How many bins should be put in a regular histogram

Issue		ESAIM: PS Volume 10, 2006


Page(s)		24 - 45
DOI		10.1051/ps:2006001
Published online		31 January 2006

ESAIM: P&S, February 2006, Vol. 10, pp. 24-45
DOI: 10.1051/ps:2006001

How many bins should be put in a regular histogram

Lucien Birgé¹ and Yves Rozenholc²

¹ UMR 7599 "Probabilités et modèles aléatoires", Laboratoire de Probabilités, boîte 188, Université Paris VI, 4 Place Jussieu, 75252 Paris Cedex 05, France; lb@ccr.jussieu.fr;
² MAP5-UMR CNRS 8145, Université Paris 5, 45 rue des Saints-Pères, 75270 Paris Cedex 06, France; yves.rozenholc@math-info.univ.paris5.fr

(Received July 7, 2003. Revised September 1, 2004 and May 11, 2005. / Published online: 31 January 2006)

Abstract
Given an n-sample from some unknown density f on [0,1], it is easy to construct an histogram of the data based on some given partition of [0,1], but not so much is known about an optimal choice of the partition, especially when the data set is not large, even if one restricts to partitions into intervals of equal length. Existing methods are either rules of thumbs or based on asymptotic considerations and often involve some smoothness properties of f. Our purpose in this paper is to give an automatic, easy to program and efficient method to choose the number of bins of the partition from the data. It is based on bounds on the risk of penalized maximum likelihood estimators due to Castellan and heavy simulations which allowed us to optimize the form of the penalty function. These simulations show that the method works quite well for sample sizes as small as 25.

Mathematics Subject Classification. 62E25, 62G05

Key words: Regular histograms, density estimation, penalized maximum likelihood, model selection.

© EDP Sciences, SMAI 2006

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