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On the consistency of the MLE for Ornstein–Uhlenbeck and other selfdecomposable processes

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In this paper we give easy to verify conditions for the strong consistency of the maximum likelihood estimator (MLE) in the case when data is sampled from a parametric family of selfdecomposable distributions. The difficulty arises from the fact that standard conditions for the consistency of the MLE are based on the pdf, which, for most selfdecomposable distributions, is not available in a closed form. Instead, our conditions are based on properties of the Lévy triplet (i.e. the Lévy measure, the Gaussian part, and the shift) of the distribution. Further, we extend out results to certain selfdecomposable stochastic processes, and, in particular, we give conditions in the case when the data is sampled from a Lévy or an Ornstein–Uhlenbeck process.

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The author wishes to thank Professor Gennady Samorodnitsky for the benefit of several discussions and the two anonymous referees whose comments led to improvements in the presentation of this paper.

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Correspondence to Michael Grabchak.

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Grabchak, M. On the consistency of the MLE for Ornstein–Uhlenbeck and other selfdecomposable processes. Stat Inference Stoch Process 19, 29–50 (2016). https://doi.org/10.1007/s11203-015-9118-9

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  • Consistent estimator
  • MLE
  • Selfdecomposable distributions
  • Ornstein–Uhlenbeck processes
  • Tempered stable distributions
  • Stable distributions

Mathematics Subject Classification

  • 62F12
  • 62M05
  • 60G51