  We revisit the family of goodness-of-fit tests for exponentiality based on
the mean residual life time proposed by Baringhaus & Henze (2008). We motivate
the test statistic by a characterisation of Shanbhag (1970) 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 derived using results on weighted Brownian bridges. In addition
we provide further asymptotic theory under fixed alternatives and derive
approximate Bahadur efficiencies, which provide an insight into the choice of
the tuning parameter with regard to the power performance of the tests.
