### 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 language | English |
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Pages (from-to) | 951-988 |

Number of pages | 38 |

Journal | Scandinavian Journal of Statistics |

Volume | 44 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2017 Dec 1 |

### Fingerprint

### 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

### Cite this

**Threshold Estimation for Stochastic Processes with Small Noise.** / Shimizu, Yasutaka.

Research output: Contribution to journal › Article

*Scandinavian Journal of Statistics*, vol. 44, no. 4, pp. 951-988. https://doi.org/10.1111/sjos.12287

}

TY - JOUR

T1 - Threshold Estimation for Stochastic Processes with Small Noise

AU - Shimizu, Yasutaka

PY - 2017/12/1

Y1 - 2017/12/1

N2 - 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.

AB - 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.

KW - 60G52

KW - 60J75

KW - drift estimation

KW - mighty convergence

KW - semimartingale noise

KW - small noise asymptotics

KW - stochastic differential equation

KW - threshold estimator MSC2010:62F12; 62M05

UR - http://www.scopus.com/inward/record.url?scp=85025076407&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85025076407&partnerID=8YFLogxK

U2 - 10.1111/sjos.12287

DO - 10.1111/sjos.12287

M3 - Article

AN - SCOPUS:85025076407

VL - 44

SP - 951

EP - 988

JO - Scandinavian Journal of Statistics

JF - Scandinavian Journal of Statistics

SN - 0303-6898

IS - 4

ER -