A sharp estimate of the binomial mean absolute deviation with applications

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Minimax estimation of discrete distributions (2015, methodology): “Let {Tn} be any estimator sequence in the experiments (PO', 8 E IRk).”
Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance (2017, unknown): “2(1 −µ(S)) + q |{i: Qk i ∩S6= ∅}|/n. Proof. Let Q(S) = {i: Qk i ∩S6= ∅}, and write S′ = [ i∈Q(S) Qk i . Since nµˆn(Qk i ) is a Binomial random variable with parameters (n,µ(Qk i )), we have the bound [BK13] IE|µ(Qk i ) −µˆn(Qki)| ≤ q µ(Qk i )/n∧2µ(Qk i ). Applying the ﬁrst bound on S′ and the second”
Minimax Estimation of the $L_1$ Distance (2017, background): “(92) The following lemma provides sharp estimates of E|q̂ − q|, where nq̂ ∼ Poi(nq), which can be viewed as an analog of the binomial case studied in [28].”