Nonnegative piecewise linear histograms*

Abstract

We introduce a modified version ƒnof the piecewiss linear hisiugrimi uf Beirlant et al. (1998) which is a true probability density, i.e., ƒn[d] 0 and [d]ƒn=1. We prove that ƒnestimates the underlying densitv ƒ strongly consistently in the L1mmn, derive large deviation inequalities for the t\\ error \\ƒn- f\\ and prove that £||/"-/|| tends to zero with the rate n -1\\3, We also show that the derivative lf'n estimates consistently in ine expected Lx error the derivative/ of sufficiently smooth density and evaluate the rate of convergence n-i/5 for Epf'n -f'% The estimator/" thus enables to approximate/in the Besov space with a guaranteed rate of convergence. Optimization of the smoothing parameter is also studied. The theoretical or experimentally approximated values of the expected errors E\\\\ƒn- f\\\\ and E||2ƒ'n-ƒ' are compared with tiie errors aCiiieveu u-y t"e histogram of Beirlant et ah, and other nonparametric methods.