In this paper we investigate the limiting behavior of the failure rate for the convolution of two or more life distributions. In a previous paper on mixtures, Block, Mi and Savits (1993) showed that the failure rate behaves like the limiting behavior of the strongest component. We show a similar result here for convolutions. We also show by example that unlike a mixture population, the ultimate direction of monotonicity does not necessarily follow that of the strongest component.

Mean residual life and failure rate functions are ubiquitously employed in reliabilityanalysis. The term of useful period of lifetime distributions of bathtub-shaped failurerate functions is referred to the flat rigion of this function and has attracted authorsand researchers in reliability, actuary, and survival analysis. In recent years,considering the change points of mean residual life and failure rate functions has beenextensively utelized in determining the optimum burn-in time. In this paper we investigatethe difference between the change points of failure rate and mean residual life functionsof some generalized gamma type distributions due to the capability of these distributionsin modeling various bathtub-shaped failure rate functions.

In this paper we introduce a variation on Glaser's method for determining the shape of the failure rate function of a mixture. It has often been seen that the shape of the failure rate depends on the mixing parameter q. Our method provides an explanation for this phenomenon. We then illustrate our technique with the mixture of an exponential and a gamma density for all possible cases.

It can be seen that a mixture of an exponential distribution and a gamma distribution with increasing failure rate for the right choice of parameters can yield a distribution with a bathtub-shaped failure rate. In this paper we consider a continuous mixture of exponentials and a continuous mixture of gammas with increasing failure rates and show that the resulting mixture has a bathtub-shaped failure rate.