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# Distribution functions invariant under residual-lifetime and length-biased sampling

• Y. Vardi
• L. A. Shepp
• B. F. Logan
Article

## Summary

The equation
$$F(qx) = \int\limits_0^x {\left( {1 - F\left( u \right)} \right)du,} {\rm{ }}x \mathbin{\lower.3ex\hbox{\buildrel>\over{\smash{\scriptstyle=}\vphantom{_x}}}} {\rm{0}}$$
(*)
where F is a distribution function (d.f.), arises when the limiting d.f. of the residual-lifetime in a renewal process is a scaled version of the general-lifetime d.f. F. The equation
$$G(qx) = \int\limits_0^x {uG\left( {du} \right),} {\rm{ }}x \mathbin{\lower.3ex\hbox{\buildrel>\over{\smash{\scriptstyle=}\vphantom{_x}}}} {\rm{0}}$$
(**)
on the other hand arises when the limiting d.f. of the total-lifetime in a renewal process is a scaled version of the general-lifetime d.f. G.

For 0<q<1 the class, F q , of all d.f.'s satisfying (*) has been recently characterized and shown to include infinitely many d.f.'s. By explicitly exhibiting all the extreme points of F q , we recharacterize F q as the convex hull of its extreme points and use this characterization to show that for q close to one the d.f. solution to (*) is “nearly unique.” For example, if q>0.8 then all the infinitely many d.f.'s in F q agree to more than 15 decimal places.

The class, G q , of all d.f. solutions to (**) is studied here, apparently for the first time, and shown to be in a one-to-one correspondence with F q ; symbolically, 1−F q (x) is the Laplace transform of G q (qx). For 0<q<1, we characterize Gq as the convex hull of its extreme points and obtain results analogous to those for F q . For q>1 we give a simple argument to show that neither (**) nor (*) has a d.f. solution. We present a complete, self-contained, unified treatment of the two dual families, G q and F q , and discuss previously known results.

A further application of the theory to graphical comparisons of two samples (Q-Q plots) is described.

## Keywords

Distribution Function Hull Stochastic Process Probability Theory Convex Hull
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer-Verlag 1981

## Authors and Affiliations

• Y. Vardi
• 1
• L. A. Shepp
• 1
• B. F. Logan
• 1
1. 1.Bell LaboratoriesMurray HillUSA