Efficient estimation of stable Levy process with symmetric jumps

TL;DR

This paper develops an efficient method for estimating parameters of a stable Levy process with symmetric jumps observed at high frequency, utilizing local asymptotic normality and a simple initial estimator.

Contribution

It introduces a novel estimation approach leveraging local asymptotic normality and a preliminary method of moments, bypassing complex likelihood optimization.

Findings

One-step estimator performs similarly to maximum likelihood estimator in finite samples.

The likelihood is shown to be locally asymptotically normal with a non-singular Fisher information matrix.

A simple method of moments can serve as an effective initial estimator.

Abstract

Efficient estimation of a non-Gaussian stable Levy process with drift and symmetric jumps observed at high frequency is considered. For this statistical experiment, the local asymptotic normality of the likelihood is proved with a non-singular Fisher information matrix through the use of a non-diagonal norming matrix. The asymptotic normality and efficiency of a sequence of roots of the associated likelihood equation are shown as well. Moreover, we show that a simple preliminary method of moments can be used as an initial estimator of a scoring procedure, thereby conveniently enabling us to bypass numerically demanding likelihood optimization. Our simulation results show that the one-step estimator can exhibit quite similar finite-sample performance as the maximum likelihood estimator.