Mittag Leffler Distributions Estimation and Autoregressive Framework

TL;DR

This paper introduces a method for estimating Mittag-Leffler distribution parameters using empirical Laplace transforms, demonstrates its effectiveness through simulations and real high-frequency trading data, and develops an autoregressive model incorporating these distributions.

Contribution

It presents a novel empirical Laplace transform-based estimation technique for Mittag-Leffler distributions and integrates these into autoregressive models for the first time.

Findings

Estimation method yields satisfactory results in simulations.

Application to high-frequency trading data demonstrates practical utility.

Autoregressive models with Mittag-Leffler marginals are feasible and effective.

Abstract

This work deals with the estimation of parameters of Mittag-Leffler (ML($\alpha, \sigma$)) distribution. We estimate the parameters of ML($\alpha, \sigma$) using empirical Laplace transform method. The simulation study indicates that the proposed method provides satisfactory results. The real life application of ML($\alpha, \sigma$) distribution on high frequency trading data is also demonstrated. We also provide the estimation of three-parameter Mittag-Leffler distribution using empirical Laplace transform. Additionally, we establish an autoregressive model of order 1, incorporating the Mittag-Leffler distribution as marginals in one scenario and as innovation terms in another. We apply empirical Laplace transform method to estimate the model parameters and provide the simulation study for the same.