Estimation of the weighted integrated square error of the Grenander estimator by the Kolmogorov-Smirnov statistic

10/08/2018

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by   Malkhaz Shashiashvili, et al.

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We consider in this paper the Grenander estimator of unbounded, in general, nonincreasing densities on the interval [0; 1] without any smoothness assumptions. For fixed number n of i.i.d. random vari- ables X1;X2; : : : ;Xn with values in [0; 1] and the nonincreasing den- sity function f(x), 0 < x < 1, we prove an inequality bounding the weighted integrated square error of the Grenander estimator with probability one by the classical Kolmogorov-Smirnov statistic. Fur- ther, we consider some interesting implications of the latter inequality