On bounds for the range of ordered variates II

Letx ₁, ,x _n be real numbers with ₁ ⁿ x _j =0, |x ₁ ||x ₂ ||x _n |, and ₁ ⁿ f(|x _i |)=A>0, wheref is a continuous, strictly increasing function on [0, ) withf(0)=0. Using a generalized Chebycheff inequality (or directly) it is easy to see that an upper bound for |x _m | isf ^–1 (A/(n–m+1)). If (n–m+1) is even, this bound is best possible, but not otherwise. Best upper bounds are obtained in case (n–m+1) is odd provided either (i)f is strictly convex on [0, ), or (ii)f is strictly concave on [0, ). Explicit best bounds are given as examples of (i) and (ii), namely the casesf(x)=x ^p forp>1 and 0<p<1 respectively.     

Some inequalities for communtators and an application to spectral variation

Summary If –I is a positive semidefinite operator andA andB are either both Hermitian or both unitary, then every unitarily invariant norm ofA–B is shown to be bounded by that ofA–B. Some related inequalities are proved. An application leads to a generalization of the Lidskii-Wielandt inequality to matrices similar to Hermitian.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth     

Connection between the Hadamard and matrix products with an application to matrix-variate Birnbaum-Saunders distributions

In this paper, we establish a connection between the Hadamard product and the usual matrix multiplication. In addition, we study some new properties of the Hadamard product and explore the inverse problem associated with the established connection, which facilitates diverse applications. Furthermore, we propose a matrix-variate generalized Birnbaum-Saunders (GBS) distribution. Three representations of the matrix-variate GBS density are provided, one of them by using the mentioned connection. The main motivation of this article is based on the fact that the representation of the matrix-variate GBS density based on element-by-element specification does not allow matrix transformations. Consequently, some statistical procedures based on this representation, such as multivariate data analysis and statistical shape theory, cannot be performed. For this reason, the primary goal of this work is to obtain a matrix representation of the matrix-variate GBS density that is useful for some statistical applications. When the GBS density is expressed by means of a matrix representation based on the Hadamard product, such a density is defined in terms of the original matrices, as is common for many matrix-variate distributions, allowing matrix transformations to be handled in a natural way and then suitable statistical procedures to be developed.     

On the spectrum of resolvable orthogonal arrays invariant under the Klein group K4

It is shown that there exists a resolvablen ² by 4 orthogonal array which is invariant under the Klein 4-groupK ₄ for all positive integersn congruent to 0 modulo 4 except possibly forn {12, 24, 156, 348}.     

Dichteschätzungen mit gleichverteilungsmethoden

Density estimates with methods of uniform distribution mod 1. Some nonparametric multivariate density estimators for continuous functions, based on the Fejér and Jackson kernel, are presented. Uniform strong consistency results are obtained with methods of uniform distribution mod 1.

     

Quaternion orthogonal designs from complex companion designs

The success of applying generalized complex orthogonal designs as space-time block codes recently motivated the definition of quaternion orthogonal designs as potential building blocks for space-time-polarization block codes. This paper offers techniques for constructing quaternion orthogonal designs via combinations of specially chosen complex orthogonal designs. One technique is used to build quaternion orthogonal designs on complex variables for any even number of columns. A second related technique is applied to maximum rate complex orthogonal designs to generate an infinite family of quaternion orthogonal designs on complex variables such that the resulting designs have no zero entries. This second technique is also used to generate an infinite family of quaternion orthogonal designs defined over quaternion variables that display a regular redundancy. The proposed constructions are theoretically important because they provide the first known direct techniques for building infinite families of orthogonal designs over quaternion variables for any number of columns.     

Efficiencies of tests and estimators forp-order autoregressive processes when the error distribution is nonnormal

Summary We considerpth order autoregressive time series where the shocks need not be normal. By employing the concept of contiguity, we obtain the sysmptotic power for tests of hypothesis concerning the autoregressive parameters. Our approach allows consideration of the double exponential and other thicker-tailed distributions for the shocks. We derive a new result in the contiguity framework that leads directly to an expression for the Pitman efficiencies of tests as well as estimators. The numerical values of the efficiencies suggest a lack of robustness for the normal theory least squares estimators when the shock distribution is thick tailed or an outlier prone mixed normal. An important alternative test statistic is proposed that competes with the normal theory tests. This research was supported by the Office of Naval Research under Grant No. N00014-78-C-0722 and by the Army Research Office.     

Proper bayes minimax estimators for a multivariate normal mean with unknown common variance under a convex loss function

Summary Let X ∼ N_p(μ,σ²I_p) and let s/σ² ∼ χ _n ² , independent ofX, where μ and σ² are unknown. This paper considers the estimation of μ (by δ) relative to a convex loss function given by (δ−μ)′[(1−α)I_p/σ²+αQ](δ−μ)/[(1−α)p/σ²+α tr (Q)], whereQ is a knownp×p diagonal matrix and 0≦α≦1. Two classes of minimax estimators are obtained for μ whenp≦3; the first is a new result and the second is a generalization of a result of Strawderman (1973,Ann. Statist.,1, 1189–1194). A proper Bayes estimator is also obtained which is shown to satisfy the conditions of the second class of minimax estimators. The paper concludes by discussing the estimation of μ relative to another convex loss function. This work was supported by the Army, Navy and Air Force under Office of Naval Research Contract No. N00014-80-C-0093. Reproduction in whole or in part is permitted for any purpose of the United States Government.     

Linear combination of concomitants of order statistics with application to testing and estimation

Summary Let (X ₁,Y ₁), (X ₂,Y ₂),…, (X _n,Y _n) be i.i.d. as (X, Y). TheY-variate paired with therth orderedX-variateX _rn is denoted byY _rn and terms the concomitant of therth order statistic. Statistics of the form are considered. The asymptotic normality ofT _n is established. The asymptotic results are used to test univariate and bivariate normality, to test independence and linearity ofX andY, and to estimate regression coefficient based on complete and censored samples.     

On two inequalities similar to Opial's inequality in two independent variables

In the present paper we establish two new integral inequalities similar to Opial's inequality in two independent variables. The inequalities established in this paper are similar to the analogues of Calvert's generalizations of Opial's inequality, in two independent variables and contains in the special case the analogue of Opial's inequality given by G. S. Yang in two independent variables.     

Lower bound of risk in linear unbiased estimation and its application

Summary Lower bound of risk in linear unbiased estimation and its connection with the existence of a uniformly minimum variance linear unbiased estimator is considered.