A Simulation Study of Estimator for the Stable Index - Innovative Management in Information and Production - page 8

A Simulation Study of Estimator
for the Stable Index

Wararit Panichkitkosolkul

Abstract Stable distributions are applied in several areas such as communications
theory, physics, biology, astronomy, finance, economics and sociology. By using the
structure of V-statistics, we develop a new estimation of stable index of the stable
distribution. Furthermore, we also compare the proposed estimator with a simple
and unbiased estimator proposed by Fan (Commun Stat-Theor Methods 35(2):
245-255, 2006). Our estimator is constructed by using the linear relation between
U-statistics and V-statistics. In simulation study, we show that proposed estimator is
more efficient than the existing estimator developed by Fan (Commun Stat-Theor
Methods 35(2):245-255, 2006) in terms of the standard deviation, interquartile
range, and mean square error of estimators.

Keywords Stable distributions ￿ Stable index ￿ Estimator ￿ U-statistics ￿

V-statistics

1 Introduction

Stable distributions are a class of probability distributions which have skewness
and heavy tails ( Rimmer and Nolan 2005 ). Stable distributions have a wide
area of applications: probability theory, communications theory, physics, biology,
astronomy, finance, economics, sociology, and so forth ( Bolov 2005 ). The parameter
estimation of stable distributions was first studied by Fama and Roll ( 1971 ). Press
( 1972 ) proposed several methods for estimating parameters of stable distributions.
His methods are based on the sample characteristic functions and depend on