Convergence Rate of Wavelet Expansions of Gaussian Random Processes

This article characterizes uniform convergence rate for general classes of wavelet expansions of stationary Gaussian random processes. The convergence in probability is considered.     

Ranking paired contestants

The ranking of paired contestants (players) after a series of contests is difficult when every player does not play every other player. In the 1975 JASA Mark Thompson presented a maximum likelihood solution based on the assumption that the probability of any one player defeating any other is a function only of the difference in their ranks. Here the linear approximation to that likelihood is shown to lead to a nonparametric measure of the efficacy of the ranking, called the net difference in ranks (NDR) , which is the sum of the differences in ranks of the paired players in the observed contests that agree with the ranking minus the sum of the differences in ranks in the observed contests that disagree with the ranking (upsets) . The subject is part of a large literature that has been consolidated by H.A. David in The Method of Paired Comparisons (1963, 1988). The method was introduced by the psychophysicist Fechner in 1860 and has been widely applied to sensory testing,     

Admissibility of Test Procedures for Linear Hypotheses in General Linear Model

This article shows that an F-test procedure is admissible for testing a linear hypothesis concerning one of the split mean vectors in a general linear model and an F-test procedure is also admissible for testing a linear hypothesis concerning another of the split mean vectors in the same model. These results are proved by showing that the critical functions of the tests are unique Bayes procedures with respect to proper prior distributions set in common for the null hypotheses and for the alternative ones, respectively.     

On the Similarity Structure of Order Statistics

In this article, we address the similarity structure between pairs of order statistics of an identically distributed independent random variables X₁,…, X_n. The overlapping coefficient (Δ) of Weitzman (1970 Weitzman , M. S. ( 1970 ). Measures of overlap of income distributions of white and Negro families in the United States . Technical Paper No. 22 , Washington , D.C. , U.S.A : Department of Commerce, Bureau of Census . [Google Scholar]), is used to assess the degree of similarity or closeness between pairs of order statistics. It appears that the degree of the similarity between any of such pairs is independent of the parent distribution. Using this notion, it is shown that for i < j, the degree of similarity between distributions of the ith and the jth order statistics decreases as i and j sunder. Some possible biometric applicability of the value of Δ are explored. In particular, the use of this measure in estimation of the number of possible strata, subgroups or natural subdivisions in a population are suggested.     

A Nonrandomized,Nonconservative Version of the Fisher Exact Test

The Fisher exact test has been unjustly dismissed by some as ‘only conditional,’ whereas it is unconditionally the uniform most powerful test among all unbiased tests, tests of size α and with power greater than its nominal level of significance α. The problem with this truly optimal test is that it requires randomization at the critical value(s) to be of size α. Obviously, in practice, one does not want to conclude that ‘with probability x the we have a statistical significant result.’ Usually, the hypothesis is rejected only if the test statistic's outcome is more extreme than the critical value, reducing the actual size considerably.

The randomized unconditional Fisher exact is constructed (using Neyman–structure arguments) by deriving a conditional randomized test randomizing at critical values c(t) by probabilities γ(t), that both depend on the total number of successes T (the complete-sufficient statistic for the nuisance parameter—the common success probability) conditioned upon.

In this paper, the Fisher exact is approximated by deriving nonrandomized conditional tests with critical region including the critical value only if γ (t) > γ₀, for a fixed threshold value γ₀, such that the size of the unconditional modified test is for all value of the nuisance parameter—the common success probability—smaller, but as close as possible to α. It will be seen that this greatly improves the size of the test as compared with the conservative nonrandomized Fisher exact test.

Size, power, and p value comparison with the (virtual) randomized Fisher exact test, and the conservative nonrandomized Fisher exact, Pearson's chi-square test, with the more competitive mid-p value, the McDonald's modification, and Boschloo's modifications are performed under the assumption of two binomial samples.     

Dirichlet models for square contingency tables

We consider the square contingency tables which arise when the same method of classification is applied twice. The hypothesis of marginal homogeneity is then relevant! and can be tested by various methods Models are discussed which contain marginal homogeneity as a special case. They include a class based on univariate and bivariate Dirichlet distributions. The question of ordered categories is briefly discussed. Applications are made to data on unaided distance vision.     

Some concepts of mult1variate dependence

Several types of positive dependence are shown to be equiv-alent to the concept of a distribution with a density which is TP₂ in pairs. Among these is the concept of m^?-positive depend-ence of Alam and Wallenius. Using this result, all relationships among many of the most important concepts of positive dependence are determined. Furthermore, an application of the equivalences of these types of positive dependence yields a results of Ahmed, Langberg, Leon and Proschan.     

Nonparametric tests for scale shift at an unknown time point

A class of linear rank tests is suggested for testing a shift in scale at an unknown time point in a sequence of independent observations. The tests,based on inverse normal scores and on ordered exponential scores,are shown to be asymptotically as efficient as their distribution-oriented competitors. Critical values and powers for these two rank tests are also discussed.     

The effects of correlation among observations on the consistency property of sample variance

It is well known that even when the sample observations are correlated and not normal the sample variance, S² converges in probability to E(S²). But the required sample size for S² to be a consistent estimator of E(S²) is an open question. Some light is shed on this question in this paper. In particular the relation between the rate of convergence and the correlation property of the observations is explored. It is shown that the retardation to the rate of convergence is not appreciable if the correlation is moderate but it can be severe for extreme correlations.     

Construction of orthogonal factorial designs controlling interaction efficiencies

This paper employs some variants of the usual Kronecker product to construct orthogonal factorial designs controlling the interaction efficiencies. The methods suggested have a fairly wide coverage and the resulting designs involve a small number of replicates.