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Near optimal thresholding estimation of a Poisson intensity on the real line. (English) Zbl 1329.62176

Summary: The purpose of this paper is to estimate the intensity of a Poisson process \(N\) by using thresholding rules. In this paper, the intensity, defined as the derivative of the mean measure of \(N\) with respect to \(ndx\) where \(n\) is a fixed parameter, is assumed to be non-compactly supported. The estimator \(\tilde f_{n,\gamma}\) based on random thresholds is proved to achieve the same performance as the oracle estimator up to a possible logarithmic term. Then, minimax properties of \(\tilde f_{n, \gamma}\) on Besov spaces \(\mathcal B^{\alpha}_{p,q}\) are established. Under mild assumptions, we prove that \[ \sup_{f\in \mathcal B^{\alpha}_{p,q}\cap \mathbb L_{\infty}} \mathbb E\|\tilde f_{n, \gamma}-f\|{2\atop 2} \leq C\bigg(\frac{\log n}{n}\bigg)^{\frac{\alpha}{\alpha+\frac{1}{2}+\left(\frac{1}{2}-\frac{1}{p}\right)_{+}}} \] and the lower bound of the minimax risk for \(\mathcal B^\alpha_{p, q}\cap \mathbb L_\infty\) coincides with the previous upper bound up to the logarithmic term. This new result has two consequences. First, it establishes that the minimax rate of Besov spaces \(\mathcal B^{\alpha}_{p,q}\) with \(p\leq 2\) when non compactly supported functions are considered is the same as for compactly supported functions up to a logarithmic term. When \(p>2\), the rate exponent, which depends on \(p\), deteriorates when \(p\) increases, which means that the support plays a harmful role in this case. Furthermore, \(\tilde f_{n, \gamma}\) is adaptive minimax up to a logarithmic term. Our procedure is based on data-driven thresholds. As usual, they depend on a tuning parameter \(\gamma \) whose optimal value is hard to estimate from the data. In this paper, we study the problem of calibrating \(\gamma\) both theoretically and practically. Finally, some simulations are provided, proving the excellent practical behavior of our procedure with respect to the support issue.

MSC:

62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
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[1] Allen, D.M. (1974). The relationship between variable selection and data augmentation and a method for prediction., Technometrics 16 125-127. · Zbl 0286.62044 · doi:10.2307/1267500
[2] Anscombe, F.J. (1948). The transformation of Poisson, binomial and negative binomial data., Biometrika 35 246-254. · Zbl 0032.03702 · doi:10.1093/biomet/35.3-4.246
[3] Arlot, S. and Massart, P. (2009). Data-driven calibration of penalties for least-squares regression., Journal of Machine Learning Research 10 245-279.
[4] Autin, F. (2006). Maxiset for density estimation on \Bbb R., Mathematical Methods of Statistics 15 (2) 123-145.
[5] Autin, F., Picard, D. and Rivoirard, V. (2006). Large variance Gaussian priors in Bayesian nonparametric estimation: a maxiset approach., Mathematical Methods of Statistics 15 (4) 349-373.
[6] Baraud, Y. and Birgé, L. (2009). Estimating the intensity of a random measure by histogram type estimators., Probability Theory and Related Fields 143 (1-2) 239-284. · Zbl 1149.62019 · doi:10.1007/s00440-007-0126-6
[7] Bertin, K. and Rivoirard, V. (2009). Maxiset in sup-norm for kernel estimators., Test 18 (3) 475-496. · Zbl 1203.62050 · doi:10.1007/s11749-008-0109-7
[8] Besbeas, P., De Feis, I. and Sapatinas, T. (2004). A Comparative Simulation Study of Wavelet Shrinkage Estimators For Poisson Counts., Internat. Statist. Rev. 72 (2) 209-237. · Zbl 1211.62055 · doi:10.1111/j.1751-5823.2004.tb00234.x
[9] Birgé, L. (2001). A new look at an old result: Fano’s Lemma., Manuscript.
[10] Birgé, L. (2007). Model selection for Poisson processes., Asymptotics: Particles, Processes and Inverse Problems: Festschrift for Piet Groeneboom (Eric A. Cator, Geurt Jongbloed, Cor Kraaikamp, Hendrik P. Lopuhaä, Jon A. Wellner, eds.) 32-64. · Zbl 1176.62082
[11] Birgé, L. and Massart, P. (2007). Minimal penalties for Gaussian model selection., Probab. Theory Related Fields 138 (1-2) 33-73. · Zbl 1112.62082 · doi:10.1007/s00440-006-0011-8
[12] Bretagnolle, J. and Huber, C. (1979). Estimation des densités: risque minimax., Z. Wahrsch. Verw. Gebiete 47 (2) 119-137. · Zbl 0413.62024 · doi:10.1007/BF00535278
[13] Cavalier, L. and Koo, J.Y. (2002). Poisson intensity estimation for tomographic data using a wavelet shrinkage approach., IEEE Trans. Inform. Theory 48 (10) 2794-2802. · Zbl 1062.92042 · doi:10.1109/TIT.2002.802632
[14] Cavalier, L. and Raimondo, M. (2006). On choosing wavelet resolution in image deblurring., Manuscript.
[15] Cohen, A., Daubechies, I. and Feauveau, J.C. (1992). Biorthogonal bases of compactly supported wavelets., Comm. Pure Appl. Math. 45 (5) 485-560. · Zbl 0776.42020 · doi:10.1002/cpa.3160450502
[16] Coronel-Brizio, H.F. and Hernandez-Montoya, A.R. (2005). On fitting the Pareto-Lévy distribution to stock market index data: Selecting a suitable cutoff value., Physica A: Statistical Mechanics and its Applications 354 437-449.
[17] Delyon, B. and Juditsky, A. (1997). On the computation of wavelet coefficients., J. Approx. Theory 88 (1) 47-79. · Zbl 0862.42023 · doi:10.1006/jath.1996.3008
[18] DeVore, R.A. and Lorentz, G.G. (1993)., Constructive approximation . Springer-Verlag, Berlin. · Zbl 0797.41016
[19] Donoho, D.L. (1993). Nonlinear wavelet methods for recovery of signals, densities, and spectra from indirect and noisy data., Different perspectives on wavelets (San Antonio, TX, 1993) , Proc. Sympos. Appl. Math., 47 , Amer. Math. Soc., Providence, RI 173-205. · Zbl 0786.62094 · doi:10.1090/psapm/047/1268002
[20] Donoho, D.L. and Johnstone, I.M. (1994). Ideal spatial adaptation by wavelet shrinkage., Biometrika 81 (3) 425-455. · Zbl 0815.62019 · doi:10.1093/biomet/81.3.425
[21] Donoho, D.L., Johnstone, I.M., Kerkyacharian, G. and Picard, D. (1996). Density estimation by wavelet thresholding., Annals of Statistics 24 (2) 508-539. · Zbl 0860.62032 · doi:10.1214/aos/1032894451
[22] Figueroa-López, J.E. and Houdré, C. (2006). Risk bounds for the non-parametric estimation of Lévy processes., IMS Lecture Notes-Monograph series High Dimensional Probability 51 96-116. · Zbl 1117.62085 · doi:10.1214/074921706000000789
[23] Geisser, S. (1975). The predictive sample reuse method with applications., J. Amer. Statist. Assoc. 70 320-328. · Zbl 0321.62077 · doi:10.2307/2285815
[24] Golubev, G.K. (1992). Nonparametric estimation of smooth densities of a distribution in \(\mathbbL _2\)., Problems Inform. Transmission 28 (1) 44-54. · Zbl 0785.62039
[25] Härdle, W., Kerkyacharian, G., Picard, D. and Tsybakov, A. (1998)., Wavelets, approximation and statistical applications . Lecture Notes in Statistics 129 . Springer-Verlag, New York. · Zbl 0899.62002
[26] Houghton, J.C. (1988). Use of the truncated shifted Pareto distribution in assessing size distribution of oil and gas fields., Mathematical geology 20 (8) 907-937.
[27] Johnson, W.B. (1985). Best Constants in Moment Inequalities for Linear Combinations of Independent and Exchangeable Random Variables., Annals of probability 13 (1) 234-253. · Zbl 0564.60020 · doi:10.1214/aop/1176993078
[28] Johnstone, I.M. (1994). Minimax Bayes, asymptotic minimax and sparse wavelet priors., Statistical decision theory and related topics. Springer, New York 303-326. · Zbl 0815.62017 · doi:10.1007/978-1-4612-2618-5_23
[29] Juditsky, A. and Lambert-Lacroix, S. (2004). On minimax density estimation on \Bbb R., Bernoulli 10 (2) 187-220. · Zbl 1076.62037 · doi:10.3150/bj/1082380217
[30] Kerkyacharian, G. and Picard, D. (2000). Thresholding algorithms, maxisets and well-concentrated bases., Test 9 283-344. · Zbl 1107.62323 · doi:10.1007/BF02595738
[31] Kingman, J.F.C. (1993)., Poisson processes. Oxford studies in Probability. · Zbl 0771.60001
[32] Kolaczyk, E.D. (1997). Non-Parametric Estimation of Gamma-Ray Burst Intensities Using Haar Wavelets., The Astrophysical Journal 483 340-349.
[33] Kolaczyk, E.D. (1999). Wavelet shrinkage estimation of certain Poisson intensity signals using corrected thresholds., Statist. Sinica 9 (1) 119-135. · Zbl 0927.62081
[34] Lebarbier, E. (2005). Detecting multiple change-points in the mean of Gaussian process by model selection., Signal Processing 85 (4) 717-736. · Zbl 1148.94403 · doi:10.1016/j.sigpro.2004.11.012
[35] Merton, R.C. (1975). Option pricing when underlying stock returns are discontinuous. Working paper, Sloan School of Management 787-795.
[36] Reynaud-Bouret, P. (2003). Adaptive estimation of the intensity of inhomogeneous Poisson processes via concentration inequalities., Probability Theory and Related Fields 126 (1) 103-153. · Zbl 1019.62079 · doi:10.1007/s00440-003-0259-1
[37] Rudemo, M. (1982). Empirical choice of histograms and density estimators., Scand. J. Statist. 9 (2) 65-78. · Zbl 0501.62028
[38] Stone, M. (1974). Cross-Validatory Choice and Assessment of Statistical Predictions., J. Roy. Stat. Soc., Ser. B 36 111-147. · Zbl 0308.62063
[39] Uhler, R.S. and Bradley, P. G. (1970). A Stochastic Model for Determining the Economic Prospects of Petroleum Exploration Over Large Regions., Journal of the American Statistical Association 65 (330) 623-630.
[40] Willett, R.M. and Nowak, R.D. (2007). Multiscale Poisson Intensity and Density Estimation., IEEE Transactions on Information Theory 53 (9) 3171-3187. · Zbl 1325.94036 · doi:10.1109/TIT.2007.903139
[41] Zhang, B., Fadili, J.M. and Starck, J.L. (2008). Wavelets, Ridgelets, and Curvelets for Poisson Noise Removal., IEEE Transactions on Image Processing 17 (7) 1093-1108. · doi:10.1109/TIP.2008.924386
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