Summary: The characteristic function \(\psi(t)=E[exp(itX)]\) of a random variable \(X\) with exponential density \(\theta^{-1}exp(-x\theta^{-1}),\;x\geq 0\), satisfies the equation \(\nu(t)-\theta t u(t)=0\), \(t\in\mathbb{R}\), where \(u(t)\) and \(\nu(t)\) denote the real and the imaginary part of \(\psi(t)\), respectively.
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.