Explicit and combined estimators for stable distributions parameters

TLDR: In this article, the stability index and scale parameter of stable random variables are estimated based on log-moments, which always exist for such random variables, and the main advantage of this estimator is that it has a simple closed form expression.

Abstract: This article focuses on the estimation of the stability index and scale parameter of stable random variables. While there is a sizable literature on this topic, no precise theoretical results seem available. We study an estimator based on log-moments, which always exist for such random variables. The main advantage of this estimator is that it has a simple closed form expression. This allows us to prove an almost sure convergence result as well as a central limit theorem. We show how to improve the accuracy of this estimator by combining it with previously defined ones. The closed form also enables us to consider the case of non identically distributed data, and we show that our results still hold provided deviations from stationarity are ”small”. Using a centro-symmetrization, we expand the previous estimators to skewed stable variables and we construct a test to check the skewness of the data. As applications, we show numerically that the stability index of multistable Levy motion may be estimated accurately and consider a financial log, namely the S&P 500, where we find that the stability index evolves in time in a way that reflects with major financial events.