  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
