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

TL;DR

This paper introduces new log-moment estimators for generalized Linnik and Mittag-Leffler distributions, demonstrating their effectiveness in financial data modeling and highlighting limitations of traditional models.

Contribution

It develops asymptotically unbiased, computationally efficient estimation procedures for the generalized distributions, with applications to stock market data.

Findings

Standard models are insufficient for current stock data

Proposed estimators are asymptotically unbiased

Methods are computationally efficient

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.