AISM 54, 816-826

© 2002 ISM

## Maximum likelihood estimation of asymmetric Laplace parameters

### Samuel Kotz^{1}, Tomasz J. Kozubowski^{2} and Krzysztof Podgórski^{3}

^{1}Department of Engineering Management and Systems
Engineering, George Washington University, Washington, D.C. 20052, U.S.A.

^{2}Department of Mathematics, University of Nevada,
Reno, NV 89557-0045, U.S.A.

^{3}Department 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.

**Source**
(TeX , DVI )