Götz Trenkler

Core inverse of matrices

This article introduces the notion of the Core inverse as an alternative to the group inverse. Several of its properties are derived with a perspective towards possible applications. Furthermore, a matrix partial ordering based on the Core inverse is introduced and extensively investigated.

Continuous univariate distributions

Continuous Distributions (General). Normal Distributions. Lognormal Distributions. Inverse Gaussian (Wald) Distributions. Cauchy Distribution. Gamma Distributions. Chi-Square Distributions Including Chi and Rayleigh. Exponential Distributions. Pareto Distributions. Weibull Distributions. Abbreviations. Indexes.

Mean squared error matrix comparisons between biased estimators — An overview of recent results

In the following we give a systematic report on mean squared error matrix comparisons of competing biased estimators. Our approach is quite general: The parameter vector to be estimated is assumed to belong to a subset of the p-dimensional Euclidean space. However, to illustrate our results, we shall pay attention to the linear regression model where biased estimation is very popular. Especially we are interested in generalized ridge and restricted least squares estimation.

Nonsingularity of the Difference of Two Oblique Projectors

For two real projectors P and Q of the same order it is shown that the difference P - Q is nonsingular if and only if the column spaces of P and Q are complementary and the row spaces of P and Q are complementary. Moreover, it is demonstrated that nonsingularity of P - Q implies nonsingularity of P + Q,I - QP, and P + Q - QP.

Generalized and hypergeneralized projectors

Abstract Generalizations of the class of projectors are considered within the classes of normal and EP matrices. Some properties are derived, and results on the sum, difference, and product of generalized and hypergeneralized projectors are given.

Quasi minimax estimation in the linear regression model

Since the explicit determination of the minimax estimator in th elinear regression model usually is difficult we propose to minimize an upper bound of the maximal risk instead. The resulting estimator coincides with the minimax estimator when the weight matrix of the risk function has rank one. Specific attention is paid to additional equality restrictions

Nonnegative and positive definiteness of matrices modified by two matrices of rank one

Abstract Necessary and sufficient conditions are given for the nonnegative and positive definiteness of matrices of the form A − a 1 a 1 ∗ − a 2 a 2 ∗ and A + a 1 a 1 ∗ − a 2 a 2 ∗ , where A is a Hermitian matrix and a 1 , a 2 are complex vectors. They are derived using some auxiliary results which seem to be of independent interest as well. Some particular cases of these conditions are also discussed, especially in the context of related results known in the literature.

MEAN SQUARE ERROR MATRIX IMPROVEMENTS AND ADMISSIBILITY OF LINEAR ESTIMATORS

Abstract In the first part of this paper, the set L (Cy + c), comprising all linear estimators of β which are as good as a given unbiased estimator Cy + c with respect to the mean square error matrix criterion in at least one point of the parameter space, is investigated under the unrestricted linear regression model M = {y, Xβ, σ2In} and the restricted model M0 = {y, Xβ¦R0β = r0, σ2/In}. In the second part, new characterizations of the sets A and A 0 of all linear estimators that are admissible for β under M and M0 with respect to the mean square error criterion are derived via appropriately reducing the sets L (β^) and L (β^0), where β^ and β^0 are the minimum dispersion linear unbiased estimators of β in these two models. The convexity of the sets L (Cy + c), A , and A 0 is also pointed out.

Characterizations of EP, normal, and Hermitian matrices

Various characterizations of EP, normal, and Hermitian matrices are obtained by exploiting an elegant representation of matrices derived by Hartwig and Spindelböck [7, Corollary 6]. One aim of the present article is to demonstrate its usefulness when investigating different matrix identities. The second aim is to extend and generalize lists of characterizations of Equal Projectors (EP), normal, and Hermitian matrices known in the literature, by providing numerous sets of equivalent conditions referring to the notions of conjugate transpose, Moore–Penrose inverse, and group inverse.

A revisitation of formulae for the Moore–Penrose inverse of modified matrices

Abstract Formulae for the Moore–Penrose inverse M + of rank-one-modifications of a given m × n complex matrix A to the matrix M = A + bc ∗ , where b and c ∗ are nonzero m ×1 and 1× n complex vectors, are revisited. An alternative to the list of such formulae, given by Meyer [SIAM J. Appl. Math. 24 (1973) 315] in forms of subtraction–addition type modifications of A + , is established with the emphasis laid on achieving versions which have universal validity and are in a strict correspondence to characteristics of the relationships between the ranks of M and A . Moreover, possibilities of expressing M + as multiplication type modifications of A + , with multipliers required to be projectors, are explored. In the particular case, where A is nonsingular and the modification of A to M reduces the rank by 1, such a possibility was pointed out by Trenkler [R.D.H. Heijmans, D.S.G. Pollock, A. Satorra (Eds.), Innovations in Multivariate Statistical Analysis. A Festschrift for Heinz Neudecker, Kluwer, London, 2000, p. 67]. Some applications of the results obtained to various branches of mathematics are also discussed.