De Finetti-type Representations for Life Distributions

Beginning with a finite population of units and the judgment of exchangeability for units with respect to lifetime, we argue that measures of similarity lead to the appropriate probabilistic models for aging. This in turn implies that Schur-concavity of the joint probability function (or, more generally, the joint survival distribution) provides the correct probabilistic description of aging. Following this argument and using the principle of indifference, we argue that the appropriate probability models for life distributions conditional on average life are in a family of distributions that we call the generalized gamma distributions. If, on the other hand, we are interested in probabilistic models for aging conditional on average lifetime maintenance cost, it follows from our development that generalized Weibull distributions are appropriate.