Maximum likelihood estimation in a mininum-type model with exponential and weibull failure modes

The existence and some properties of maximum likelihood estimators (MLE’s) are studied for a minimum-type distribution function corresponding to a minimum of two independent random variables having exponential and Weibull distributions. It is shown that if all three parameters are unknown, then there is a path in the parameter space along which the likelihood function (LF) tends to infinity. It is also proved that if the Weibull shape parameter is known, then the LF is concave, the MLE’s exist, and they can be found by solving the set of likelihood equations. Properties of the MLE’s for this case are illustrated by a Monte Carlo experiment. A sufficient condition for the existence of MLE’s is given for the case of known Weibull scale parameter.