The test of exponentiality based on the mean residual life function revisited

We revisit the family of goodness-of-fit tests for exponentiality based on the mean residual life time proposed by Baringhaus and Henze [(2008), ‘A New Weighted Integral Goodness-of-Fit Statistic for Exponentiality’, Statistics & Probability Letters, 78(8), 1006–1016]. We motivate the test statistic by a characterisation of Shanbhag [(1970), ‘The Characterizations for Exponential and Geometric Distributions’, Journal of the American Statistical Association, 65(331), 1256–1259] and provide an alternative representation, which leads to simple and short proofs for the known theory and an easy to access covariance structure of the limiting Gaussian process under the null hypothesis. Explicit formulas for the eigenvalues and eigenfunctions of the operator associated with the limit covariance are given using results on weighted Brownian bridges. In addition, we derive further asymptotic theory under fixed alternatives as well as approximate Bahadur efficiencies, which provide an insight into the choice of the tuning parameter with regard to the power performance of the tests.