Adaptive tests of homogeneity for a Poisson process
Annales de l'I.H.P. Probabilités et statistiques, Volume 47 (2011) no. 1, p. 176-213

We propose to test the homogeneity of a Poisson process observed on a finite interval. In this framework, we first provide lower bounds for the uniform separation rates in -norm over classical Besov bodies and weak Besov bodies. Surprisingly, the obtained lower bounds over weak Besov bodies coincide with the minimax estimation rates over such classes. Then we construct non-asymptotic and non-parametric testing procedures that are adaptive in the sense that they achieve, up to a possible logarithmic factor, the optimal uniform separation rates over various Besov bodies simultaneously. These procedures are based on model selection and thresholding methods. We finally complete our theoretical study with a Monte Carlo evaluation of the power of our tests under various alternatives.

Nous proposons de tester l'homogénéité d'un processus de Poisson observé sur un intervalle borné. Nous établissons tout d'abord des bornes inférieures pour les vitesses de séparation uniformes relativement à la norme sur des Besov bodies classiques ou faibles. De façon surprenante, nous obtenons des bornes inférieures sur les Besov bodies faibles qui coïncident avec les vitesses minimax d'estimation sur ce type de classe. Ensuite, nous construisons des procédures de tests non asymptotiques et non paramétriques qui sont adaptatives, au sens où elles atteignent, à un facteur logarithmique près dans certains cas, les vitesses de séparation optimales sur plusieurs classes d'alternatives simultanément. Ces procédures sont basées sur des méthodes de sélection de modèles et de seuillage. Enfin, nous complétons cette étude théorique par des simulations afin d'estimer par la méthode de Monte Carlo la puissance de nos tests sous diverses alternatives.

DOI : https://doi.org/10.1214/10-AIHP367
Classification:  62G10,  62G20
Keywords: Poisson process, adaptive hypotheses testing, uniform separation rate, minimax separation rate, model selection, thresholding rule
@article{AIHPB_2011__47_1_176_0,
     author = {Fromont, M. and Laurent, B. and Reynaud-Bouret, P.},
     title = {Adaptive tests of homogeneity for a Poisson process},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {47},
     number = {1},
     year = {2011},
     pages = {176-213},
     doi = {10.1214/10-AIHP367},
     zbl = {1207.62161},
     mrnumber = {2779402},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2011__47_1_176_0}
}
Fromont, M.; Laurent, B.; Reynaud-Bouret, P. Adaptive tests of homogeneity for a Poisson process. Annales de l'I.H.P. Probabilités et statistiques, Volume 47 (2011) no. 1, pp. 176-213. doi : 10.1214/10-AIHP367. http://www.numdam.org/item/AIHPB_2011__47_1_176_0/

[1] D. J. Aldous. Exchangeability and related topics. In Ecole d'été de probabilité de Saint-Flour XIII 1-198. Lecture Notes in Math. 1117. Springer, Berlin, 1985. | MR 883646 | Zbl 0562.60042

[2] L. J. Bain, M. Engelhardt and F. T. Wright. Tests for an increasing trend in the intensity of a Poisson process: A power study. J. Amer. Statist. Assoc. 80 (1985) 419-422.

[3] Y. Baraud. Non asymptotic minimax rates of testing in signal detection. Bernoulli 8 (2002) 577-606. | MR 1935648 | Zbl 1007.62042

[4] Y. Baraud, S. Huet and B. Laurent. Adaptive tests of linear hypotheses by model selection. Ann. Statist. 31 (2003) 225-251. | MR 1962505 | Zbl 1018.62037

[5] M. Bhattacharjee, J. V. Deshpande and U. V. Naik-Nimbalkar. Unconditional tests of goodness of fit for the intensity of time-truncated nonhomogeneous Poisson processes. Technometrics 46 (2004) 330-338. | MR 2082502

[6] C. Butucea and K. Tribouley. Nonparametric homogeneity tests. J. Statist. Plann. Inference 136 (2006) 597-639. | MR 2181971 | Zbl 1077.62031

[7] A. Cohen and H. B. Sackrowitz. Evaluating tests for increasing intensity of a Poisson process. Technometrics 35 (1993) 446-448. | MR 1242781 | Zbl 0792.62073

[8] D. R. Cox. Some statistical methods connected with series of events. J. Roy. Statist. Soc. Ser. B 17 (1955) 129-164. | MR 92301 | Zbl 0067.37403

[9] L. H. Crow. Reliability and analysis for complex repairable systems. In Reliability and Biometry (F. Proschan and R. J. Serfling, eds.) 379-410. Society for Industrial and Applied Mathematics, Philadelphia, 1974. | MR 336857

[10] S. Dachian and Y. A. Kutoyants. Hypotheses testing: Poisson versus self-exciting. Scand. J. Statist. 33 (2006) 391-408. | MR 2279649 | Zbl 1125.62083

[11] K. Fazli. Second-order efficient test for inhomogeneous Poisson processes. Stat. Inference Stochastic Proc. 10 (2007) 181-208. | MR 2273329 | Zbl 1110.62109

[12] K. Fazli and Y. A. Kutoyants. Two simple hypotheses testing for Poisson process. Far East J. Theor. Stat. 15 (2005) 251-290. | MR 2163930 | Zbl 1069.62060

[13] M. Fromont and B. Laurent. Adaptive goodness-of-fit tests in a density model. Ann. Statist. 34 (2006) 680-720. | MR 2281881 | Zbl 1096.62040

[14] G. Gusto and S. Schbath. FADO: A statistical method to detect favored or avoided distances between motif occurrences using the Hawkes' model. Stat. Appl. Genet. Mol. Biol. 4 (2005) Article 24. | MR 2170440 | Zbl 1095.62126

[15] C. Houdré and P. Reynaud-Bouret. Exponential inequalities, with constants, for U-statistics of order 2. In Stochastic Inequalities and Applications 55-69. Progr. Probab. 56. Birkhauser, Basel, 2003. | MR 2073426 | Zbl 1036.60015

[16] Y. I. Ingster. Asymptotically minimax testing for nonparametric alternatives I-II-III. Math. Methods Statist. 2 (1993) 85-114, 171-189, 249-268. | MR 1257978 | Zbl 0798.62059

[17] Y. I. Ingster. Adaptive chi-square tests. J. Math. Sci. 99 (2000) 1110-1119. | MR 1700386 | Zbl 0949.62031

[18] Y. I. Ingster and Y. A. Kutoyants. Nonparametric hypothesis testing for intensity of the Poisson process. Math. Methods Statist. 16 (2007) 217-245. | MR 2356819 | Zbl 1231.62081 | Zbl pre05614391

[19] G. Kerkyacharian and D. Picard. Thresholding algorithms, maxisets and well-concentrated bases. Test 9 (2000) 283-344. | Zbl 1107.62323

[20] B. Laurent. Adaptive estimation of a quadratic functional of a density by model selection. ESAIM Probab. Stat. 9 (2005) 1-18. | Numdam | MR 2148958 | Zbl 1136.62333

[21] P. Reynaud-Bouret. Adaptive estimation of the intensity of inhomogeneous Poisson processes via concentration inequalities. Probab. Theory Related Fields 126 (2003) 103-153. | MR 1981635 | Zbl 1019.62079

[22] P. Reynaud-Bouret and V. Rivoirard. Near optimal thresholding estimation of a Poisson intensity on the real line. Electron J. Statist. 4 (2010) 172-238. | MR 2645482

[23] V. Rivoirard. Nonlinear estimation over weak Besov spaces and minimax Bayes method. Bernoulli 12 (2006) 609-632. | MR 2248230 | Zbl 1125.62001

[24] S. Robin, F. Rodolphe and S. Schbath. DNA Words and Models. Cambridge Univ. Press, Cambridge, 2005. | MR 2214200 | Zbl 1185.92047

[25] V. G. Spokoiny. Adaptive hypothesis testing using wavelets. Ann. Statist. 24 (1996) 2477-2498. | MR 1425962 | Zbl 0898.62056

[26] V. G. Spokoiny. Adaptive and spatially hypothesis testing of a nonparametric hypothesis. Math. Methods Statist. 7 (1998) 245-273. | MR 1651777 | Zbl 1103.62345

[27] G. S. Watson. Estimating the intensity of a Poisson process. In Applied Time Series Analysis (1st Proceeding, Tulsa, 1976) 325-345. Academic Press, New York, 1978. | MR 507854 | Zbl 0464.62085