Tight bounds on the sup metric between the exponential distribution and new better (worse) than used in expectation (NBUE, NWUE) distributions are established in terms of the proximity of the second moments of the distributions concerned. Real variable methods are used to identify the extremal distributions that attain the bounds. Similar methods establish similar results for the harmonic NBUE and NWUE classes of distributions.
D. J. Daley. "Tight Bounds on the Exponential Approximation of Some Aging Distributions." Ann. Probab. 16 (1) 414 - 423, January, 1988. https://doi.org/10.1214/aop/1176991912