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

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


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.


Distribution Function Hull Stochastic Process Probability Theory Convex Hull 
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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

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