are not identically distributed under the null hypothesis. The $\xi_1,\xi_2,\ldots,\xi_n$ and $\zeta_1,\zeta_2,\ldots,\zeta_n$ are obtained experimentally and $F_{\zeta_i}(\xi_i)$ can be calculated but a PDF for $\zeta_i$, that is $m(\cdot)$, has no easy expression. We still want to perform a statistics test that is similar to the KS test even in such situations where the distribution $m(\cdot)$ of $\zeta$ is unknown. First we define what we call the total null hypothesis and the conditional null hypothesis.

Let $\xi_1, \xi_2, \ldots, \xi_n$ be $n$ empirical observations of a random variable $\xi$. The total null hypothesis is that $\xi$ is distributed with the CDF $F(\cdot)$. The conditional null is that each $\xi_i$ is distributed with the conditional CDF $F_{\zeta_i}(\cdot)$.

A new statistical test is developed, which is similar to the KS test.

Suppose the conditional null hypothesis is true. Let $F_{\zeta_i}(\cdot)$ be continuous. Let $S_n$ be the statistic given by $$\begin{aligned}
S_n=\sup_{x\in[0,1]}
\left|
\frac{1}{n}
\sum^n_{i=1}
\mathbf{1}_{[0,x]}
\left(
F_{\zeta_i}(\xi_i)
\right)
-x
\right|\end{aligned}$$ then $S_n$ is KS distributed.

Adiabatic Large Deviation
-------------------------

We have to stress that this paper is built on three approximations, which form the backbone of all the research presented. These are small noise approximation, adiabatic approximation and perfect phase approximation.

Perfect phase 