# zbMATH — the first resource for mathematics

Nonparametric Bayesian analysis of the compound Poisson prior for support boundary recovery. (English) Zbl 1452.62155
The paper presents the performance of compound Poisson (CPP) as nonparametric priors. It is proved that under (CPP) priors, optimal posterior contraction rates are attained for Hölder functions and for monotone functions. In Section 2, the contraction rates for compound Poisson processes and subordinator priors are investigated. A general description of the asymptotic posterior shape in which the results thereafter can be embedded is presented in Section 3. In Section 4, Bernstein-von Mises-type theorems and results on the frequentist coverage of credible sets for CPP priors are presented. The proofs of the new theorems are presented in two appendices.

##### MSC:
 62C10 Bayesian problems; characterization of Bayes procedures 62G05 Nonparametric estimation 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
Full Text:
##### References:
 [1] Bontemps, D. (2011). Bernstein-von Mises theorems for Gaussian regression with increasing number of regressors. Ann. Statist. 39 2557-2584. · Zbl 1231.62061 [2] Castillo, I. and Rousseau, J. (2015). A Bernstein-von Mises theorem for smooth functionals in semiparametric models. Ann. Statist. 43 2353-2383. · Zbl 1327.62302 [3] Castillo, I., Schmidt-Hieber, J. and van der Vaart, A. (2015). Bayesian linear regression with sparse priors. Ann. Statist. 43 1986-2018. · Zbl 06502640 [4] Castillo, I. and van der Vaart, A. (2012). Needles and straw in a haystack: Posterior concentration for possibly sparse sequences. Ann. Statist. 40 2069-2101. · Zbl 1257.62025 [5] Chatterjee, S., Guntuboyina, A. and Sen, B. (2015). On risk bounds in isotonic and other shape restricted regression problems. Ann. Statist. 43 1774-1800. · Zbl 1317.62032 [6] Chernozhukov, V. and Hong, H. (2004). Likelihood estimation and inference in a class of nonregular econometric models. Econometrica 72 1445-1480. · Zbl 1091.62135 [7] Chipman, H. A., George, E. I. and McCulloch, R. E. (2010). BART: Bayesian additive regression trees. Ann. Appl. Stat. 4 266-298. · Zbl 1189.62066 [8] Coram, M. and Lalley, S. P. (2006). Consistency of Bayes estimators of a binary regression function. Ann. Statist. 34 1233-1269. · Zbl 1113.62006 [9] Denison, D. G. T., Mallick, B. K. and Smith, A. F. M. (1998). A Bayesian CART algorithm. Biometrika 85 363-377. · Zbl 1048.62502 [10] Embrechts, P., Klüppelberg, C. and Mikosch, T. (2003). Modelling Extremal Events: For Insurance and Finance. Applications of Mathematics (New York) 33. Springer, New York. · Zbl 0873.62116 [11] Frick, K., Munk, A. and Sieling, H. (2014). Multiscale change point inference. J. R. Stat. Soc. Ser. B. Stat. Methodol. 76 495-580. · Zbl 1411.62065 [12] Gao, C., Han, F. and Zhang, C.-H. (2017). On estimation of isotonic piecewise constant signals. Preprint. Available at arXiv:1705.06386. Ann. Statist. (to appear). [13] Ghosal, S. (1999). Asymptotic normality of posterior distributions in high-dimensional linear models. Bernoulli 5 315-331. · Zbl 0948.62007 [14] Ghosal, S. (2000). Asymptotic normality of posterior distributions for exponential families when the number of parameters tends to infinity. J. Multivariate Anal. 74 49-68. · Zbl 1118.62309 [15] Ghosal, S., Ghosh, J. K. and Samanta, T. (1995). On convergence of posterior distributions. Ann. Statist. 23 2145-2152. · Zbl 0858.62024 [16] Ghosal, S. and van der Vaart, A. (2017). Fundamentals of Nonparametric Bayesian Inference. Cambridge Series in Statistical and Probabilistic Mathematics 44. Cambridge Univ. Press, Cambridge. · Zbl 1376.62004 [17] Gijbels, I., Mammen, E., Park, B. U. and Simar, L. (1999). On estimation of monotone and concave frontier functions. J. Amer. Statist. Assoc. 94 220-228. · Zbl 1043.62105 [18] Holmes, C. C. and Heard, N. A. (2003). Generalized monotonic regression using random change points. Stat. Med. 22 623-638. [19] Jirak, M., Meister, A. and Reiß, M. (2014). Adaptive function estimation in nonparametric regression with one-sided errors. Ann. Statist. 42 1970-2002. · Zbl 1305.62172 [20] Kim, Y. and Lee, J. (2004). A Bernstein-von Mises theorem in the nonparametric right-censoring model. Ann. Statist. 32 1492-1512. · Zbl 1047.62043 [21] Kleijn, B. and Knapik, B. (2012). Semiparametric posterior limits under local asymptotic exponentiality. Preprint. Available at arXiv:1210.6204. [22] Korostelëv, A. P. and Tsybakov, A. B. (1993). Minimax Theory of Image Reconstruction. Lecture Notes in Statistics 82. Springer, New York. · Zbl 0833.62039 [23] Li, M. and Ghosal, S. (2017). Bayesian detection of image boundaries. Ann. Statist. 45 2190-2217. · Zbl 06821123 [24] Mariucci, E., Ray, K. and Szabo, B. (2017). A Bayesian nonparametric approach to log-concave density estimation. Preprint. Available at arXiv:1703.09531. · Zbl 07166557 [25] Meister, A. and Reiß, M. (2013). Asymptotic equivalence for nonparametric regression with non-regular errors. Probab. Theory Related Fields 155 201-229. · Zbl 1257.62045 [26] Panov, M. and Spokoiny, V. (2015). Finite sample Bernstein-von Mises theorem for semiparametric problems. Bayesian Anal. 10 665-710. · Zbl 1335.62057 [27] Reiß, M. and Schmidt-Hieber, J. (2017). Posterior contraction rates for support boundary recovery. Preprint. Available at arXiv:1703.08358. [28] Reiß, M. and Schmidt-Hieber, J. (2019). Supplement to “Nonparametric Bayesian analysis of the compound Poisson prior for support boundary recovery.” https://doi.org/10.1214/19-AOS1853SUPP. [29] Reiß, M. and Selk, L. (2017). Efficient estimation of functionals in nonparametric boundary models. Bernoulli 23 1022-1055. · Zbl 1380.62177 [30] Rivoirard, V. and Rousseau, J. (2012). Bernstein-von Mises theorem for linear functionals of the density. Ann. Statist. 40 1489-1523. · Zbl 1257.62036 [31] Rockova, V. and van der Pas, S. (2017). Posterior concentration for Bayesian regression trees and their ensembles. Preprint. Available at arXiv:1708.08734. Ann. Statist. (to appear). [32] Salomond, J.-B. (2014). Concentration rate and consistency of the posterior distribution for selected priors under monotonicity constraints. Electron. J. Stat. 8 1380-1404. · Zbl 1298.62064 [33] Sato, K. (2013). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge Univ. Press, Cambridge. [34] Scricciolo, C. (2007). On rates of convergence for Bayesian density estimation. Scand. J. Stat. 34 626-642. · Zbl 1150.62018 [35] Simon, T. (2004). Small ball estimates in $$p$$-variation for stable processes. J. Theoret. Probab. 17 979-1002. · Zbl 1074.60055 [36] van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics 3. Cambridge Univ. Press, Cambridge. [37] van der Vaart, A. · Zbl 0862.60002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.