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

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