Universal optimality of experimental designs: Definitions and a criterion

Several definitions of universal optimality of experimental designs are found in the Literature; we discuss the interrelations of these definitions using a recent characterization due to Friedland of convex functions of matrices. An easily checked criterion is given for a design to satisfy the main definition of universal optimality; this criterion says that a certain set of linear functions of the eigenvalues of the information matrix is maximized by the information matrix of a design if and only if that design is universally optimal. Examples are given; in particular we show that any universally optimal design is (M, S)-optimal in the sense of K. Shah.     

General bounds and inequalities in order statistics

This article is primarily a review of recent work on inequalities requiring no or only mild ssumptions.As feature,unified approaches are given for the derivation of algebraic inequalities involving linear functions of order statistics. Various inequalities for the expected value of such functions are presented. A new result is that a class of these inequalities can be improved by use of a theorem for ordered sums. Some other recent results are noted. Applications are indicated throughout.     

Barnard Convex Sets

In calculating significance levels for statistical non inferiority tests, the critical regions that satisfy the Barnard convexity condition have a central role. According to a theorem proved by Röhmel and Mansmann (1999 Röhmel , J. , Mansmann , U. ( 1999 ). Unconditional nonasymptotic one sided tests for independent binomial proportions when the interest lies in showing noninferiority and or superiority . Biometr. J. 2 : 149 – 170 .[Crossref], [Web of Science ®] , [Google Scholar]), when the critical regions satisfy this condition, the significance level for non inferiority tests can be calculated much more efficiently. In this study, the sets that fulfil the Barnard convexity condition are called Barnard convex sets, and because of their relevance, we studied their properties independently of the context from which the sets originated. Among several results, we found that Barnard convex sets are a convex geometry and that each Barnard convex set has a unique basis. Also, we provide an algorithm for calculating the Barnard convex hull of any set. Finally, we present some applications of the concept of the Barnard convex hull of a set for non inferiority tests.