In one-sample Kolmogorov-Smirnov test one tests whether sample $X = (X_1,\dots,X_n)$ come from theoretical distribution by computing $D_n = \sup_{x} |F_n(x) - F(x)|$, where $F_n(x)$ denotes the empirical cumulative distribution function of $X$ and $F(x)$ denotes the cumulative distribution function of a prespecified theoretical distribution.
It is known that in case when $F(x)$ is continuous, the distribution of $D_n$ does not depend on $F(x)$. In classical KS-test it is assumed that distibutions are continuous. It is possible, however, to perform test for purely discrete $F(x)$. In that case it's hard to compute $P(D_n \leq q |H_0)$ (p-value), where $H_0$ is a null hypothesis that $X$ comes from $F(x)$.
Dimitrova, D. S et al. study how to compute p-value for continuous, discrete or mixed $F(x)$ - http://openaccess.city.ac.uk/id/eprint/18541/. Let's focus on purely discrete CDF. In that case $F(x)$ must be non-decreasing and right-continuous, with countable number of jumps. If it's true, one could compute $D_n$ and corresponding p-value for purely discrete $F(x)$ using function KSgeneral::disc_ks_test() from package KSgeneral (implemented by the same authors) in R.
Is it possible to use this implementation for two-sample KS-test, when we get samples $X = (X_1,\dots,X_n)$ and $Y = (Y_1,\dots,Y_m)$, to test whether these two samples come from the same discrete distribution? Let $F(x)$ be empirical cumulative distribution function of $Y$. Then, I believe, KSgeneral::disc_ks_test() is going to perform two-sample test for $X$ and $Y$, am I wrong?
