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Mittag Leffler Distributions Estimation and Autoregressive Framework

Monika S. Dhull

TL;DR

The paper addresses parameter estimation for Mittag-Leffler distributions and AR models driven by ML marginals or innovations. It introduces an empirical Laplace transform method, leveraging the theoretical transform $\phi_M(s)=\dfrac{1}{1+(\sigma s)^{\alpha}}$ (and its Prabhakar extension) to estimate $\alpha,\sigma$ (and $\gamma$) from data, with validation via simulations and real oil-futures data. It also develops an AR(1) framework with ML($\alpha,1$) marginals and derives the distributional form of the innovations, including a non-time-reversible joint transform, and demonstrates accurate parameter recovery through simulation. Overall, the approach provides a practical, scalable toolkit for modeling heavy-tailed fractional processes in finance and beyond, with demonstrated applicability to high-frequency inter-arrival times and Prabhakar-type processes.

Abstract

This work deals with the estimation of parameters of Mittag-Leffler (ML($α, σ$)) distribution. We estimate the parameters of ML($α, σ$) using empirical Laplace transform method. The simulation study indicates that the proposed method provides satisfactory results. The real life application of ML($α, σ$) 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.

Mittag Leffler Distributions Estimation and Autoregressive Framework

TL;DR

The paper addresses parameter estimation for Mittag-Leffler distributions and AR models driven by ML marginals or innovations. It introduces an empirical Laplace transform method, leveraging the theoretical transform (and its Prabhakar extension) to estimate (and ) from data, with validation via simulations and real oil-futures data. It also develops an AR(1) framework with ML() marginals and derives the distributional form of the innovations, including a non-time-reversible joint transform, and demonstrates accurate parameter recovery through simulation. Overall, the approach provides a practical, scalable toolkit for modeling heavy-tailed fractional processes in finance and beyond, with demonstrated applicability to high-frequency inter-arrival times and Prabhakar-type processes.

Abstract

This work deals with the estimation of parameters of Mittag-Leffler (ML()) distribution. We estimate the parameters of ML() using empirical Laplace transform method. The simulation study indicates that the proposed method provides satisfactory results. The real life application of ML() 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.
Paper Structure (8 sections, 4 theorems, 20 equations, 6 figures, 6 tables)

This paper contains 8 sections, 4 theorems, 20 equations, 6 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Mittag-Leffler ($\alpha, \sigma$) distribution
  3. Estimation of ML($\alpha, \sigma$) distribution
  4. Real data application
  5. Mittag-Leffler($\alpha, \sigma,\gamma$) or Prabhakar distribution
  6. Autoregressive models
  7. Simulation study
  8. Conclusion

Key Result

Corollary 1

The Laplace transform of Prabhakar distribution $M_{\alpha}^{\sigma, \gamma}$ is, where ${}_2F_1(a,b,c;z)$ is the Gauss hypergeometric function.

Figures (6)

  • Figure 1: The boxplots of ML($\alpha,\sigma$) distribution with true parameters $(\alpha_1, \sigma_1) =(0.3,4)$, ($\alpha_2, \sigma_2)= (0.5, 4)$, ($\alpha_3, \sigma_3)= (0.7, 10)$ and ($\alpha_4, \sigma_4)= (0.9, 10)$. The simulated data has $500$ trajectories from ML($\alpha,\sigma$) each of length $1000.$
  • Figure 2: Survival functions for oil futures high frequency trading data compared with the corresponding exponential distribution's survival functions.
  • Figure 3: The boxplots of ML($\alpha,\sigma, \gamma$) distribution with true parameters $(\alpha_1, \sigma_1, \gamma_1) =(0.8,1.4,1)$, ($\alpha_2, \sigma_2, \gamma_2)= (0.5, 2,3)$, and ($\alpha_3, \sigma_3, \gamma_3)= (0.2,1.8,5)$. The simulated data has $100$ trajectories from ML($\alpha,\sigma, \gamma$) each of length $1000.$
  • Figure 4: Contour plot with branch point at $O=(0,0)$ in anti-clockwise direction.
  • Figure 5: The plots of time series $\{Y_t\}$ with ML($\alpha,1$) marginals with true parameters (a) $\alpha_1 = 0.4,\, \rho_1=0.4$ (c) $\alpha_2 = 0.6,\, \rho_2 = 0.8$ and the corresponding innovation series $\{\epsilon_t\}$ with true parameters (b) $\alpha_1 = 0.4, \rho_1 = 0.4$ (d) $\alpha_2 = 0.6, \rho_2 = 0.8$.
  • ...and 1 more figures

Theorems & Definitions (8)

  • Definition 1
  • Corollary 1
  • Proposition 1
  • proof
  • Theorem 2
  • proof
  • Proposition 3
  • proof