Explicit and combined estimators for parameters of stable distributions. Issue 4 (4th July 2021) - Lévy Véhel, J.; Philippe, A.; Robet, C.
Abstract : This article focuses on the estimation of the stability index and scale parameter of stable random variables. We study an estimator based on log-moments, . 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 Lévy motion may be estimated accurately and consider a financial log&.