Nonnegative piecewise linear histograms * |
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Authors: | Alain Berlinet Igor Vajda |
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Affiliation: | 1. University of Montpellier II , France;2. Czech Academy of Sciences , Czech Republic |
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Abstract: | 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. |
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Keywords: | Density estimation Density derivative estimation Histogram Modified histogram Mean integrated absolute errors Asymptotics of errors Optimization |
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