Tail behavior of the failure rate functions of mixtures

The tail behavior of the failure rate of mixtures of lifetime distributions is studied. A typical result is that if the failure rate of the strongest component of the mixture decreases to a limit, then the failure rate of the mixture decreases to the same limit. For a class of distributions containing the gamma distributions this result can be improved in the sense that the behavior of the failure rate of the mixture asymptotically mirrors that of the strongest component in whether it decreases or increases to a limit.