Consider the empirical and population CDF, $$ F_n(t) = \frac{1}{n} \sum_{i=1}^n 1\{X_i \leq t\} \quad \mbox{and} \quad F(t) = \mathbb{E} [F_n(t)], $$ where above $X_1, \dots, X_n$ are iid, real-valued random variables and the expectation is taken with respect to the same distribution.

It is well-known (the DKW inequality with sharp constants due to Massart) that $$ \mathbb{P}\big(\sqrt{n} \|F_n - F\|_\infty > \lambda\big) \leq 2 \exp(-2\lambda^2), \quad \mbox{for all}~\lambda > 0.$$ Above, $\|\cdot\|_\infty$ denotes the supremum norm.

I am wondering if there is a normalized version of this inequality available. Specifically, define $$ \widehat{ \sigma}_n(t) = F_n(t) (1 - F_n(t)) $$ This can be seen as a sample approximation to the true variance since $\widehat{ \sigma}_n(t) \to F(t)(1- F(t))$ a.s. by continuous mapping and the strong law as $n \to \infty$. The quantity $F(t) (1-F(t))$ is the variance of the random variable $B(t) := 1\{X \leq t\}$.

Question: Consider the normalized quantity $$ \sup_{t} \frac{|F_n(t) - F(t)|}{\sqrt{\widehat{\sigma}_n(t)}} \equiv \| (\widehat{\sigma}_n)^{-1/2}\cdot (F_n - F)\|_\infty. $$

Is it possible to give an exponential tail bound for the quantity $$ \mathbb{P}\big(\sqrt{n}\| (\widehat{\sigma}_n)^{-1/2}\cdot (F_n - F)\|_\infty > \lambda) $$ for all $\lambda > 0$?

The closest I could find is a paper by Bercu et al 2002 (Theorem 1.1) but their result does not have explicit constants and has a slightly different normalization.

Consider the empirical and population CDF, $$ F_n(t) = \frac{1}{n} \sum_{i=1}^n 1\{X_i \leq t\} \quad \mbox{and} \quad F(t) = \mathbb{E} [F_n(t)], $$ where above $X_1, \dots, X_n$ are iid, real-valued random variables and the expectation is taken with respect to the same distribution.

It is well-known (the DKW inequality with sharp constants due to Massart) that $$ \mathbb{P}\big(\sqrt{n} \|F_n - F\|_\infty > \lambda\big) \leq 2 \exp(-2\lambda^2), \quad \mbox{for all}~\lambda > 0.$$ Above, $\|\cdot\|_\infty$ denotes the supremum norm.

I am wondering if there is a normalized version of this inequality available. Specifically, define $$ \widehat{ \sigma}_n(t) = F_n(t) (1 - F_n(t)) $$ This can be seen as a sample approximation to the true variance since $\widehat{ \sigma}_n(t) \to F(t)(1- F(t))$ a.s. by continuous mapping and the strong law as $n \to \infty$. The quantity $F(t) (1-F(t))$ is the variance of the random variable $B(t) := 1\{X \leq t\}$.

Question: Consider the normalized quantity $$ \sup_{t} \frac{|F_n(t) - F(t)|}{\sqrt{\widehat{\sigma}_n(t)}} $$ Does a similar inequality exist?

Consider the empirical and population CDF, $$ F_n(t) = \frac{1}{n} \sum_{i=1}^n 1\{X_i \leq t\} \quad \mbox{and} \quad F(t) = \mathbb{E} [F_n(t)], $$ where above $X_1, \dots, X_n$ are iid, real-valued random variables and the expectation is taken with respect to the same distribution.

It is well-known (the DKW inequality with sharp constants due to Massart) that $$ \mathbb{P}\big(\sqrt{n} \|F_n - F\|_\infty > \lambda\big) \leq 2 \exp(-2\lambda^2), \quad \mbox{for all}~\lambda > 0.$$ Above, $\|\cdot\|_\infty$ denotes the supremum norm.

I am wondering if there is a normalized version of this inequality available. Specifically, define $$ \widehat{ \sigma}_n(t) = F_n(t) (1 - F_n(t)) $$ This can be seen as a sample approximation to the true variance since $\widehat{ \sigma}_n(t) \to F(t)(1- F(t))$ a.s. by continuous mapping and the strong law as $n \to \infty$. The quantity $F(t) (1-F(t))$ is the variance of the random variable $B(t) := 1\{X \leq t\}$.

Question: Consider the normalized quantity $$ \sup_{t} \frac{|F_n(t) - F(t)|}{\sqrt{\widehat{\sigma}_n(t)}} \equiv \| (\widehat{\sigma}_n)^{-1/2}\cdot (F_n - F)\|_\infty. $$

Is it possible to give an exponential tail bound for the quantity $$ \mathbb{P}\big(\sqrt{n}\| (\widehat{\sigma}_n)^{-1/2}\cdot (F_n - F)\|_\infty > \lambda) $$ for all $\lambda > 0$?

The closest I could find is a paper by Bercu et al 2002 (Theorem 1.1) but their result does not have explicit constants.

Consider the empirical and population CDF, $$ F_n(t) = \frac{1}{n} \sum_{i=1}^n 1\{X_i \leq t\} \quad \mbox{and} \quad F(t) = \mathbb{E} [F_n(t)], $$ where above $X_1, \dots, X_n$ are iid, real-valued random variables and the expectation is taken with respect to the same distribution.

It is well-known (the DKW inequality with sharp constants due to Massart) that $$ \mathbb{P}\big(\sqrt{n} \|F_n - F\|_\infty > \lambda\big) \leq 2 \exp(-2\lambda^2), \quad \mbox{for all}~\lambda > 0.$$ Above, $\|\cdot\|_\infty$ denotes the supremum norm.

I am wondering if there is a normalized version of this inequality available. Specifically, define $$ \widehat{ \sigma}_n(t) = F_n(t) (1 - F_n(t)) $$ This can be seen as a sample approximation to the true variance since $\widehat{ \sigma}_n(t) \to F(t)(1- F(t))$ a.s. by continuous mapping and the strong law as $n \to \infty$. The quantity $F(t) (1-F(t))$ is the variance of the random variable $B(t) := 1\{X \leq t\}$.

Question: Consider the normalized quantity $$ \sup_{t} \frac{|F_n(t) - F(t)|}{\sqrt{\widehat{\sigma}_n(t)}} \equiv \| (\widehat{\sigma}_n)^{-1/2}\cdot (F_n - F)\|_\infty. $$

Is it possible to give an exponential tail bound for the quantity $$ \mathbb{P}\big(\sqrt{n}\| (\widehat{\sigma}_n)^{-1/2}\cdot (F_n - F)\|_\infty > \lambda) $$ for all $\lambda > 0$?

The closest I could find is a paper by Bercu et al 2002 (Theorem 1.1) but their result does not have explicit constants and has a slightly different normalization.