Estimating a discrete distribution via histogram selection

Nathalie Akakpo

ESAIM: Probability and Statistics (2011)

  • Volume: 15, page 1-29
  • ISSN: 1292-8100

Abstract

top
Our aim is to estimate the joint distribution of a finite sequence of independent categorical variables. We consider the collection of partitions into dyadic intervals and the associated histograms, and we select from the data the best histogram by minimizing a penalized least-squares criterion. The choice of the collection of partitions is inspired from approximation results due to DeVore and Yu. Our estimator satisfies a nonasymptotic oracle-type inequality and adaptivity properties in the minimax sense. Moreover, its computational complexity is only linear in the length of the sequence. We also use that estimator during the preliminary stage of a hybrid procedure for detecting multiple change-points in the joint distribution of the sequence. That second procedure still satisfies adaptivity properties and can be implemented efficiently. We provide a simulation study and apply the hybrid procedure to the segmentation of a DNA sequence.

How to cite

top

Akakpo, Nathalie. "Estimating a discrete distribution via histogram selection." ESAIM: Probability and Statistics 15 (2011): 1-29. <http://eudml.org/doc/277143>.

@article{Akakpo2011,
abstract = {Our aim is to estimate the joint distribution of a finite sequence of independent categorical variables. We consider the collection of partitions into dyadic intervals and the associated histograms, and we select from the data the best histogram by minimizing a penalized least-squares criterion. The choice of the collection of partitions is inspired from approximation results due to DeVore and Yu. Our estimator satisfies a nonasymptotic oracle-type inequality and adaptivity properties in the minimax sense. Moreover, its computational complexity is only linear in the length of the sequence. We also use that estimator during the preliminary stage of a hybrid procedure for detecting multiple change-points in the joint distribution of the sequence. That second procedure still satisfies adaptivity properties and can be implemented efficiently. We provide a simulation study and apply the hybrid procedure to the segmentation of a DNA sequence.},
author = {Akakpo, Nathalie},
journal = {ESAIM: Probability and Statistics},
keywords = {adaptive estimator; approximation result; categorical variable; change-point detection; minimax estimation; model selection; nonparametric estimation; penalized least-squares estimation},
language = {eng},
pages = {1-29},
publisher = {EDP-Sciences},
title = {Estimating a discrete distribution via histogram selection},
url = {http://eudml.org/doc/277143},
volume = {15},
year = {2011},
}

TY - JOUR
AU - Akakpo, Nathalie
TI - Estimating a discrete distribution via histogram selection
JO - ESAIM: Probability and Statistics
PY - 2011
PB - EDP-Sciences
VL - 15
SP - 1
EP - 29
AB - Our aim is to estimate the joint distribution of a finite sequence of independent categorical variables. We consider the collection of partitions into dyadic intervals and the associated histograms, and we select from the data the best histogram by minimizing a penalized least-squares criterion. The choice of the collection of partitions is inspired from approximation results due to DeVore and Yu. Our estimator satisfies a nonasymptotic oracle-type inequality and adaptivity properties in the minimax sense. Moreover, its computational complexity is only linear in the length of the sequence. We also use that estimator during the preliminary stage of a hybrid procedure for detecting multiple change-points in the joint distribution of the sequence. That second procedure still satisfies adaptivity properties and can be implemented efficiently. We provide a simulation study and apply the hybrid procedure to the segmentation of a DNA sequence.
LA - eng
KW - adaptive estimator; approximation result; categorical variable; change-point detection; minimax estimation; model selection; nonparametric estimation; penalized least-squares estimation
UR - http://eudml.org/doc/277143
ER -

References

top
  1. [1] M. Aerts and N. Veraverbeke, Bootstrapping a nonparametric polytomous regression model. Math. Meth. Statist.4 (1995) 189–200. Zbl0832.62031MR1335154
  2. [2] Y. Baraud and L. Birgé, Estimating the intensity of a random measure by histogram type estimators. Prob. Theory Relat. Fields143 (2009) 239–284. Zbl1149.62019MR2449129
  3. [3] A. Barron, L. Birgé and P. Massart, Risk bounds for model selection via penalization. Prob. Theory Relat. Fields113 (1999) 301–413. Zbl0946.62036MR1679028
  4. [4] C. Bennett and R. Sharpley, Interpolation of operators, volume 129 of Pure and Applied Mathematics. Academic Press Inc., Boston, M.A. (1988). Zbl0647.46057MR928802
  5. [5] L. Birgé, Model selection via testing: an alternative to (penalized) maximum likelihood estimators. Ann. Inst. H. Poincaré Probab. Statist.42 (2006) 273–325. Zbl1333.62094MR2219712
  6. [6] L. Birgé, Model selection for Poisson processes, in Asymptotics: Particles, Processes and Inverse Problems, Festschrift for Piet Groeneboom. IMS Lect. Notes Monograph Ser. 55. IMS, Beachwood, USA (2007) 32–64. Zbl1176.62082MR2459930
  7. [7] L. Birgé and P. Massart, Minimal penalties for Gaussian model selection. Prob. Theory Relat. Fields138 (2007) 33–73. Zbl1112.62082MR2288064
  8. [8] J.V. Braun and H.-G. Müller, Statistical methods for DNA sequence segmentation. Stat. Sci.13 (1998) 142–162. Zbl0960.62121
  9. [9] J.V. Braun, R.K. Braun and H.-G. Müller, Multiple changepoint fitting via quasilikelihood, with application to DNA sequence segmentation. Biometrika87 (2000) 301–314. Zbl0963.62067MR1782480
  10. [10] T.H. Cormen, C.E. Leiserson, R.L. Rivest and C. Stein, Introduction to algorithms. Second edition. MIT Press, Cambridge, MA (2001). Zbl1187.68679MR1848805
  11. [11] M. Csűrös, Algorithms for finding maximum-scoring segment sets, in Proc. of the 4th international workshop on algorithms in bioinformatics 2004. Lect. Notes Comput. Sci. 3240. Springer, Berlin, Heidelberg (2004) 62–73. MR2155594
  12. [12] R.A. DeVore and G.G. Lorentz, Constructive approximation. Springer-Verlag, Berlin, Heidelberg (1993). Zbl0797.41016MR1261635
  13. [13] R.A. DeVore and R.C. Sharpley, Maximal functions measuring smoothness. Mem. Amer. Math. Soc. 47 (1984) 293. Zbl0529.42005MR727820
  14. [14] R.A. DeVore and X.M. Yu, Degree of adaptive approximation. Math. Comp.55 (1990) 625–635. Zbl0723.41015MR1035930
  15. [15] C. Durot, E. Lebarbier and A.-S. Tocquet, Estimating the joint distribution of independent categorical variables via model selection. Bernoulli15 (2009) 475–507. Zbl1200.62024MR2543871
  16. [16] Y.-X. Fu and R.N. Curnow, Maximum likelihood estimation of multiple change points. Biometrika77 (1990) 562–565. Zbl0724.62025MR1087847
  17. [17] S. GeyS. and E. Lebarbier, Using CART to detect multiple change-points in the mean for large samples. SSB preprint, Research report No. 12 (2008). 
  18. [18] M. Hoebeke, P. Nicolas and P. Bessières, MuGeN: simultaneous exploration of multiple genomes and computer analysis results. Bioinformatics19 (2003) 859–864. 
  19. [19] E. Lebarbier, Quelques approches pour la détection de ruptures à horizon fini. Ph.D. thesis, Université Paris Sud, Orsay, 2002. 
  20. [20] E. Lebarbier and E. Nédélec, Change-points detection for discrete sequences via model selection. SSB preprint, Research Report No. 9 (2007). 
  21. [21] P. Massart, Concentration inequalities and model selection. Lectures from the 33rd Summer School on Probability Theory held in Saint-Flour, July 6–23, 2003. Lect. Notes Math. 1896. Springer, Berlin, Heidelberg (2007). Zbl1170.60006MR2319879
  22. [22] P. Nicolas et al., Mining Bacillus subtilis chromosome heterogeneities using hidden Markov models. Nucleic Acids Res.30 (2002) 1418–1426. 
  23. [23] W. Szpankowski, L. Szpankowski and W. Ren, An optimal DNA segmentation based on the MDL principle. Int. J. Bioinformatics Res. Appl.1 (2005) 3–17. 

NotesEmbed ?

top

You must be logged in to post comments.