Empirical Processes , and the Kolmogorov – Smirnov Statistic Math 6070 , Spring 2006

Empirical processes in probabilistic number theory : the LIL for the discrepancy of ( n k ω ) mod 1

We prove a law of the iterated logarithm for the Kolmogorov-Smirnov statistic, or equivalently, the discrepancy of sequences (nkω) mod 1. Here (nk) is a sequence of integers satisfying a subHadamard growth condition and such that linear Diophantine equations in the variables nk do not have too many solutions. The proof depends on a martingale embedding of the empirical process; the number-theor...

Elements of Density Estimation Math 6070 , Spring 2006

A comparison of the discrete Kolmogorov-Smirnov statistic and the Euclidean distance

Goodness-of-fit tests gauge whether a given set of observations is consistent (up to expected random fluctuations) with arising as independent and identically distributed (i.i.d.) draws from a user-specified probability distribution known as the “model.” The standard gauges involve the discrepancy between the model and the empirical distribution of the observed draws. Some measures of discrepan...

Fuzzy Empirical Distribution Function: Properties and Application

The concepts of cumulative distribution function and empirical distribution function are investigated for fuzzy random variables. Some limit theorems related to such functions are established. As an application of the obtained results, a method of handling fuzziness upon the usual method of Kolmogorov–Smirnov one-sample test is proposed. We transact the α-level set of imprecise observations in ...

A critique of the use of the Kolmogorov-Smirnov (KS) statistic for the analysis of BOLD fMRI data.

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...