GOODNESS-OF-FIT TESTS BASED ON A NEW CHARACTERIZATION OF THE EXPONENTIAL DISTRIBUTION

The characteristic function $\psi(t)$ = E[exp(itX)] of a random variable X with exponential density exp(-x/0)/0, $\geq$ 0, satisfies the equation $_{v(t) - 0tu(t) = 0, t \epsilon R}$, where u(t) and v(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 [${v_n(t) - \theta_ntu_n(t)}]$$^2$, where ${\theta_n}$ is the maximum-likelihood estimate of θ, 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 two nonzero components of Neyman's smooth test for exponentiality. The new tests are compared with other omnibus tests for exponentiality.