  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.
