Nonparametric Bayesian inference for Gamma-type Lévy subordinators

Denis Belomestny, Shota Gugushvili, Moritz Schauer, Peter Spreij

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1 Citation (Scopus)

Abstract

Given discrete time observations over a growing time interval, we consider a nonparametric Bayesian approach to estimation of the Lévy density of a Lévy process belonging to a exible class of in finite activity subordinators. Posterior inference is performed via MCMC, and we circumvent the problem of the intractable likelihood via the data augmentation device, that in our case relies on bridge process sampling via Gamma process bridges. Our approach also requires the use of a new in fiite-dimensional form of a reversible jump MCMC algorithm. We show that our method leads to good practical results in challenging simulation examples. On the theoretical side, we establish that our nonparametric Bayesian procedure is consistent: in the low frequency data setting, with equispaced in time observations and intervals between successive observations remaining fixed, the posterior asymptotically, as the sample size n → ∞, concentrates around the Lévy density under which the data have been generated. Finally, we test our method on a classical insurance dataset.

Original languageEnglish
Pages (from-to)781-816
Number of pages36
JournalCommunications in Mathematical Sciences
Volume17
Issue number3
DOIs
Publication statusPublished - 30 Aug 2019

Keywords

  • Bridge sampling
  • Data augmentation
  • Gamma process
  • Lévy density
  • Lévy process
  • MCMC
  • Metropolis-Hastings algorithm
  • Nonparametric Bayesian estimation
  • Posterior consistency
  • Reversible jump MCMC
  • Subordinator
  • θ-subordinator

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