator(ME)when the stochastic restrictions are available and the multicollinearity presents.The conditions of supe-riority of the proposed estimator over the ordinary least square estimator,ME,ridge estimator,Liu estimator,almost unbiased Liu estimator,stochastic restricted Liu esti-mator and MAULE in the mean squared error matrix sense are obtained.Finally,a numerical example and a Monte Carlo simulation are given to illustrate the theoretical findings.

Quadratic Loss of Isotonic Normal Means under Simultaneous Order Restrictions

For two normal populations with unknown means μ and unknown variances σ2, assume that there are simple order restrictions among the means and variances: μ1 < μ2 and σ12 >σ22 > 0. This case is said to be simultaneous order restriction by Shi (Maximum likelihood estimation of means and variances from normal populations under simultaneous order restrictions, J. Multivariate Anal., 50(1994), 282-293.) and an iterative algorithm of computing the order restricted maximum likelihood estimates of μi and σi2 was given in that paper. This paper shows that the restricted maximum likelihood estimate of μi has smaller mean square loss than the usual estimate xi under some conditions.

Estimation of Ordered Means of Two Normal Distributions with Ordered Variances

The authors consider the problem of estimating the ordered means of two normal distributions with unknown ordered variances. The authors discuss the estimation of two ordered means, individually, in terms of stochastic domination and MSE （mean squared error）. The authors show that in estimating the mean with larger variance, the usual estimator under order restriction on means can be improved upon. However, in estimating the mean with smaller variance, the usual estimator can＇t be improved upon even under MSE. The authors also discuss simultaneous estimation problem of two ordered means when unknown variances are ordered.

Redundancy of Moment Conditions in Restricted GMM Estimation

Using a novel approach to calculating the rank of the difference of two asymptotic variance matrices, The author derives the necessary and sufficient conditions for an extra set of moment conditions to be redundant given a set of moment conditions in GMM estimation with general nonlinear restrictions. The necessary and sufficient conditions derived in this paper include as a special case the redundancy of moment conditions for GMM estimation without restrictions that was first derived by Breusch et al. （1999）. Therefore this paper advances the research on redundancy of moment conditions from unrestricted GMM estimation to a larger class of GMM estimation. To show their usefulness, the main results of the current paper are applied to instrumental variables estimation of linear regression models and the efficient estimation of seemingly unrelated regressions models, subject to restrictions.

An equivalence result for moment equations when data are missing at random

We consider general statistical models defined by moment equations when data are missing atrandom. Using the inverse probability weighting, such a model is shown to be equivalent with amodel for the observed variables only, augmented by a moment condition defined by the missing mechanism. Our framework covers a large class of parametric and semiparametric modelswhere we allow for missing responses, missing covariates and any combination of them. Theequivalence result is stated under minimal technical conditions and sheds new light on variousaspects of interest in the missing data literature, as for instance the efficiency bounds and theconstruction of the efficient estimators, the restricted estimators and the imputation.

An Off-Diagonal Marcinkiewicz Interpolation Theorem on Lorentz Spaces

Let （X, μ） be a measure space. In this paper, using some ideas from Grafakos and Kalton, the authors establish an off-diagonal Marcinkiewicz interpolation theorem for a quasilinear operator T in Lorentz spaces L^P,q（X） with p, q ￡ （0, ∞], which is a corrected version of Theorem 1.4.19 in [Grafakos, L.： Classical Fourier Analysis, Second Edition, Graduate Texts in Math., No. 249, Springer, New York, 2008] and which, in the case that T is linear or nonnegative sublinear, p ￡ [1, ∞） and q ￡ [l,∞）, was obtained by Stein and Weiss [Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, N.J., 1971].

Analysis of φ-divergence for Loglinear Models with Constraints under Product-multinomial Sampling

The loglinear model under product-multinomial sampling with constraints is considered. The asymptotic expansion and normality of the restricted minimum C-divergence estimator （RMDE） which is a generalization of the maximum likelihood estimator is presented. Then various statistics based on C-divergence and RMCDE are used to test various hypothesis test problems under the model considered. These statistics contain the classical loglikelihood ratio test statistics and Pearson chi-squared test statistics. Ia the last section, a simulation study is implemented.