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 $$ R(\theta,\hat{\theta}) = E_\theta[L(\theta,\hat\theta(X_1,\ldots,X_n))]? $$


1 Answer 1


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 $\Theta\subset\mathcal{F}$.

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