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 language | English |
|---|---|
| Pages (from-to) | 525-565 |
| Number of pages | 41 |
| Journal | Electronic Journal of Statistics |
| Volume | 4 |
| DOIs | |
| Publication status | Published - Jan 1 2010 |
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All Science Journal Classification (ASJC) codes
- Statistics and Probability
Cite this
Approximate self-weighted LAD estimation of discretely observed ergodic ornstein-uhlenbeck processes. / Masuda, Hiroki.
In: Electronic Journal of Statistics, Vol. 4, 01.01.2010, p. 525-565.Research output: Contribution to journal › Article
}
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 -