### Abstract

We prove the local asymptotic normality for the full parameters of the normal inverse Gaussian Lévy process X, when we observe high-frequency data X_{Δ}n,X 2Δ_{n},...,X nΔ_{n} with sampling mesh Δ_{n} → 0 and the terminal sampling time nΔ_{n} → â̂ž. The rate of convergence turns out to be (aš nΔ_{n}, ǎš nΔ_{n}, ǎš n, ǎš n) for the dominating parameter (α,β,δ,μ), where α stands for the heaviness of the tails, β the degree of skewness, δ the scale, and μ the location. The essential feature in our study is that the suitably normalized increments of X in small time is approximately Cauchy-distributed, which specifically comes out in the form of the asymptotic Fisher information matrix.

Original language | English |
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Pages (from-to) | 13-32 |

Number of pages | 20 |

Journal | ESAIM - Probability and Statistics |

Volume | 17 |

DOIs | |

Publication status | Published - Jan 1 2013 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability

### Cite this

**Local asymptotic normality for normal inverse Gaussian Lévy processes with high-frequency sampling δ.** / Kawai, Reiichiro; Masuda, Hiroki.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Local asymptotic normality for normal inverse Gaussian Lévy processes with high-frequency sampling δ

AU - Kawai, Reiichiro

AU - Masuda, Hiroki

PY - 2013/1/1

Y1 - 2013/1/1

N2 - We prove the local asymptotic normality for the full parameters of the normal inverse Gaussian Lévy process X, when we observe high-frequency data XΔn,X 2Δn,...,X nΔn with sampling mesh Δn → 0 and the terminal sampling time nΔn → â̂ž. The rate of convergence turns out to be (aš nΔn, ǎš nΔn, ǎš n, ǎš n) for the dominating parameter (α,β,δ,μ), where α stands for the heaviness of the tails, β the degree of skewness, δ the scale, and μ the location. The essential feature in our study is that the suitably normalized increments of X in small time is approximately Cauchy-distributed, which specifically comes out in the form of the asymptotic Fisher information matrix.

AB - We prove the local asymptotic normality for the full parameters of the normal inverse Gaussian Lévy process X, when we observe high-frequency data XΔn,X 2Δn,...,X nΔn with sampling mesh Δn → 0 and the terminal sampling time nΔn → â̂ž. The rate of convergence turns out to be (aš nΔn, ǎš nΔn, ǎš n, ǎš n) for the dominating parameter (α,β,δ,μ), where α stands for the heaviness of the tails, β the degree of skewness, δ the scale, and μ the location. The essential feature in our study is that the suitably normalized increments of X in small time is approximately Cauchy-distributed, which specifically comes out in the form of the asymptotic Fisher information matrix.

UR - http://www.scopus.com/inward/record.url?scp=84870895217&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84870895217&partnerID=8YFLogxK

U2 - 10.1051/ps/2011101

DO - 10.1051/ps/2011101

M3 - Article

AN - SCOPUS:84870895217

VL - 17

SP - 13

EP - 32

JO - ESAIM - Probability and Statistics

JF - ESAIM - Probability and Statistics

SN - 1292-8100

ER -