7Parameter Estimation of α-Stable Distributions

7.1. Introduction

The advantages and the potential benefits of α-stable probability distributions are now widely recognized by researchers and the reverse engineering community. For the sake of the argument, Lévy-Véhel and Walter [LEV 02] have already brought forward the advantages of α-stable distributions in the modeling of financial markets. As a result, the non-normality of probability distributions is now an accepted fact. Mandelbrot [MAN 63] and Fama in the 1960s [FAM 65] suggested a possible alternative consisting of introducing α-stable distributions. The second most striking example is the proof of asymmetry in stock markets returns. In effect, Fieletz and Smith [FIE 72], on the one hand, and Leitch and Paulson [LEI 75], on the other, have defended the requirement for the relaxation of the Fama/Rolls symmetry hypothesis [FAM 65, FAM 71]. Due to its symmetry and its excessively blurred tail, Gaussian probability distributions fail in modeling most physical phenomena having a scale-invariant nature. α-stable distributions then generalize the analytical framework.

The current challenge is then to obtain an adequate estimate of these probability distributions. Most of these methods aiming to estimate the parameters of α-stable distributions (equation [7.2]) are time-intensive and/or often centered on only two parameters (α, γ) or with restrictions on parameters. Examples include estimators based on distribution quantiles ...

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