Estimation of location and scale in Cauchy distributions using the empirical characteristic function

Abstract

The iterative method for the parameter estimation of stable laws proposed by Koutrouvelis (1981) is modified to handle the case where the distribution is assumed to belong to a Cauchy family with location and/or scale parameter unknown. It is shown that the determination of optimum values at which the empirical characteristic function needs to be evaluated reduces in this case to the determination of the asymptotically optimum quantiles for the parameter estimation of an exponential distribution by linear functions of order statistics. The estimators of the scale and location parameters are asymptotically independent and normally distributed, and achieve asymptotic efficiencies close to unity. The proposed procedure is compared to a method of estimation using linear functions of order statistics.