Estimation of the Weighted Integrated Square Error of the Grenander Estimator by the Kolmogorov-Smirnov Statistic

Author: Malkhaz Shashiashvili
Keywords: Maximum likelihood estimators, consistent estimation, Grenander estimator, nonparametric estimation of nonincreasing densities, the weighted integrated square error, Kolmogorov-Smirnov statistic
Annotation:

We consider in this paper the Grenander estimator of unbounded, in general, nonincreasing densities on the interval [0; 1] without any smoothness assumptions. For fixed 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.

Lecture files:

Malkhaz Shashiashvili,ENG [ka]
Malkhaz Shashiashvili,GEO [ka]