Exponential Approximations for Two Classes of Aging Distributions

摘要

If a random variable is exponentially distributed with microns = EX and microns 2 = EX2, then microns 2 = 2 microns squared. Defining rho determinant u(2)/2 micron squared minus 1, it is tempting to conjecture that under mild restrictions a distribution with small rho is approximately exponential. That restrictions are needed is seen by the example, Pr(X = 0) = Pr(X - 1) = 1/2, for which rho = 0. The scale invariant quantity, rho, was suggested by Keilson. It has an interesting interpretation. Define G(x) = micron 1/minus x integral (x,0)/minus negative F (s)ds, the stationary renewal distribution corresponding to F. Then microns G = microns 2/2 microns and rho = determinant micron(G)/micron minus 1. The parameter rho is thus the scaled (by micron) distance between micron and micron G. For F exponential, F = G and thus micron = micron G.