William H. Kruskal and the Development of Coordinate-Free Methods

Discussion of ``The William Kruskal Legacy: 1919--2005'' by Stephen E. Fienberg, Stephen M. Stigler and Judith M. Tanur [arXiv:0710.5063]Comment: Published in at http://dx.doi.org/10.1214/088342306000000367 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

A maximization problem and its application to canonical correlation

AbstractLet Σ be an n × n positive definite matrix with eigenvalues λ1 ≥ λ2 ≥ … ≥ λn > 0 and let M = {x, y | x ϵ Rn, y ϵ Rn, x ≠ 0, y ≠ 0, x′y = 0}. Then for x, y in M, we have that x′Σy(x′Σxy′Σy)12 ≤ (λ1 − λn)(λ1 + λn) and the inequality is sharp. If ∑=∑11∑12∑21∑22 is a partitioning of Σ, let θ1 be the largest canonical correlation coefficient. The above result yields θ1 ≤ (λ1 − λn)(λ1 + λn)

Dutch book in simple multivariate normal prediction: Another look

In this expository paper we describe a relatively elementary method of establishing the existence of a Dutch book in a simple multivariate normal prediction setting. The method involves deriving a nonstandard predictive distribution that is motivated by invariance. This predictive distribution satisfies an interesting identity which in turn yields an elementary demonstration of the existence of a Dutch book for a variety of possible predictive distributions.Comment: Published in at http://dx.doi.org/10.1214/193940307000000356 the IMS Collections (http://www.imstat.org/publications/imscollections.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

Evaluation of Formal posterior distributions via Markov chain arguments

We consider evaluation of proper posterior distributions obtained from improper prior distributions. Our context is estimating a bounded function $\phi$ of a parameter when the loss is quadratic. If the posterior mean of $\phi$ is admissible for all bounded $\phi$, the posterior is strongly admissible. We give sufficient conditions for strong admissibility. These conditions involve the recurrence of a Markov chain associated with the estimation problem. We develop general sufficient conditions for recurrence of general state space Markov chains that are also of independent interest. Our main example concerns the $p$-dimensional multivariate normal distribution with mean vector $\theta$ when the prior distribution has the form $g(\|\theta\|^2) d\theta$ on the parameter space $\mathbb{R}^p$. Conditions on $g$ for strong admissibility of the posterior are provided.Comment: Published in at http://dx.doi.org/10.1214/07-AOS542 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

A condition for null robustness

AbstractSufficient conditions are given that certain statistics have a common distribution under a wide class of underlying distributions. Invariance methods are the primary technical tool in establishing the theoretical results. These results are applied to MANOVA problems, problems involving canonical correlations, and certain statistics associated with the complex normal distribution

A Maximization Problem and its Application to Canonical Correlation

1 online resource (PDF, 9 pages

A Probability Inequality for Linear Combinations of Bounded Random Variables

1 online resource (PDF, 10 pages

Orderings Induced on Rn By Compact Groups of Linear Transformations with Applications to Probability Inequalities (Preliminary Report)

1 online resource (PDF, 19 pages