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.

Keywords: regular histograms, density estimation, penalized maximum likelihood, model selection

@article{PS_2006__10__24_0, author = {Birg\'e, Lucien and Rozenholc, Yves}, title = {How many bins should be put in a regular histogram}, journal = {ESAIM: Probability and Statistics}, pages = {24--45}, publisher = {EDP-Sciences}, volume = {10}, year = {2006}, doi = {10.1051/ps:2006001}, mrnumber = {2197101}, zbl = {1136.62329}, language = {en}, url = {http://www.numdam.org/articles/10.1051/ps:2006001/} }

TY - JOUR AU - Birgé, Lucien AU - Rozenholc, Yves TI - How many bins should be put in a regular histogram JO - ESAIM: Probability and Statistics PY - 2006 SP - 24 EP - 45 VL - 10 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2006001/ DO - 10.1051/ps:2006001 LA - en ID - PS_2006__10__24_0 ER -

Birgé, Lucien; Rozenholc, Yves. How many bins should be put in a regular histogram. ESAIM: Probability and Statistics, Volume 10 (2006), pp. 24-45. doi : 10.1051/ps:2006001. http://www.numdam.org/articles/10.1051/ps:2006001/

[1] A new look at the statistical model identification. IEEE Trans. Automatic Control 19 (1974) 716-723. | Zbl

,[2] Risk bounds for model selection via penalization. Probab. Theory Relat. Fields 113 (1999) 301-415. | Zbl

, and .[3] From model selection to adaptive estimation, in Festschrift for Lucien Le Cam: Research Papers in Probability and Statistics, D. Pollard, E. Torgersen and G. Yang, Eds., Springer-Verlag, New York (1997) 55-87. | Zbl

and ,[4] Gaussian model selection. J. Eur. Math. Soc. 3 (2001) 203-268. | Zbl

and ,[5] Modified Akaike's criterion for histogram density estimation. Technical Report. Université Paris-Sud, Orsay (1999).

,[6] Sélection d'histogrammes à l'aide d'un critère de type Akaike. CRAS 330 (2000) 729-732. | Zbl

,[7] The construction of optimal histograms. Commun. Stat., Theory Methods 17 (1988) 2921-2931. | Zbl

,[8] A Course in Density Estimation. Birkhäuser, Boston (1987). | MR | Zbl

,[9] Nonparametric Density Estimation: The ${L}_{1}$ View. John Wiley, New York (1985). | MR | Zbl

, and ,[10] Combinatorial Methods in Density Estimation. Springer-Verlag, New York (2001). | MR | Zbl

and ,[11] On the histogram as a density estimator: ${L}_{2}$ theory. Z. Wahrscheinlichkeitstheor. Verw. Geb. 57 (1981) 453-476. | Zbl

and ,[12] Akaike's information criterion and Kullback-Leibler loss for histogram density estimation. Probab. Theory Relat. Fields 85 (1990) 449-467. | Zbl

,[13] On stochastic complexity and nonparametric density estimation. Biometrika 75 (1988) 705-714. | Zbl

and ,[14] Selecting the number of bins in a histogram: A decision theoretic approach. J. Stat. Plann. Inference 61 (1997) 49-59. | Zbl

and ,[15] Some new methods for wavelet density estimation. Sankhya, Series A 63 (2001) 394-411.

, and ,[16] On two recent papers of Y. Kanazawa. Statist. Probab. Lett. 24 (1995) 269-271. | Zbl

,[17] Hellinger distance and Akaike's information criterion for the histogram. Statist. Probab. Lett. 17 (1993) 293-298. | Zbl

,[18] Asymptotic Methods in Statistical Decision Theory. Springer-Verlag, New York (1986). | MR | Zbl

,[19] Asymptotics in Statistics: Some Basic Concepts. Second Edition. Springer-Verlag, New York (2000). | MR | Zbl

and ,[20] Stochastic complexity and the MDL principle. Econ. Rev. 6 (1987) 85-102. | Zbl

,[21] Empirical choice of histograms and kernel density estimators. Scand. J. Statist. 9 (1982) 65-78. | Zbl

,[22] On optimal and databased histograms. Biometrika 66 (1979) 605-610. | Zbl

,[23] The choice of a class interval. J. Am. Stat. Assoc. 21 (1926) 65-66.

,[24] Akaike's information criterion and the histogram. Biometrika. 74 (1987) 636-639. | Zbl

,[25] The maximal smoothing principle in density estimation. J. Am. Stat. Assoc. 85 (1990) 470-477.

,[26] Data-based choice of histogram bin width. Am. Statistician 51 (1997) 59-64.

,*Cited by Sources: *