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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 smoothnessassumptions. For fixed number n of i.i.d. random vari- ables X1;X2; : : : ;Xnwith 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 theGrenander estimator with probability one by the classical Kolmogorov-Smirnovstatistic. Fur- ther, we consider some interesting implications of the latterinequality

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