Electronic Journal of Statistics

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

Richard Nickl and Jakob Söhl

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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
Received: May 2019
First available in Project Euclid: 1 October 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1569895282

Digital Object Identifier
doi:10.1214/19-EJS1609

Mathematical Reviews number (MathSciNet)
MR4013745

Zentralblatt MATH identifier
07113725

Keywords
Bayesian nonlinear inverse problems compound Poisson processes Lévy processes asymptotics of nonparametric Bayes procedures

Rights
Creative Commons Attribution 4.0 International License.

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


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