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Zbl 1008.62043
Henze, Norbert; Meintanis, Simos G.
Goodness-of-fit tests based on a new characterization of the exponential distribution. (English)
[J] Commun. Stat., Theory Methods 31, No. 9, 1479-1497 (2002). ISSN 0361-0926; ISSN 1532-415X/e

Summary: The characteristic function $\psi(t)=E[exp(itX)]$ of a random variable $X$ with exponential density $\theta^{-1}exp(-x\theta^{-1}),\ x\ge 0$, satisfies the equation $\nu(t)-\theta t u(t)=0$, $t\in\bbfR$, where $u(t)$ and $\nu(t)$ denote the real and the imaginary part of $\psi(t)$, respectively.\par We study a new class of consistent tests for exponentiality based on a suitably weighted integral of $[\nu_n(t)-\hat\theta_n tu_n(t)]^2$, where $\hat\theta_n$ is the maximum-likelihood estimate of $\theta$, and $u_n$ and $\nu_n$ denote the empirical counterparts of $u(t)$ and $\nu(t)$, respectively. As the decay of the weight function tends to infinity, the test statistic approaches the square of a linear combination of the first two nonzero components of Neyman's smooth test for exponentiality. The new tests are compared with other omnibus tests for exponentiality.

MSC 2000:: *62G10 Nonparametric hypothesis testing
62E10 Structure theory of statistical distributions

Keywords: tests for exponentiality; characterization of exponentiality; characteristic functions; empirical characteristic functions; smooth tests

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