A new weighted integral goodness-of-fit statistic for exponentiality

We propose a new weighted integral goodness-of-fit statistic for exponentiality. The statistic is motivated by a characterization of the exponential distribution via the mean residual life function. Its limit null distribution is the same as that of a certain weighted integral of the squared Brownian bridge. The Laplace transform and cumulants of the latter are expressible in terms of Bessel functions.