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The problem of establishing an upper bound for the volume of a parallelepiped is considered by utilizing an original approach involving a skew-symmetric matrix of order four (along with its Moore–Penrose inverse). It is shown that the commonly known inequality characterizing the bound can be virtually sharpened. Similarly, a sharpening is established with respect to the Cauchy–Schwarz inequality. General properties of the Moore–Penrose inverse of a skew-symmetric matrix are discussed as well.

The paper considers diagonalization of the cross-product matrices, i.e., skew-symmetric matrices of order three. A procedure to determine a nonsingular matrix, which yields the diagonalization is indicated. Furthermore, a method to derive the inverse of a diagonalizing matrix is proposed by means of a formula for the Moore–Penrose inverse of any matrix, which is columnwise partitioned into two matrices having disjoint ranges. This rather nonstandard method to obtain the inverse of a nonsingular matrix is appealing, as it can be applied to any diagonalizing matrix, and not only of those originating from diagonalization of the cross-product matrices. The paper provides also comments and examples demonstrating applicability of the diagonalization procedure to calculate roots of a cross-product matrix.

SummaryThe numerical range, easy to understand but often tedious to compute, provides useful information about a matrix. Here we describe the numerical range of a 3 × 3 magic square. Applying our results to one of the most famous of those squares, the Luoshu, it turns out that its numerical range is a piece of cake—almost.

Abstract: It is well known that dropping variables in regression analysis decreases the variance of the least squares (LS) estimator of the remaining parameters. However, after elimination estimates of these parameters are biased, if the full model is correct. In his recent paper, Boscher (1991) showed that the LS-estimator in the special case of a mean shift model (cf. Cook and Weisberg, 1982) which assumes no “outliers” can be considered in the framework of a linear regression model where some variables are deleted. He derived conditions under which this estimator outperforms the LS-estimator of the full model in terms of the mean squared error (MSE)-matrix criterion. We demonstrate that this approach can be extended to the general set-up of dropping variables. Necessary and sufficient conditions for the MSE-matrix superiority of the LS-estimator in the reduced model over that in the full model are derived. We also provide a uniformly most powerful F-statistic for testing the MSE-improvement.

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.

In this paper we show that for two projectors (idempotents) P1 and P2, idempotency of the product P1P2 is equivalent to the coincidence of the range of P1 P2 and a certain subspace, which only depends on the onto-and along-spaces of P1 and P2. Some further investigations are made, and necessary and sufficient conditions for the commutativity of two projectors are given.

The results of Stahlecker and Schmidt (1987) are considered for a more general class of transformations of the dependent variable. Mean square error matrix dominance of the so-called naive estimator over the usual OLS estimator is investigated and the related testing procedures are discussed. Some remarks on the admissibility of the naive estimator are given.

It is shown that a necessary and sufficient condition derived by Farebrother (1984)for a generalized ridge estimator to dominate the ordinary least-squares estimator with respect to the mean-square-error-matrix criterion in the linear regression model admits a similar interpretation as the well known criterion of Toro-Viz-carrondo and Wallace (1968)for the dominance of a restricted least-squares estimator over the ordinary least-squares estimator. Two other properties of the generalized ridge estimators, referring to the concept of admissibility, are also pointed out.

Summary: This article deals with the linear Bayes estimator (LBE) for incorrect prior assumptions under the misspecified linear regression model. In order to obtain the mean square error (MSE)-matrix properties of theLBE, we compare theMSE-matrix of theLBE averaging over the correct prior assumptions with that averaging over incorrect prior assumptions. We also compare theMSE-matrices of the ordinary least square estimator (OLSE) and theLBE under correct prior assumptions, given that the original prior assumptions are incorrect. Finally, we examine theMSE-matrix properties of theLBE in prediction problems.

We consider an approach yielding a minimax estimator in the linear regression model with a priori information on the parameter vector, e.g., ellipsoidal restrictions. This estimator is computed directly from the loss function and can be motivated by the general Pitman nearness criterion. It turns out that this approach coincides with the projection estimator which is obtained by projecting an initial arbitrary estimate on the subset defined by the restrictions.

The purpose of this paper is to extend the results achieved by Hsuan (1981) to a wide class of biased estimators. It is shown that in the case of multicollinearity ridge and Iteration estimator can be made very close to the principal component estimator whereas the shrunken estimator does not have this property.

If a priori information given by interval or ellipsoidal constraints on the unknown coefficients is used by (approximate) Minimax estimators, it is not ensured, that the estimates lie within the restricted set. In the present paper, the probability of such events is investigated on the basis of multivariate two-sided normal probability integrals.

We present an excursion with Jeff Hunter, visiting some of his research topics. Specifically, we will present some facts about certain people whose work seems to have influenced Jeff in his scientific career; we illustrate our presentation with postage stamps that have been issued in honour of these people. Our main guide is Hunter’s two-volume book entitled Mathematical Techniques of Applied Probability (Academic Press, 1983).

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