# Kolmogorov–Smirnov “with perturbation”

Let $F$ be known continuous CDF of a continuous R.V. and $F_n$ represent the empirical CDF for sample of size $n$, hypothesized to be drawn from $F$. The Kolmogorov–Smirnov statistic is $D_n :=sup |F_n(x) - F(x)|$. In applying the Kolmogorov–Smirnov test, the key is that $\sqrt{n}D_n$ converges in distribution to the so-called Kolmogorov distribution, and this fact is then used to compute p-values.

Now I wish to consider a slight variation. Supposed my samples are hypothesized to be drawn from $F$, but $x_i$ has been independently perturbed by an error $E_i$ (each $E_i$ has different distribution) of mean 0 before being recorded. If I construct the same empirical CDF using the perturbed values and attempt to apply the KS test, first of all, is it true that $\sqrt{n}D_n$ still converges in distribution to the Kolmogorov distribution (or some modification of it)?

[Note: If $E_i$ are iid~$F'$, then I can just use the original KS test on convolution of $F$ with $F'$. But what happens otherwise?]

[Maybe a proof of the original convergence theorem will also help to answer the new version with perturbation.]

• You will need some kind of control over the error distributions, for otherwise they could dominate the situation. For instance, make the first $10$ of the $E_i$ standard normal, then the next $100$ of them normal with variance $2$, then the next $1000$ normal with variance $3$, etc. As $n$ grows larger and if $F$ has finite variance, the variance of the empirical distribution will grow arbitrarily large, so $\sqrt{n}D_n$ could not possibly converge to the Kolmogorov distribution. – whuber Jun 18 '14 at 14:56
• How about if we use some form of control that is typical for CLT to happen? – renrenthehamster Jun 18 '14 at 15:17
• That's the right spirit--but the way to do it is not to assume particular conditions and see whether they work, but to investigate what conditions are needed on the $E_i$ in order for the desired convergence to hold (because there seems little reason for the CLT conditions to be applicable here). Because that could be a difficult thing to do, if it hasn't already been done (and somebody can quote the research), consider narrowing your question to focus on the data you actually have. – whuber Jun 18 '14 at 15:25