What is an easy to understand step by step procedure on how to compute a distance between a cumulative distribution function and an empirical distribution function given a random sample using Kolmogorov-Smirnov distance.
An actual illustrative implementaton in R would be helpful.
Your problem of computing the KS distance, i.e. $$d_{KS}(F_1,F_2)=\sup\{ |F_1(x) - F_2(x)| \mbox{ for } x\in\mathbb{R}\}$$ is very much simplified for empirical data because the emprical CDF is a step function with a finite number of steps. You thus simply need to compare the empirical CDF with the other CDF at the observed data points. Example:
x <- rnorm(50)
# build empirical CDF (beware: returns a FUNCTION)
x.cdf <- ecdf(x)
# compute the maximum of absolute difference at the observed data points
d.KS <- max(abs(x.cdf(x) - pnorm(x)))
print(d.KS)
# for comparison (should yield the same value):
ks.test(x, pnorm)$statistic
If you want to compare two empirical CDFs of two random variables $X$ and $Y$, you must evaluate the difference at the union of data points observed for $X$ and $Y$. Example:
x <- rnorm(50)
y <- rnorm(50)
xy <- c(x,y)
d.KS <- max(abs(ecdf(x)(xy) - ecdf(y)(xy)))
The way to compare empirical distributions is to make cumulative distribution functions out of the raw data [use ecdf()] then compare them by plotting [via qqplot()]. And then use KS. Discrete distributions can have direct functions called probability mass functions. Continuous can not because any single realization instance (say 3.57867363474624) can not have a probability, only an interval can (say 3-4). You could compare histograms but ecdf are used far far more often.
