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Chung-Smirnov property for Bernstein estimators of distribution functions. (English) Zbl 1157.62016
Summary: We show that the Chung-Smirnov property [K.-L. Chung, Trans. Am. Math. Soc. 67, 36–50 (1949; Zbl 0034.22602), N. V. Smirnov, Ups. Mat. Nauk. 10, 179–206 (1944; Zbl 0063.07087)] holds for Bernstein estimators of distribution functions under different conditions on the underlying distribution to be estimated. In doing so, we obtain general results that characterise the closeness between these Bernstein estimators and the empirical distribution function \(F_n\).

MSC:
62G07 Density estimation
60F15 Strong limit theorems
62G30 Order statistics; empirical distribution functions
62G20 Asymptotic properties of nonparametric inference
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