Notes on estimating inverse-Gaussian and gamma subordinators under high-frequency sampling

Abstract

We study joint efficient estimation of two parameters dominating either the inverse-Gaussian or gamma subordinator, based on discrete observations sampled at ${(t^{n}_{i})_{i=1}^{n}}$ satisfying ${h_{n}:=\max_{i\le n}(t^{n}_{i}-t^{n}_{i-1}) \to 0}$ as ${n \to \infty}$ . Under the condition that ${T_{n}:=t^{n}_{n} \to \infty}$ as ${n\to\infty}$ we have two kinds of optimal rates, ${\sqrt{n}}$ and ${\sqrt{T_{n}}}$ . Moreover, as in estimation of diffusion coefficient of a Wiener process the ${\sqrt{n}}$ -consistent component of the estimator is effectively workable even when T _n does not tend to infinity. Simulation experiments are given under several h _n ’s behaviors.