Estimating the intensity of a random measure by histogram type estimators

Probability Theory and Related Fields volume 143, pages239–284(2009)Cite this article

Abstract

The purpose of this paper is to estimate the intensity of some random measure N on a set \({\mathcal{X}}\) by a piecewise constant function on a finite partition of \({\mathcal{X}}\) . Given a (possibly large) family \({\mathcal{M}}\) of candidate partitions, we build a piecewise constant estimator (histogram) on each of them and then use the data to select one estimator in the family. Choosing the square of a Hellinger-type distance as our loss function, we show that each estimator built on a given partition satisfies an analogue of the classical squared bias plus variance risk bound. Moreover, the selection procedure leads to a final estimator satisfying some oracle-type inequality, with, as usual, a possible loss corresponding to the complexity of the family \({\mathcal{M}}\) . When this complexity is not too high, the selected estimator has a risk bounded, up to a universal constant, by the smallest risk bound obtained for the estimators in the family. For suitable choices of the family of partitions, we deduce uniform risk bounds over various classes of intensities. Our approach applies to the estimation of the intensity of an inhomogenous Poisson process, among other counting processes, or the estimation of the mean of a random vector with nonnegative components.

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References

  1. 1.

    Andersen P., Borgan O., Gill R. and Keiding N. (1993). Statistical Models Based on Counting Processes. Springer, New York

    Google Scholar 

  2. 2.

    Antoniadis A. (1989). A penalty method for nonparametric estimation of the intensity function of a counting process. Ann. Inst. Statist. Math. 41: 781–807

    MATH  Article  MathSciNet  Google Scholar 

  3. 3.

    Antoniadis A., Besbeas P. and Sapatinas T. (2001). Wavelet shrinkage for natural exponential families with cubic variance functions. Sankhya 63: 309–327

    MathSciNet  Google Scholar 

  4. 4.

    Antoniadis A. and Sapatinas T. (2001). Wavelet shrinkage for natural exponential families with quadratic variance functions. Biometrika 88: 805–820

    MATH  Article  MathSciNet  Google Scholar 

  5. 5.

    Barron A.R., Birgé L. and Massart P. (1999). Risk bounds for model selection via penalization. Probab. Theory Relat. Fields 113: 301–415

    MATH  Article  Google Scholar 

  6. 6.

    Barron A.R. and Cover T.M. (1991). Minimum complexity density estimation. IEEE Trans. Inf. Theory 37: 1034–1054

    Article  MathSciNet  MATH  Google Scholar 

  7. 7.

    Birgé L. (1983). Approximation dans les espaces métriques et théorie de l’estimation. Z. Wahrscheinlichkeitstheorie Verw. Geb. 65: 181–237

    MATH  Article  Google Scholar 

  8. 8.

    Birgé L. (1986). On estimating a density using Hellinger distance and some other strange facts. Probab. Theory Relat. Fields 71: 271–291

    MATH  Article  MathSciNet  Google Scholar 

  9. 9.

    Birgé L. (2006). Model selection via testing : an alternative to (penalized) maximum likelihood estimators. Ann. Inst. Henri Poincaré Probab. Statist. 42: 273–325

    MATH  Article  Google Scholar 

  10. 10.

    Birgé, L.: Model selection for Poisson processes, in Asymptotics: particles, processes and inverse problems, Festschrift for Piet Groeneboom. Cator, E., Jongbloed, G., Kraaikamp, C., Lopuhaä, R., Wellner, J. (eds.) IMS Lecture Notes—Monograph Series, vol. 55 (2007)

  11. 11.

    Birgé L. and Massart P. (1998). Minimum contrast estimators on sieves: exponential bounds and rates of convergence. Bernoulli 4: 329–375

    MATH  Article  MathSciNet  Google Scholar 

  12. 12.

    Birgé L. and Massart P. (2001). Gaussian model selection. J. Eur. Math. Soc. 3: 203–268

    MATH  Article  Google Scholar 

  13. 13.

    Birgé L. and Rozenholc Y. (2006). How many bins should be put in a regular histogram. ESAIM-Probab. Statist. 10: 24–45

    MATH  Article  Google Scholar 

  14. 14.

    Breiman L., Friedman J.H., Olshen R.A. and Stone C.J. (1984). Classification and Regression Trees. Wadsworth, Belmont

    Google Scholar 

  15. 15.

    Brown L.D. and Low M.G. (1996). Asymptotic equivalence of nonparametric regression and white noise. Ann. Statist. 24: 2384–2398

    MATH  Article  MathSciNet  Google Scholar 

  16. 16.

    Brown L.D., Carter A.V., Low M.G. and Zhang C.-H. (2004). Equivalence theory for density estimation, Poisson processes, and Gaussian white noise with drift. Ann. Statist. 32: 2074–2097

    MATH  Article  MathSciNet  Google Scholar 

  17. 17.

    Castellan, G.: Modified Akaike’s criterion for histogram density estimation. Technical Report 99.61. Université Paris-Sud, Orsay (1999)

  18. 18.

    Castellan G. (2000). Sélection d’histogrammes à l’aide d’un critère de type Akaike. C.R.A.S. 330: 729–732

    MATH  MathSciNet  Google Scholar 

  19. 19.

    DeVore R.A. (1998). Nonlinear approximation. Acta Numer. 7: 51–150

    MathSciNet  Google Scholar 

  20. 20.

    DeVore R.A. and Lorentz G.G. (1993). Constructive Approximation. Springer, Berlin

    Google Scholar 

  21. 21.

    DeVore R.A. and Yu X.M. (1990). Degree of adaptive approximation. Math. Comp. 55: 625–635

    MATH  Article  MathSciNet  Google Scholar 

  22. 22.

    Donoho, D.L., Liu, R.C.: Geometrizing rates of convergence I. Technical report 137. Department of Statistics, University of California, Berkeley (1987)

  23. 23.

    Donoho D.L. and Liu R.C. (1991). Geometrizing rates of convergence II. Ann. Statist. 19: 633–667

    MATH  Article  MathSciNet  Google Scholar 

  24. 24.

    Donoho D.L. and Liu R.C. (1991). Geometrizing rates of convergence III. Ann. Statist. 19: 668–701

    MATH  Article  MathSciNet  Google Scholar 

  25. 25.

    Gey S. and Nédélec E. (2005). Model selection for CART regression trees. IEEE Trans. Inf. Theory 51: 658–670

    Article  Google Scholar 

  26. 26.

    Grégoire G. and Nembé J. (2000). Convergence rates for the minimum complexity estimator of counting process intensities. J. Nonparametr. Statist. 12: 611–643

    MATH  Article  MathSciNet  Google Scholar 

  27. 27.

    Kerkyacharian G. and Picard D. (2000). Thresholding algorithms, maxisets and well-concentrated bases. Test 9: 283–344

    MATH  Article  MathSciNet  Google Scholar 

  28. 28.

    Kolaczyk E. (1999). Wavelet shrinkage estimation of certain Poisson intensity signals using corrected threshold. Stat. Sin. 9: 119–135

    MATH  MathSciNet  Google Scholar 

  29. 29.

    Kolaczyk E. and Nowak R. (2004). Multiscale likelihood analysis and complexity penalized estimation. Ann. Statist. 32: 500–527

    MATH  Article  MathSciNet  Google Scholar 

  30. 30.

    Laurent B. and Massart P. (2000). Adaptive estimation of a quadratic functional by model selection. Ann. Statist. 28: 1302–1338

    MATH  Article  MathSciNet  Google Scholar 

  31. 31.

    Le Cam, L.M.: Limits of experiments. In: Proceedings of 6th Berkeley Symposium on Math. Stat. and Prob. I, 245–261 (1972)

  32. 32.

    Le Cam L.M. (1973). Convergence of estimates under dimensionality restrictions. Ann. Statist. 1: 38–53

    MATH  Article  MathSciNet  Google Scholar 

  33. 33.

    Le Cam L.M. (1986). Asymptotic methods in statistical decision theory. Springer, New York

    Google Scholar 

  34. 34.

    Lepskii O.V. (1991). Asymptotically minimax adaptive estimation I: Upper bounds. Optimally adaptive estimates. Theory Probab. Appl. 36: 682–697

    Article  MathSciNet  Google Scholar 

  35. 35.

    Massart P. (2000). Some applications of concentration inequalities to Statistics. Ann. Fac. Sci. Toulouse IX: 245–303

    MathSciNet  Google Scholar 

  36. 36.

    Nussbaum M. (1996). Asymptotic equivalence of density estimation and Gaussian white noise. Ann. Statist. 24: 2399–2430

    MATH  Article  MathSciNet  Google Scholar 

  37. 37.

    Patil P.N. and Wood A.T. (2004). A counting process intensity estimation by orthogonal wavelet methods. Bernoulli 10: 1–24

    MATH  Article  MathSciNet  Google Scholar 

  38. 38.

    Reynaud-Bouret P. (2003). Adaptive estimation of the intensity of inhomogeneous Poisson processes via concentration inequalities. Probab. Theory Relat. Fields 126: 103–153

    MATH  Article  MathSciNet  Google Scholar 

  39. 39.

    Reynaud-Bouret P. (2006). Penalized projection estimators of the Aalen multiplicative intensity. Bernoulli 12: 633–661

    MATH  Article  MathSciNet  Google Scholar 

  40. 40.

    Stanley R.P. (1999). Enumerative Combinatorics, Vol. 2. Cambridge University Press, Cambridge

    Google Scholar 

  41. 41.

    van de Geer S. (1995). Exponential inequalities for martingales, with application to maximum likelihood estimation for counting processes. Ann. Statist. 23: 1779–1801

    MATH  Article  MathSciNet  Google Scholar 

  42. 42.

    Wu S.S. and Wells M.T. (2003). Nonparametric estimation of hazard functions by wavelet methods. J. Nonparametr. Statist. 15: 187–203

    MATH  Article  MathSciNet  Google Scholar 

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Author information

Affiliations

  1. Laboratoire J-A Dieudonné, Université de Nice Sophia-Antipolis, Parc Valrose, 06108, Nice Cedex 02, France

    Yannick Baraud

  2. Laboratoire de Probabilités et Modèles Aléatoires, boîte 188, Université Paris VI, 4 place Jussieu, 75252, Paris Cedex 05, France

    Lucien Birgé

Authors
  1. Yannick Baraud
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  2. Lucien Birgé
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Correspondence to Lucien Birgé.

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Baraud, Y., Birgé, L. Estimating the intensity of a random measure by histogram type estimators. Probab. Theory Relat. Fields 143, 239–284 (2009). https://doi.org/10.1007/s00440-007-0126-6

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Keywords

  • Model selection
  • Histogram
  • Discrete data
  • Poisson process
  • Intensity estimation
  • Adaptive estimation

Mathematics Subject Classification (2000)

  • 62G05