Marshall and Proschan (1965) showed that the MLE for a life distribution with increasing failure rate is strongly consistent. In this note we show that the MLE for a life distribution with increasing failure rate average is not consistent; in fact, maximum likelihood estimation in the IFRA case yields estimators of the average failure rate and of the distribution function which, in general, converge a.s. to values other than the true values. In the decreasing failure rate average case, the MLE fails to exist.
"Inconsistency of the Maximum Likelihood Estimator of a Distribution Having Increasing Failure Rate Average." Ann. Statist. 13 (1) 413 - 417, March, 1985. https://doi.org/10.1214/aos/1176346602