On piecewise linear density estimators J. Beirlant* Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-300 Leuven, Belgium A. Berlinet Universite  Montpellier II, Place Bataillon, 34095 Montpellier Cedex, France L. GyoÈ r® Department of Mathematics and Computer Science, Technical University of Budapest, 1521 Stoczek u. 2, Budapest, Hungary We study piecewise linear density estimators from the L 1 point of view: the frequency polygons investigated by SCOTT (1985) and JONES et al. (1997), and a new piecewise linear histogram. In contrast to the earlier proposals, a unique multivariate generalization of the new piecewise linear histogram is available. All these estimators are shown to be universally L 1 strongly consistent. We derive large deviation inequalities. For twice dierentiable densities with compact support their expected L 1 error is shown to have the same rate of convergence as have kernel density estimators. Some simulated examples are presented. Key Words and Phrases: nonparametric density estimation, histogram, asymptotics. 1 Introduction We consider the problem of estimating consistently an unknown probability density function in L 1 from a sample of independent and identically distributed (i.i.d.) random variables by means of histogram-based estimators. The histogram is attract- ive from a computational point of view, but for twice dierentiable densities it has a worse rate of convergence than the kernel density estimator. The frequency polygon (SCOTT, 1985a), the edge frequency polygon (JONES et al., 1998) and the piecewise linear histogram introduced next combine the two advantages: they are computa- tionally ecient (quickly evaluated and updated) and therefore well adapted to on- line high data speed signal processing. Theyalso have a good rate of convergence for smooth densities. In order to estimate a univariate density f we are given X 1 ,..., X n , a sequence of i.i.d. random variables with common density f. Let PfA nj g and RfA  nj ; A  nj g be #VVS, 1999. Published by Blackwell Publishers, 108 Cowley Road, Oxford OX4 1JF, UK and 350 Main Street, Malden, MA 02148, USA. Statistica Neerlandica (1999) Vol. 53, nr. 3, pp. 287±308 287 * e-mail: jan.beirlant@wis.kuleuven.ac.be