Morris L. Eaton

Hotelling's T2 Test Under Symmetry Conditions

Abstract The familiar Hotellings T 2 statistic used for testing the null hypothesis that a collection of observed vectors has mean vector 0 is discussed outside the usual framework of independently and identically distributed multivariate normal observations. A wider null hypothesis called “orthant symmetry” is introduced, and certain robustness properties of the T 2 statistic within this null hypothesis are demonstrated. (In particular, orthant symmetry includes the case where the vectors are generated independently and normally with mean vector 0, but each with a different covariance matrix.) Relationships of T 2 with various rank statistic tests are discussed.

A Confidence Interval for the Maximal Mean QT Interval Change Caused by Drug Effect

A statistical problem of primary interest in a thorough QT/QTc study is that of deciding if a drug is noninferior to placebo in terms of QT/QTc prolongation. A standard way of approaching this problem is to construct a 90% two-sided (or a 95% one-sided) confidence interval, using the t distribution, at each time point in the study for the difference in mean QTc between drug and placebo and to conclude that the drug is noninferior to placebo if the upper end points of all of these confidence intervals is less than a prespecified constant, such as 10 ms. Under standard normality assumptions, this procedure corresponds to both an intersection-union test and the likelihood ratio test of size .05. It is not without its drawbacks, however. It is conservative in that the probability of a type I error may be smaller than the intended level .05. It is also biased, which means that the power function, for some values of parameters in the alternative space, takes values less than .05. The May 12, 2005, draft of the International Conference on Harmonisation E14 guidance states: “a negative ‘thorough QT/QTc study’ is one in which the upper bound of the 95% one-sided confidence interval for the largest time-matched mean effect of the drug on the QTc interval excludes 10 ms.” In this article, we show how an approximate confidence interval can be constructed for the largest difference in population mean QT/QTc between drug and placebo. The interval is approximate in the sense that, as sample sizes increase, the asymptotic probability of coverage is at least as large as intended. The results of simulations on a proposed one-sided 95% confidence interval are provided and discussed. Situations in which this interval works well, and does not work well, are delineated.

Consistency and strong inconsistency of group-invariant predictive inferences

Consider a statistical model which is invariant under a group of transformations that acts transitively on the parameter space. In this situation, the problem of constructing invariant predictive distributions is considered. It is shown, under certain assumptions, that Fisherian pivoting and the use of right Haar measure as an improper prior distribution both yield the same invariant predictive distribution. Furthermore, it is shown that any other invariant predictive distribution is strongly inconsistent in the sense of Stone.

Concentration inequalities for Gauss-Markov estimators

Let M be the regression subspace and [gamma] the set of possible covariances for a random vector Y. The linear model determined by M and [gamma] is regular if the identity is in [gamma] and if [Sigma](M)[subset, double equals]M for all [Sigma][set membership, variant][gamma]. For such models, concentration inequalities are given for the Gauss-Markov estimator of the mean vector under various distributional and invariance assumptions on the error vector. Also, invariance is used to establish monotonicity results relative to a natural group induced partial ordering.

Some Remarks on Scheffé's Solution to the Behrens-Fisher Problem

Abstract The problem considered here is a multivariate generalization of the classical Behrens-Fisher problem. Based on samples of size ni , i = 1, …, k, from k multivariate normal populations (of perhaps different dimension), a test statistic is proposed to test null hypotheses about linear relationships between components of the mean vectors of the k populations. Under the null hypothesis, the test statistic has a central F distribution and under alternatives, a non-central F distribution. For three different specializations of the above problem, the proposed test statistic reduces to tests proposed by Scheffe, Bennett, and Anderson respectively.

Admissibility in quadratically regular problems and recurrence of symmetric Markov chains: Why the connection?

Abstract In the expository paper, a sufficient condition is discussed for the almost admissibility of formal Bayes rules in quadratically regular problems. This sufficient condition is equivalent to a recurrence property of a natural symmetric Markov chain constructed from the model and the improper prior. Some simple examples involving translation parameter models illustrate the results.

The formal posterior of a standard flat prior in MANOVA is incoherent

A standard improper prior for the parameters of a MANOVA model is shown to yield an inference that is incoherent in the sense of Heath and Sudderth. The proof of incoherence is based on the fact that the formal Bayes estimate, sayδ 0 , of the covariance matrix based on the improper prior and a certain bounded loss function is uniformly inadmissible in that there is another estimatorδ l and an ɛ>0 such that the risk functions satisfyR(δ l ,Σ)⩽R δ 0 ,Σ)−e for all values of the covariance matrix Σ. The estimatorδ I is formal Bayes for an alternative improper prior which leads to a coherent inference.

Group invariant inference and right Haar measure

Every group invariant inference is strongly inconsistent except possibly for the formal Bayes inference that uses an improper prior induced by right Haar measure. The proof uses a number of technical assumptions which hold for many group invariant statistical models including examples in MANOVA.

Concentration inequalities for multivariate distributions: I. multivariate normal distributions

Let X ~ Np(0, [Sigma]), the p-variate normal distribution with mean 0 and positive definite covariance matrix [Sigma]. Anderson (1955) showed that if [Sigma]2 - [Sigma]1 is positive semidefinite then P[Sigma]1(C) [greater-or-equal, slanted] P[Sigma]2(C) for every centrally symmetric (- C = C) convex set C[subset, double equals]p. Fefferman, Jodeit and Perlman (1972) extended this result to elliptically contoured distributions. In the present study similar multivariate concentration inequalities are investigated for convex sets C that satisfy a more general symmetry condition, namely invariance under a group G of orthogonal transformations on p, as well as for non-convex sets C that are monotonically decreasing with respect to a pre-ordering determined by G. Both new results and counterexamples are presented. Concentration inequalities may be used to convert classical efficiency comparisons, expressed in terms of covariance matrices, into comparisons of probabilities of multivariate regions.

A condition for null robustness

Sufficient 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.