# Kolmogorov-Smirnov and bootstrap

The KS test uses the statistic $$D_n=\sup_x |\hat{F}_n(x)-F_0(x)|$$ where $F_0(x)$ is the distribution to be tested and $\hat{F}_n(x)$ the empirical distribution. Under the null hypothesis $D_n$ is distributed according to the Kolmogorov distribution and the KS-test is performed testing $D_n$ against the Kolmogorov distribution.

My question concerns the bootstrapped version , i.e., $$D^*_n=\sup_x |\hat{F}_n^*(x)-F_0(x)|$$ where $\hat{F}_n^*(x)$ is obtained through bootstrapping from $\hat{F}_n(x)$. Given the data, which is the distribution of $D^*_n$ generated by bootstrap? Which is the relationship between this distribution and the kolmogorov distribution for $n\rightarrow \infty$?

Can I derive a KS-test based on the distribution $D^*_n$ given the data? Is there some known result about that?

Thank you for any help.

• I am curious as to why you are interested in this. Generally speaking, the bootstrap is about constructing an approximate sampling distribution with which to do inference. Here, the distribution of $D_n$ is fixed (invariant to the underlying data) and known under quite general hypotheses. It can be computed pretty readily as well. – cardinal Apr 10 '13 at 11:58
• I'm trying to implement a KS test based on the Dirichlet process and I ended with this problem. To compare the classic KS test and mine, I need to understand the relationship between the distributions of $D_n$ and $D_n^*$. – vatna Apr 10 '13 at 12:57