On a variance dependent Dvoretzky-Kiefer-Wolfowitz inequality

08/09/2023

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by   Daniel Bartl, et al.

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Let X be a real-valued random variable with distribution function F. Set X_1,…, X_m to be independent copies of X and let F_m be the corresponding empirical distribution function. We show that there are absolute constants c_0 and c_1 such that if Δ≥ c_0loglog m/m, then with probability at least 1-2exp(-c_1Δ m), for every t∈ℝ that satisfies F(t)∈[Δ,1-Δ], |F_m(t) - F(t) | ≤√(Δmin{F(t),1-F(t)}) . Moreover, this estimate is optimal up to the multiplicative constants c_0 and c_1.