Page 437 - Contributed Paper Session (CPS) - Volume 2

CPS1917 Trijya S.

Estimation of parameters of a mixture of two
exponential distributions
Trijya Singh
Le Moyne College, Syracuse, U.S.A

Abstract
For estimating the parameters of a mixture of two exponential distributions,
the method of moments, which uses roots of a quadratic equation involving
the estimates of the first three raw moments, has been used in the past.
Because of poor estimates of these moments, in many situations roots of the
quadratic equation turnout to be complex and hence the method fails. In this
paper, a methodology based on a quadrature formula of numerical integration
is proposed for estimation of the moments. The peak and tail characteristics
of a distribution are explained by the standardized fourth central moment, that
is, the coefficient of kurtosis. To incorporate information about these
characteristics, a methodology based on the first four sample moments is also
proposed here. We have applied the proposed methodology to obtain initial
estimates of parameters of an extremely useful model used in
pharmacokinetic analysis and illustrated this using a drug concentration data
set. It has been shown that methods using all four moments perform better
than those based on only the first three moments. We have also demonstrated
the superiority of the proposed methods over an existing method of finding
initial estimates for an exponential mixture distribution.

Keywords
Trapezoidal curvature formula; method of moments; two-compartment
model; pharmacokinetic analysis

1. Introduction
A mixture of two exponential distributions as an underlying distribution of
a data generating process may arise in many areas of application. In life testing
and reliability analysis, when failures of components occur due to two
prominent causes, the distribution of failure times in some cases turns out to
be the mixture of two exponential distributions. We may also encounter this
mixture model in actuarial studies for the distribution of losses (settled claims)
in the case of insured events if the accidents have two major causes like road
accidents and fires as shown in Hogg & Klugman (1984). Harrison & Millard
(1991) reported yet another interesting example from hospital management.
They found that the distribution of duration of hospital stay of geriatric
patients suffering from two levels of severity of a disease was a mixture of two

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