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A CLASS OF CONSISTENT TESTS FOR EXPONENTIALITY

BASED ON THE EMPIRICAL LAPLACE TRANSFORM

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LUDWIG BARINGHAUS AND NORBERT HENZE

*Institut für Mathematische Stochastik, Universität Hannover,*

Welfengarten 1, D-3000 Hannover 1, FRG
(Received November 6, 1989; revised May 14, 1990)

**Abstract.**
The Laplace transform *psi*(*t*) = *E*[exp(-*tX*)]
of a random variable with exponential density *lambda* exp(-*lambda*
*x*), *x* __>__ 0, satisfies the differential equation
(*lambda* + *t*)*psi*'(*t*) + *psi*(*t*) = 0, *t* __>__ 0. We study the behaviour of a
class of consistent (``omnibus'') tests for exponentiality based on a
suitably weighted integral of
[(^{^}*lambda*_{n} + *t*)*psi*'_{n}(*t*) + *psi*_{n}(*t*)]^{2}, where ^{^}*lambda*_{n} is
the maximum-likelihood-estimate of *lambda* and *psi*_{n} is the
empirical Laplace transform, each based on an i.i.d. sample
*X*_{1},...., *X*_{n}.

*Key words and phrases*:
Exponential distribution,
goodness-of-fit test, empirical Laplace transform, consistency.

**Source**
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