Goodness-of-fit tests based on a new characterization of the exponential distribution

Henze, Norbert
Meintanis, Simos

facul_01; 21

Date: 19. 07. 2001
Abstract:
The characteristic function p(t) = E(exp(itX)) of a random
variable X with exponential density a exp(-ax), x > 0,
satisfies the equation v(t) - atu(t) = 0 for all real t,
where u(t) and v(t) denote the real and the imaginary part
of p(t), respectively. We study a new class of consistent
tests for exponentiality based on a suitably weighted integral
of (v_n(t)- a_ntu_n(t))^2, where a_n is the maximum likelihood
estimate of a, and u_n and v_n denote the empirical counterparts
of u(t) and v(t), respectively. As the decay of the weight
function tends to infinity, the test statistic approaches the
square of a linear combination of the first nonzero component
of Neyman's smooth test for exponentiality. The new tests are
compared with other omnibus tests for exponentiality.

Primary MSC: 62G10 Hypothesis testing
Secondary MSC: 62F05 Asymptotic properties of tests
Keywords: Goodness-of-fit test, Test for exponentiality, Characterization of exponentiality, Characteristic function, Empirical characteristic function, Smooth test