## Electronic Journal of Statistics

### Bernstein–von Mises theorems for statistical inverse problems II: compound Poisson processes

#### Abstract

We study nonparametric Bayesian statistical inference for the parameters governing a pure jump process of the form $Y_{t}=\sum_{k=1}^{N(t)}Z_{k},~~~t\ge 0,$ where $N(t)$ is a standard Poisson process of intensity $\lambda$, and $Z_{k}$ are drawn i.i.d. from jump measure $\mu$. A high-dimensional wavelet series prior for the Lévy measure $\nu =\lambda\mu$ is devised and the posterior distribution arises from observing discrete samples $Y_{\Delta},Y_{2\Delta},\dots,Y_{n\Delta}$ at fixed observation distance $\Delta$, giving rise to a nonlinear inverse inference problem. We derive contraction rates in uniform norm for the posterior distribution around the true Lévy density that are optimal up to logarithmic factors over Hölder classes, as sample size $n$ increases. We prove a functional Bernstein–von Mises theorem for the distribution functions of both $\mu$ and $\nu$, as well as for the intensity $\lambda$, establishing the fact that the posterior distribution is approximated by an infinite-dimensional Gaussian measure whose covariance structure is shown to attain the information lower bound for this inverse problem. As a consequence posterior based inferences, such as nonparametric credible sets, are asymptotically valid and optimal from a frequentist point of view.

#### Article information

Source
Electron. J. Statist., Volume 13, Number 2 (2019), 3513-3571.

Dates
First available in Project Euclid: 1 October 2019

https://projecteuclid.org/euclid.ejs/1569895282

Digital Object Identifier
doi:10.1214/19-EJS1609

#### Citation

Nickl, Richard; Söhl, Jakob. Bernstein–von Mises theorems for statistical inverse problems II: compound Poisson processes. Electron. J. Statist. 13 (2019), no. 2, 3513--3571. doi:10.1214/19-EJS1609. https://projecteuclid.org/euclid.ejs/1569895282

#### References

• [1] Denis Belomestny, Fabienne Comte, Valentine Genon-Catalot, Hiroki Masuda, and Markus Reiß. Lévy matters IV. Lecture Notes in Mathematics. Springer, 2015.
• [2] Denis Belomestny and Markus Reiß. Spectral calibration of exponential Lévy models. Finance Stoch., 10(4):449–474, 2006.
• [3] Boris Buchmann and Rudolf Grübel. Decompounding: an estimation problem for Poisson random sums. Ann. Statist., 31(4):1054–1074, 2003.
• [4] Ismaël Castillo. On Bayesian supremum norm contraction rates. Ann. Statist., 42(5):2058–2091, 2014.
• [5] Ismaël Castillo. Pólya tree posterior distributions on densities. Ann. Inst. Henri Poincaré Probab. Stat., 53(4):2074–2102, 2017.
• [6] Ismaël Castillo and Richard Nickl. Nonparametric Bernstein–von Mises theorems in Gaussian white noise. Ann. Statist., 41(4):1999–2028, 2013.
• [7] Ismaël Castillo and Richard Nickl. On the Bernstein–von Mises phenomenon for nonparametric Bayes procedures. Ann. Statist., 42(5):1941–1969, 2014.
• [8] Ismaël Castillo and Judith Rousseau. A Bernstein–von Mises theorem for smooth functionals in semiparametric models. Ann. Statist., 43(6):2353–2383, 2015.
• [9] Alberto J. Coca. Adaptive nonparametric estimation for compound Poisson processes robust to the discrete-observation scheme. arXiv:1803.09849, 2018.
• [10] Alberto J. Coca. Efficient nonparametric inference for discretely observed compound Poisson processes. Probab. Theory Related Fields, 170(1-2):475–523, 2018.
• [11] Masoumeh Dashti and Andrew Stuart. The Bayesian approach to inverse problems. In: Handbook of Uncertainty Quantification, Eds. Ghanem et al., Springer, 2016.
• [12] Richard M. Dudley. Real analysis and probability. Cambridge Univ. Press, 2002.
• [13] Richard M. Dudley. Uniform central limit theorems. Cambridge Univ.Press, 2014.
• [14] Gerald B. Folland. Real analysis. Wiley, second edition, 1999.
• [15] Subhashis Ghosal, Jayanta K. Ghosh, and Aad W. van der Vaart. Convergence rates of posterior distributions. Ann. Statist., 28(2):500–531, 2000.
• [16] Subhashis Ghosal and Aad W. van der Vaart. Fundamentals of nonparametric Bayesian inference. Cambridge University Press, New York, 2017.
• [17] Evarist Giné and Richard Nickl. Rates of contraction for posterior distributions in $L^{r}$-metrics, $1\leq r\leq \infty$. Ann. Statist., 39(6):2883–2911, 2011.
• [18] Evarist Giné and Richard Nickl. Mathematical foundations of infinite-dimensional statistical models. Cambridge University Press, 2016.
• [19] Matteo Giordano and Hanne Kekkonen. Bernstein-von Mises theorems and uncertainty quantification for linear inverse problems. arXiv preprint arXiv:1811.04058, 2018.
• [20] Shota Gugushvili, Frank van der Meulen, and Peter Spreij. Nonparametric Bayesian inference for multidimensional compound Poisson processes. Mod. Stoch. Theory Appl., 2(1):1–15, 2015.
• [21] François Monard, Richard Nickl, and Gabriel P. Paternain. Efficient nonparametric Bayesian inference for $X$-ray transforms. Ann. Statist., 47(2):1113–1147, 2019.
• [22] Michael H. Neumann and Markus Reiß. Nonparametric estimation for Lévy processes from low-frequency observations. Bernoulli, 15(1):223–248, 2009.
• [23] Richard Nickl. Donsker-type theorems for nonparametric maximum likelihood estimators. Probab. Theory Related Fields, 138(3-4):411–449, 2007.
• [24] Richard Nickl. Bernstein–von Mises theorems for statistical inverse problems I: Schrödinger equation. J. Eur. Math. Soc. (JEMS), to appear, 2018.
• [25] Richard Nickl and Kolyan Ray. Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions. Ann. Statist., to appear.
• [26] Richard Nickl and Markus Reiß. A Donsker theorem for Lévy measures. J. Funct. Anal., 263(10):3306–3332, 2012.
• [27] Richard Nickl, Markus Reiß, Jakob Söhl, and Mathias Trabs. High-frequency Donsker theorems for Lévy measures. Probab. Th. Rel. Fields, 164:61–108, 2016.
• [28] Richard Nickl and Jakob Söhl. Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions. Ann. Statist., 45(4):1664–1693, 2017.
• [29] Kalyanapuram R. Parthasarathy. Probability measures on metric spaces. Probability and Mathematical Statistics, No. 3. Academic Press, Inc., New York-London, 1967.
• [30] Kolyan Ray. Bayesian inverse problems with non-conjugate priors. Electron. J. Stat., 7:2516–2549, 2013.
• [31] Kolyan Ray. Adaptive Bernstein–von Mises theorems in Gaussian white noise. Ann. Statist., 45(6):2511–2536, 2017.
• [32] Andrew M. Stuart. Inverse problems: a Bayesian perspective. Acta Numer., 19:451–559, 2010.
• [33] Mathias Trabs. Information bounds for inverse problems with application to deconvolution and Lévy models. Ann. Inst. H. Poincaré, 51(4):1620–1650, 2015.
• [34] Hans Triebel. Theory of function spaces. Birkhäuser Verlag, Basel, 1983.
• [35] Aad W. van der Vaart. Asymptotic statistics. Cambridge University Press, 1998.
• [36] Aad W. van der Vaart and Jon A. Wellner. Weak convergence and empirical processes. Springer Series in Statistics. Springer-Verlag, New York, 1996.
• [37] Bert van Es, Shota Gugushvili, and Peter Spreij. A kernel type nonparametric density estimator for decompounding. Bernoulli, 13(3):672–694, 2007.