# Efficient estimation of stable Lévy process with symmetric jumps

## Abstract

Efficient estimation of a non-Gaussian stable Lévy 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.

## Notes

### Acknowledgements

The authors also thank the reviewers and the associated editor for their valuable comments, which in particular led to substantial improvements of the arguments in Sects. 3.2 and 3.4. HM especially thanks Professor Jean Jacod for letting him notice the mistake in Masuda (2009), which has been fixed in the present paper.

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