Making the Cauchy work

Abstract

A truncated version of the Cauchy distribution is introduced. Unlike the Cauchy distribution, this possesses finite moments of all orders and could therefore be a better model for certain practical situations. More than 10 practical situations where the truncated distribution could be applied are discussed. Explicit expressions are derived for the moments, L moments, mean deviations, moment generating function, characteristic function, convolution properties, Bonferroni curve, Lorenz curve, entropies, order statistics and the asymptotic distribution of the extreme order statistics. Estimation procedures are detailed by the method of moments and the method of maximum likelihood and expressions derived for the associated Fisher information matrix. Simulation issues are discussed. Finally, an application is illustrated for consumer price indices from the six major economics.