Nonnegative piecewise linear histograms *

We introduce a modified version ?_nof the piecewiss linear hisiugrimi uf Beirlant et al. (1998) which is a true probability density, i.e., ?_n[d] 0 and [d]?_n=1. We prove that ?_nestimates the underlying densitv ? strongly consistently in the L₁mmn, derive large deviation inequalities for the t error ?_n- f and prove that £||/"-/|| tends to zero with the rate n ^-13, 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?_n- f and E||2?'_n-?' are compared with tiie errors aCiiieveu u-y t"e histogram of Beirlant et ah, and other nonparametric methods.