On nonparametric maximum likelihood estimation of a distribution uniformly stochastically smaller than a standard

The class of distributions F which are uniformly stochastically smaller than a known standard G arises naturally when life testing experiments are conducted in an environment more severe or stressful than the environment in which the system has been pretested. The problem of estimating a distribution in this class via nonparametric maximum likelihood is considered. Under the assumption that the standard lifetime distribution G is continuous and strictly increasing, the NPMLE of F is derived in closed form, and its inconsistency is demonstrated.