Log-Moment Estimators for the Generalized Linnik and Mittag-Leffler Distributions with Applications to Financial Modeling

We propose formal estimation procedures for the parameters of the generalized, heavy-tailed three-parameter Linnik gL(α, µ, δ) and Mittag-Leffler gML(α, µ, δ) distributions. The paper also aims to provide guidance about the different inference procedures for the different two-parameter Linnik and Mittag-Leffler distributions in the current literature. 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 two-parameter Linnik and Mittag-Leffler models are not flexible enough to accurately model the current stock market data.