Approximate self-weighted LAD estimation of discretely observed ergodic ornstein-uhlenbeck processes

Research output: Contribution to journalArticle

16 Citations (Scopus)

Abstract

We consider drift estimation of a discretely observed OrnsteinUhlenbeck process driven by a possibly heavy-tailed symmetric Lévy process with positive activity index β. Under an infill and large-time sampling design, we first establish an asymptotic normality of a self-weighted least absolute deviation estimator with the rate of convergence being √ nh1−1/β n, where n denotes sample size and hn > 0 the sampling mesh satisfying that hn → 0 and nhn → ∞. This implies that the rate of convergence is determined by the most active part of the driving Lévy process; the presence of a driving Wiener part leads to √ nhn, which is familiar in the context of asymptotically efficient estimation of diffusions with compound Poisson jumps, while a pure-jump driving Lévy process leads to a faster one. Also discussed is how to construct corresponding asymptotic confidence regions without full specification of the driving Lévy process. Second, by means of a polynomial type large deviation inequality we derive convergence of moments of our estimator under additional conditions.

Original languageEnglish
Pages (from-to)525-565
Number of pages41
JournalElectronic Journal of Statistics
Volume4
DOIs
Publication statusPublished - Jan 1 2010

Fingerprint

Ergodic Processes
Ornstein-Uhlenbeck Process
Jump
Rate of Convergence
Deviation Inequalities
Least Absolute Deviation
Estimator
Compound Poisson
Sampling Design
Efficient Estimation
Confidence Region
Asymptotic Normality
Large Deviations
Sample Size
Ornstein-Uhlenbeck process
Mesh
Specification
Denote
Moment
Imply

All Science Journal Classification (ASJC) codes

  • Statistics and Probability

Cite this

@article{c7a063fe91ca42a2b1f66d9e182b6579,
title = "Approximate self-weighted LAD estimation of discretely observed ergodic ornstein-uhlenbeck processes",
abstract = "We consider drift estimation of a discretely observed OrnsteinUhlenbeck process driven by a possibly heavy-tailed symmetric L{\'e}vy process with positive activity index β. Under an infill and large-time sampling design, we first establish an asymptotic normality of a self-weighted least absolute deviation estimator with the rate of convergence being √ nh1−1/β n, where n denotes sample size and hn > 0 the sampling mesh satisfying that hn → 0 and nhn → ∞. This implies that the rate of convergence is determined by the most active part of the driving L{\'e}vy process; the presence of a driving Wiener part leads to √ nhn, which is familiar in the context of asymptotically efficient estimation of diffusions with compound Poisson jumps, while a pure-jump driving L{\'e}vy process leads to a faster one. Also discussed is how to construct corresponding asymptotic confidence regions without full specification of the driving L{\'e}vy process. Second, by means of a polynomial type large deviation inequality we derive convergence of moments of our estimator under additional conditions.",
author = "Hiroki Masuda",
year = "2010",
month = "1",
day = "1",
doi = "10.1214/10-EJS565",
language = "English",
volume = "4",
pages = "525--565",
journal = "Electronic Journal of Statistics",
issn = "1935-7524",
publisher = "Institute of Mathematical Statistics",

}

TY - JOUR

T1 - Approximate self-weighted LAD estimation of discretely observed ergodic ornstein-uhlenbeck processes

AU - Masuda, Hiroki

PY - 2010/1/1

Y1 - 2010/1/1

N2 - We consider drift estimation of a discretely observed OrnsteinUhlenbeck process driven by a possibly heavy-tailed symmetric Lévy process with positive activity index β. Under an infill and large-time sampling design, we first establish an asymptotic normality of a self-weighted least absolute deviation estimator with the rate of convergence being √ nh1−1/β n, where n denotes sample size and hn > 0 the sampling mesh satisfying that hn → 0 and nhn → ∞. This implies that the rate of convergence is determined by the most active part of the driving Lévy process; the presence of a driving Wiener part leads to √ nhn, which is familiar in the context of asymptotically efficient estimation of diffusions with compound Poisson jumps, while a pure-jump driving Lévy process leads to a faster one. Also discussed is how to construct corresponding asymptotic confidence regions without full specification of the driving Lévy process. Second, by means of a polynomial type large deviation inequality we derive convergence of moments of our estimator under additional conditions.

AB - We consider drift estimation of a discretely observed OrnsteinUhlenbeck process driven by a possibly heavy-tailed symmetric Lévy process with positive activity index β. Under an infill and large-time sampling design, we first establish an asymptotic normality of a self-weighted least absolute deviation estimator with the rate of convergence being √ nh1−1/β n, where n denotes sample size and hn > 0 the sampling mesh satisfying that hn → 0 and nhn → ∞. This implies that the rate of convergence is determined by the most active part of the driving Lévy process; the presence of a driving Wiener part leads to √ nhn, which is familiar in the context of asymptotically efficient estimation of diffusions with compound Poisson jumps, while a pure-jump driving Lévy process leads to a faster one. Also discussed is how to construct corresponding asymptotic confidence regions without full specification of the driving Lévy process. Second, by means of a polynomial type large deviation inequality we derive convergence of moments of our estimator under additional conditions.

UR - http://www.scopus.com/inward/record.url?scp=79251601168&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79251601168&partnerID=8YFLogxK

U2 - 10.1214/10-EJS565

DO - 10.1214/10-EJS565

M3 - Article

AN - SCOPUS:79251601168

VL - 4

SP - 525

EP - 565

JO - Electronic Journal of Statistics

JF - Electronic Journal of Statistics

SN - 1935-7524

ER -