Technical Reports

A new one-sided test for stochastic order of k > = 3 ordered multinomial populations is offered. The test has desirable monotonicity properties in the sense that it is monotone in practical directions and admissible directions. A practical direction is one for which stochastic order is more readily convincing. An admissible direction is a requirement for a test to be admissible. The test is based on a directed chi-square statistic. In addition to the monotonicity properties the test is exact (nonasymptotic), readily calculated (a program is offered), and has favorable power properties when compared with competitors. The test can also be used when some data are censored.

2005-002: More on the inadmissibility of step-up, by Arthur Cohen and Harold B. Sackrowitz [4/15/05]

Cohen and Sackrowitz [1] proved that the step-up multiple testing procedure is inadmissible for a multivariate normal model with unknown mean vector and known intraclass covariance matrix. The hypotheses tested are each mean is zero vs each mean is positive. The risk function is a 2 x 1 vector where one component is average size and the other component is one minus average power. In this paper we extend the inadmissibility result to several different models, to two-sided alternatives, and to other risk functions. The models include one-parameter exponential families, independent t-variables, independent chi-square variables, t-tests arising from the analysis of variance, and t-tests arising from testing treatments against a control. The additional risk functions are linear combinations where one component is the false discovery rate (FDR).

2005-003: A proof of the asymptotic equivalence of two tail probability approximations, by John E. Kolassa [4/21/05]

This manuscript considers asymptotic approximations to tail probabilities of a random variable whose distribution depends on a parameter n heuristically representing sample size. Random variables considered have cumulant generating functions with properties similar to that of sums of independent and indentically distributed random variables. Probability approximations of Robinson (1982) and Lugannani and Rice (1980) are shown to be equivalent to a relative size O(1/n), under regularity conditions no stronger than the weaker of those necessary to prove either of the two approximations. Applications to permutation testing are discussed.

2005-004: Bayes and empirical Bayes approaches to controlling the false discovery rate, by Weihua Tang and Cun-Hui Zhang [4/28/05]

False discovery rate has been widely used in large scale multiple testing problems as it seems to attain a balanced compromise between the more liberal per comparison error rate and the more conservative familywise error rate. We formulate a Bayes optimization problem as the maximization of the total amount of statistical discovery subject to a preassigned level of false discovery rate conditionally on test statistics, and propose an empirical Bayes approach based on the solution to it. Asymptotic optimality of certain empirical Bayes procedures is proved and the results of a simulation study are presented.

2005-005: Continuous generalized gradient descent, by Cun-Hui Zhang [12/23/05]

We derive characterizations and computational algorithms for continuous general gradient descent trajectories in high-dimensional parameter spaces for statistical model selection, prediction and classification. The discrete threshold gradient descent, a nested version of it, and their limits are considered in detail. Additional examples include robust LASSO, LARS, kernel SVM, and more. In all these problems, general gradient descent trajectories are continuous piecewise analytic vector-valued curves as solutions to matrix differential equations. Approximations of continuous solutions via infinite series expansions are proved to be computationally more efficient and accurate compared with discretization methods.