An inequality concerning the deviation between theoretical and empirical distributions (Q1117634)

An exponential inequality is proved for the uniform deviation \[ \sup_{A\in {\mathcal A}}| \quad \mu_ n(A)-\mu (A)| \] between a theoretical distribution \(\mu\) in \({\mathbb{R}}^ d\) and the corresponding empirical distribution \(\mu_ n\) when \({\mathcal A}\) is a Vapnik- Chervonenkis class of Borel sets with \(\sup_{A\in {\mathcal A}}\mu (A)\leq \delta \leq 1/8\). This result is applied to obtain a bound for the rate of the uniform consistency of the nearest neighbor density estimator proposed by \textit{D. O. Loftsgaaden} and \textit{C. P. Quesenberry} [Ann. Math. Statistics 36, 1049-1051 (1965; Zbl 0132.389)].