We prove the following conjecture of Narayana: there are no nontrivial dominance refinements of the Smirnov two-sample test if and only if the two sample sizes are relatively prime. We also count the number of natural significance levels of the Smirnov two-sample test in terms of the sample sizes and relate this to the Narayana conjecture. In particular, Smirnov tests with relatively prime samp...

We prove the following conjecture of Narayana: there are no nontrivial dominance reene-ments of the Smirnov two-sample test if and only if the two sample sizes are relatively prime. We also count the number of natural signiicance levels of the Smirnov two-sample test in terms of the sample sizes and relate this to the Narayana conjecture. In particular, Smirnov tests with relatively prime sampl...

Introduction: The Nagata-Smirnov Metrization theorem gives a full characterization of metrizable topological spaces. In other words, the theorem describes the necessary and sufficient conditions for a topology on a space to be defined using some metric. As a motivational example, consider the discrete topology on some space (every subset of the space is open). Though it might not be apparent to...

The use of the Kolmogorov-Smirnov (KS) statistic for testing hypotheses regarding activation in blood oxygenation level-dependent functional MRI data is critiqued on both theoretical and empirical grounds. Theoretically, it is argued that the KS test is formally unable to support inferences of interest to most neuro-imaging studies and has reduced sensitivity compared with parametric alternativ...

The Kolmogorov-Smirnov test determines the consistency of empirical data with a particular probability distribution. Often, parameters in the distribution are unknown, and have to be estimated from the data. In this case, the Kolmogorov-Smirnov test depends on the form of the particular probability distribution under consideration, even when the estimated parameter-values are used within the di...

A theory of matrix-valued functions from the matricial Smirnov class Nn (D) is systematically developed. In particular, the maximum principle of V.I.Smirnov, inner-outer factorization, the Smirnov-Beurling characterization of outer functions and an analogue of Frostman’s theorem are presented for matrix-valued functions from the Smirnov class Nn (D). We also consider a family Fλ = F − λI of fun...

G08CDF also returns the standardized statistic Z 1⁄4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 þ n2 n1n2 r D, where D may be Dn1;n2 , D þ n1;n2 or D n1;n2 depending on the choice of the alternative hypothesis. The distribution of this statistic converges asymptotically to a distribution given by Smirnov as n1 and n2 increase; see Feller (1948), Kendall and Stuart (1973), Kim and Jenri...