Research On The Monotonicity Of Failure Rate And Mean Residual Life Function Of Lifetime Distributions

The models with bathtub failure rate (BFR) are often used to simulate reliability datas, but there are few distributions with BFR in usual reliability models. Thus mixing two or more usual distributions to obtain BFR models is very useful. The mixture of an exponential distribution and a distribution with linear failure rate is studied in the second chapter. Under some reasonable conditions, the mixture distribution is verified to have a practical BFR by a numerical example. The failure rate of reliability models often has three or more critical points. The third chapter considers the monotonicity of failure rates for these models using the monotonicity of a transitional function, and gives explicit conclusions for how deduce the monotonicity of failure rates by the monotonicity of transitional functions. The forth chapter studies the mixture model of three exponential families of densities, gives the explicit shape of the transitional function and discusses the monotonicity of failure rates for a few mixture models of exponential families of densities, such as mixture models of two Weibull distributions and a exponential distribution, mixture models of three Weibull distributions and mixture models of three gamma distributions. In the fifth chapter, the failure rate function and mean residual life function and their monotonicity of two new lifetime distributions are studied, and the association of two function's change points of bathtub failure rate model is analyzed by studing a new lifetime distribution. The association is that the change point for mean residual life can be much earlier than that of failure rate function usually, and when the difference between the change points is larger, the portion of flatness of failure rate is longer.