https://www.jstor.org/stable/pdfplus/1912517.pdf

Likelihood ratio statistics for autoregressive time series with a unit root

Given C samples, with n i observations in the ith sample, a test of the hypothesis that the samples are from the same population may be made by ranking the observations from from 1 to Σn i (giving each observation in a group of ties the mean of the ranks tied for), finding the C sums of ranks, and computing a statistic H. Under the stated hypothesis, H is distributed approximately as χ2(C – 1), unless the samples are too small, in which case special approximations or exact tables are provided. One of the most important applications of the test is in detecting differences among the population means.* * Based in part on research supported by the Office of Naval Research at the Statistical Research Center, University of Chicago.

Use of Ranks in One-Criterion Variance Analysis

This study describes and validates a new method for metagenomic biomarker discovery by way of class comparison, tests of biological consistency and effect size estimation. This addresses the challenge of finding organisms, genes, or pathways that consistently explain the differences between two or more microbial communities, which is a central problem to the study of metagenomics. We extensively validate our method on several microbiomes and a convenient online interface for the method is provided at http://huttenhower.sph.harvard.edu/lefse/.

/pdf/metagenomic-biomarker-discovery-and-explanation-x530ilzkca.pdf

Metagenomic biomarker discovery and explanation

Upper bounds are derived for the probability that the sum S of n independent random variables exceeds its mean ES by a positive number nt. It is assumed that the range of each summand of S is bounded or bounded above. The bounds for Pr {S – ES ≥ nt} depend only on the endpoints of the ranges of the summands and the mean, or the mean and the variance of S. These results are then used to obtain analogous inequalities for certain sums of dependent random variables such as U statistics and the sum of a random sample without replacement from a finite population.

/pdf/probability-inequalities-for-sums-of-bounded-random-1jzyi7tbo3.pdf

Probability Inequalities for sums of Bounded Random Variables

Purpose: This paper discusses partial least squares path modeling (PLS), a powerful structural equation modeling technique for research on international marketing. While a significant body of research provides guidance for the use of covariance-based structural equation modeling (CBSEM) in international marketing, there are no subject-specific guidelines for the use of PLS so far.Methodology/approach: A literature review of the use of PLS in international marketing reveals the increasing application of this methodology.Findings: This paper reveals the strengths and weaknesses of PLS in the context of research on international marketing, and provides guidance for multi-group analysis.Originality/value of paper: The paper assists researchers in making well-grounded decisions regarding the application of PLS in certain research situations and provides specific implications for an appropriate application of the methodology.

The Use of Partial Least Squares Path Modeling in International Marketing

Let $x$ and $y$ be two random variables with continuous cumulative distribution functions $f$ and $g$. A statistic $U$ depending on the relative ranks of the $x$'s and $y$'s is proposed for testing the hypothesis $f = g$. Wilcoxon proposed an equivalent test in the Biometrics Bulletin, December, 1945, but gave only a few points of the distribution of his statistic. Under the hypothesis $f = g$ the probability of obtaining a given $U$ in a sample of $n x's$ and $m y's$ is the solution of a certain recurrence relation involving $n$ and $m$. Using this recurrence relation tables have been computed giving the probability of $U$ for samples up to $n = m = 8$. At this point the distribution is almost normal. From the recurrence relation explicit expressions for the mean, variance, and fourth moment are obtained. The 2rth moment is shown to have a certain form which enabled us to prove that the limit distribution is normal if $m, n$ go to infinity in any arbitrary manner. The test is shown to be consistent with respect to the class of alternatives $f(x) > g(x)$ for every $x$.

/pdf/on-a-test-of-whether-one-of-two-random-variables-is-3kx3ebntjq.pdf

Henry B. Mann

Papers

On a Test of Whether one of Two Random Variables is Stochastically Larger than the Other

Nonparametric tests against trend

Contributions to Probability and Statistics: Essays in Honor of Harold Hotelling

On the Statistical Treatment of Linear Stochastic Difference Equations

Contributions to Probability and Statistics: Essays in Honor of Harold Hotelling.