MAXIMUM LIKELIHOOD ESTIMATION OF ASYMMETRIC LAPLACE PARAMETERS

SAMUEL KOTZ^{1}, TOMASZ J. KOZUBOWSKI^{2} AND KRZYSZTOF PODGÓRSKI^{3}

1Department of Engineering Management and Systems Engineering, George Washington University,

Washington, D.C. 20052, U.S.A. 2Department of Mathematics, University of Nevada, Reno, NV 89557-0045, U.S.A.

3Department of Mathematical Sciences, Indiana University - Purdue University Indianapolis, Indianapolis, IN 46202, U.S.A.

(Received March 26, 2001; revised October 1, 2001)

Abstract. Maximum likelihood estimators (MLE’s) are presented for the parameters of a univariate asymmetric Laplace distribution for all possible situations related to known or unknown parameters. These estimators admit explicit form in all but two cases. In these exceptions effective algorithms for computing the estimators are provided. Asymptotic distributions of the estimators are given. The asymptotic normality and consistency of the MLE’s for the scale and location parameters are derived directly via representations of the relevant random variables rather than from general sufficient conditions for asymptotic normality of the MLE’s.

Key words and phrases: Double exponential distribution, geometric stable law, Laplace distribution, mathematical finance, random summation, skew Laplace distribution.