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Nonparametric Bayesian inference for multidimensional compound Poisson processes. (English) Zbl 1349.62115
Summary: Given a sample from a discretely observed multidimensional compound Poisson process, we study the problem of nonparametric estimation of its jump size density \(r_0\) and intensity \(\lambda_0\). We take a nonparametric Bayesian approach to the problem and determine posterior contraction rates in this context, which, under some assumptions, we argue to be optimal posterior contraction rates. In particular, our results imply the existence of Bayesian point estimates that converge to the true parameter pair \((r_0,\lambda_0)\) at these rates. To the best of our knowledge, construction of nonparametric density estimators for inference in the class of discretely observed multidimensional Lévy processes, and the study of their rates of convergence is a new contribution to the literature.

62G07 Density estimation
60G51 Processes with independent increments; Lévy processes
62F15 Bayesian inference
62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
62M30 Inference from spatial processes
Full Text: DOI arXiv
[1] Bücher, A.; Vetter, M., Nonparametric inference on Lévy measures and copulas, Ann. Stat., 41, 3, 1485-1515, (2013) · Zbl 1273.62067
[2] Buchmann, B.; Grübel, R., Decompounding: an estimation problem for Poisson random sums, Ann. Stat., 31, 4, 1054-1074, (2003) · Zbl 1105.62309
[3] Buchmann, B.; Grübel, R., Decompounding Poisson random sums: recursively truncated estimates in the discrete case, Ann. Inst. Stat. Math., 56, 4, 743-756, (2004) · Zbl 1078.62020
[4] Comte, F.; Genon-Catalot, V., Non-parametric estimation for pure jump irregularly sampled or noisy Lévy processes, Stat. Neerl., 64, 3, 290-313, (2010) · Zbl 1201.62042
[5] Comte, F.; Genon-Catalot, V., Estimation for Lévy processes from high frequency data within a long time interval, Ann. Stat., 39, 2, 803-837, (2011) · Zbl 1215.62084
[6] Comte, F.; Duval, C.; Genon-Catalot, V., Nonparametric density estimation in compound Poisson processes using convolution power estimators, Metrika, 77, 1, 163-183, (2014) · Zbl 1282.62088
[7] Csiszár, I., Eine informationstheoretische ungleichung und ihre anwendung auf den beweis der ergodizität von markoffschen ketten, Magy. Tud. Akad. Mat. Kut. Intéz. Közl., 8, 85-108, (1963) · Zbl 0124.08703
[8] Donnet, S.; Rivoirard, V.; Rousseau, J.; Scricciolo, C.
[9] Duval, C., Density estimation for compound Poisson processes from discrete data, Stoch. Process. Appl., 123, 11, 3963-3986, (2013) · Zbl 1320.62079
[10] Embrechts, P.; Klüppelberg, C.; Mikosch, T., Modelling Extremal Events: For Insurance and Finance, 33, 645, (1997), Springer, New York · Zbl 0873.62116
[11] Ferguson, T.S., A Bayesian analysis of some nonparametric problems, Ann. Stat., 1, 209-230, (1973) · Zbl 0255.62037
[12] Ferguson, T.S., Recent Advances in Statistics, 287-302, (1983), Academic Press, New York
[13] Ghosal, S., Bayesian Nonparametrics, 35-79, (2010), Cambridge Univ. Press, Cambridge
[14] Ghosal, S.; van der Vaart, A.W., Entropies and rates of convergence for maximum likelihood and Bayes estimation for mixtures of normal densities, Ann. Stat., 29, 5, 1233-1263, (2001) · Zbl 1043.62024
[15] Ghosal, S.; Ghosh, J.K.; van der Vaart, A.W., Convergence rates of posterior distributions, Ann. Stat., 28, 2, 500-531, (2000) · Zbl 1105.62315
[16] Gugushvili, S.
[17] Kutoyants, Y.A., Statistical Inference for Spatial Poisson Processes, 134, 276, (1998), Springer · Zbl 0904.62108
[18] Lo, A.Y., On a class of Bayesian nonparametric estimates. I. density estimates, Ann. Stat., 12, 1, 351-357, (1984) · Zbl 0557.62036
[19] Neumann, M.H.; Reiß, M., Nonparametric estimation for Lévy processes from low-frequency observations, Bernoulli, 15, 1, 223-248, (2009) · Zbl 1200.62095
[20] Pollard, D., A User’s Guide to Measure Theoretic Probability, 8, 351, (2002), Cambridge University Press, Cambridge · Zbl 0992.60001
[21] Prabhu, N.U., Stochastic Storage Processes: Queues, Insurance Risk, Dams, and Data Communication, 15, 206, (1998), Springer, New York · Zbl 0888.60073
[22] Shen, W.; Tokdar, S.T.; Ghosal, S., Adaptive Bayesian multivariate density estimation with Dirichlet mixtures, Biometrika, 100, 3, 623-640, (2013) · Zbl 1284.62183
[23] Skorohod, A.V., Random Processes with Independent Increments, (1964), Nauka, Moscow
[24] Van Es, B.; Gugushvili, S.; Spreij, P., A kernel type nonparametric density estimator for decompounding, Bernoulli, 13, 3, 672-694, (2007) · Zbl 1129.62030
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