In this paper we obtain characterizations of large classes of nonparametric life distributions, such as the increasing (decreasing) failure rate, increasing (decreasing failure rate average, new better (worse) than used, etc., classes. The methods used differ from the usual functional equation methods used for the far more common characterizations of parametric families of life distributions.
Naftali A. Langberg. Ramon V. Leon. Frank Proschan. "Characterization of Nonparametric Classes of Life Distributions." Ann. Probab. 8 (6) 1163 - 1170, December, 1980. https://doi.org/10.1214/aop/1176994577