Approximate self-weighted LAD estimation of discretely observed ergodic Ornstein-Uhlenbeck processes

Abstract

We consider drift estimation of a discretely observed Ornstein-Uhlenbeck 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 $\sqrt{n}h_{n}^{1-1/\beta}$, where n denotes sample size and h_n>0 the sampling mesh satisfying that h_n→0 and nh_n→∞. 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 $\sqrt{nh_{n}}$, 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.