Threshold Estimation for Stochastic Processes with Small Noise

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Abstract

Consider a process satisfying a stochastic differential equation with unknown drift parameter, and suppose that discrete observations are given. It is known that a simple least squares estimator (LSE) can be consistent but numerically unstable in the sense of large standard deviations under finite samples when the noise process has jumps. We propose a filter to cut large shocks from data and construct the same LSE from data selected by the filter. The proposed estimator can be asymptotically equivalent to the usual LSE, whose asymptotic distribution strongly depends on the noise process. However, in numerical study, it looked asymptotically normal in an example where filter was chosen suitably, and the noise was a Lévy process. We will try to justify this phenomenon mathematically, under certain restricted assumptions.

Original languageEnglish
Pages (from-to)951-988
Number of pages38
JournalScandinavian Journal of Statistics
Volume44
Issue number4
DOIs
Publication statusPublished - 2017 Dec

Keywords

  • 60G52
  • 60J75
  • drift estimation
  • mighty convergence
  • semimartingale noise
  • small noise asymptotics
  • stochastic differential equation
  • threshold estimator MSC2010:62F12; 62M05

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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