Commentary: Failure-rate functions for doubly-truncated random variables

Navarro and Ruiz [1] express the nonparametric maximum likelihood estimator (NPMLE) of the distribution of a failure-time random variable as a function of the NPMLE of generalized failure-rate functions. These generalized failure-rate functions are equal to the probability density functions of a doubly-truncated failure-time random variable at the endpoints of the truncating interval. Readers can infer from this paper that this simple estimator can be applied to a doubly-truncated sample of failure times. This commentary explains why that estimator does not apply to the general setting in which the observed failure times are doubly-truncated with subject-specific truncating intervals. A doubly-truncated sample of times to brain tumor progression illustrates the deviation of that estimator [1] from the NPMLE for these data. Definitions Quasar: unusually luminous objects found in remote areas of the universe; stellar object: nonquasar object; redshift: the shift in an object'S spectral lines toward the red end, indicating how fast the object is moving away from the observer.