# Confidence interval for the distance between two CDF

Let $X$ and $Y$ be two real random variables with cumulative distribution functions $F_X$ and $F_Y$, respectively.

Define $d = \sup(F_X(x) - F_Y(x))$. If $\hat F_X$ and $\hat F_Y$ are empirical versions of $F_X$ and $F_Y$ respectively, then an estimator of $d$ is $\hat d = \sup(\hat F_X(x) - \hat F_Y(x))$

Does anyone know how to build a confidence interval for $d$ based on $\hat d$?

• This looks like the test statistic of the Kolmogorov Smirnov test for the equality of two distributions . So check the derivation. The confidence interval will be the acceptance region of the test. – Placidia Sep 15 '17 at 22:01
• K-S acceptance region is constructed under the assumption that $F_X = F_Y$. I don't know how to take into consideration the possible inequality of the cdf's – Mur1lo Sep 15 '17 at 22:53
• Are you looking for one number to quantify the distance between two CDFs or a whole confidence region where you allow for some uncertainty at every point along the CDF? – Dave Jul 30 at 9:53

You can construct a confidence interval by bootstrap sampling

First sample from your empirical distributions $X$ and $Y$ to generate populations of CDFs $\left\{(\hat F_X)_1,\dots,(\hat F_X)_n\right\}$ and $\left\{(\hat F_Y)_1,\dots,(\hat F_Y)_n\right\}$

Next sample pairs of these CDFs to compute a population of bootstrap estimates of $\hat d$: $\left\{\hat d_1,\dots,\hat d_m\right\} = \left\{\sup\left((\hat F_X)_{i_1}-(\hat F_Y)_{j_1}\right),\dots,\sup\left((\hat F_X)_{i_m}-(\hat F_Y)_{j_m}\right)\right\}$

Then compute the differences between these bootstrap estimates $\hat d_*$ and the true $\hat d$: $\{\delta_1,\dots,\delta_m\} = \{\hat d_1 - \hat d,\dots,\hat d_m - \hat d\}$

And finally construct your $\alpha$ confidence interval: $\left[\hat d-\delta_{\lceil m\cdot(1-\alpha/2)\rceil}, \hat d-\delta_{\lfloor m\cdot\alpha /2 \rfloor }\right]$

This construction is equivalent to the pivot confidence interval

Note that you will need to make your numbers of samples $n$ and $m$ sufficiently large for this confidence interval to be reliable.