# Bounds for the expected value of the Kolmogorov-Smirnoff loss function

Let $$\mathcal{F}=\{F:\mathbb{R}\longrightarrow\mathbb{R}: \text{F is the CDF of some probability measure on \mathbb{R}}\}.$$ Consider the loss function, $$L:\mathcal F\times\mathcal F\to\mathbb R$$, defined by $$L(F,G) = \sup_{x\in\mathbb R}|F(x) - G(x)|.$$ (This is just the metric induced by the $$L^\infty$$-norm.)

Let $$\Theta$$ be a subset of $$\mathcal{F}$$. For $$\theta\in\Theta$$ and a random sample $$X_1,\ldots,X_n$$ drawn from $$\theta$$, let $$\hat{\theta}(X_1,\ldots,X_n)$$ be the associated empirical distribution function: $$\hat\theta(X_1,\ldots,X_n)(x) = \frac 1 n\sum_{i=1}^n I(X_i\leq x)$$ Are there "large" sets $$\Theta$$ for which one can one explicitly compute, or bound, the risk $$R$$ associated to the loss function $$L$$ in terms of $$n$$ and $$\Theta$$, where $$R$$ is defined by $$R(\theta,\hat{\theta}) = E_\theta[L(\theta,\hat\theta(X_1,\ldots,X_n))]?$$

It is easy to get an upper bound for $$\Theta = \mathcal{F}$$ using the Dvoretzky-Kiefer-Wolfowitz (DKW) inequality and $$EL = \int_0^\infty\Pr(L>u)du$$, see https://en.wikipedia.org/wiki/Dvoretzky–Kiefer–Wolfowitz_inequality#The_DKW_inequality
The DKW inequality gives us $$R(\theta,\hat\theta) = \int_0^\infty \Pr\left(L(\theta,\hat\theta)>u\right)du \leq 2\int_0^\infty e^{-2nu^2}du.$$
I'm not sure if tighter bounds exist for $$\Theta\subset\mathcal{F}$$.