On Characterization of Some Distributions by Truncation Properties

Abstract

This article draws attention to some elementary results—that belong to the folklore of actuarial science—on the characterization of a reliability function S(x) by the “life expectancy” ℓ_y at age y and other expectations and moments of the residual life of a y survivor, whether the residual life is measured in the usual scale of time or a new scale. These provide closed-form expressions for models based on life expectancy or other truncated or generalized moments hypotheses. The often neglected notion of life potential is emphasized and the notion of H-potential or life utility introduced. Poisson processes and “negative” (inverse) distributions are used as expository background in view of their heuristic unifying value. A test for exponentiality is also considered.