Log-moment estimators for the generalized Linnik and Mittag-Leffler distributions with applications to financial modeling

Abstract

We propose formal estimation procedures for the parameters of the generalized, three-parameter Linnik $gL(\alpha,\mu, \delta)$ and Mittag-Leffler $gML(\alpha,\mu, \delta)$ distributions. The estimators are derived from the moments of the log-transformed random variables, and are shown to be asymptotically unbiased. The estimation algorithms are computationally efficient and the proposed procedures are tested using the daily S\&P 500 and Dow Jones index data. The results show that the standard two-parameter Linnik and Mittag-Leffler models are not flexible enough to accurately model the current stock market data.