Question

from nonparametric statistics with applications to science and engineering book

6.7 EXERCISES 6.1. Derive the exact distribution of the Kolmogorov test statistic D, for the case n = 1.

Answer 1

Exact Distribution of the kolmogorov test statistic Dn for the case n=1 The empirical distribution function Fn for n independent & identically distributed ci-i-d) ordered observations Xi is defined as Fr(e) = 7 5 7 -20, xj (x," ) Where I 1-00 x (xi) is the indicator function equal to 1 it Xiex and equal to o 1 1-60 , xj xi) = { 1 it xizx it xi<a for n=1 Fi(x)= I [-00m (xi) = xi The kolmogorov-Smirnov statistic for given cumulative distribution function F(X) is Do = sup Fn () - F(X)) Cischscanned with the supremum of the set of distances CamScanner

for n=1 Die Sup] Fi(x)- F(X)] Nid 14 sample comes from dis+FCx) then Do Converges to o almost surely in the limit when n goes infinity In on n-sao, sup I B(F(+)) Where Blt) is the Brownian bridge kolmogorov theorem Distribution, which does Jn Dn converges to Kolmogoro ns decandech with CamScanner