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 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Barlow, R.E., Proschan, P.: Statistical Theory of Reliability and Life Testing. New York: Holt, Rinehart and Winston, Inc. 1975Google Scholar
  2. 2.
    Cox, D.R.: Renewal Theory. London: Methuen and Co. Ltd. 1962Google Scholar
  3. 3.
    DeBruijn, N.G.: The difference-differential equation F′(x) = exp(α x + β) F (x − 1). Indag. Math. 15, 449–458 (1953)Google Scholar
  4. 4.
    Feller, W.: An Introduction to Probability Theory and Its Applications, Vol. II, 2nd ed. New York: John Wiley 1971Google Scholar
  5. 5.
    Harkness, W., Shantaram, R.: Convergence of a sequence of transformations of distribution functions. Pacific J. Math. 31, 403–415 (1969)Google Scholar
  6. 6.
    Heyde, C.C.: On a property of the lognormal distribution. J. R. Statist. Soc. B 25, 392–393 (1963)Google Scholar
  7. 7.
    Shantaram, R., Harkness, W.: On a certain class of limit distributions. Ann. Math. Statist. 43, 2067–2071 (1972)Google Scholar
  8. 8.
    van Beek, P., Braat, J.: The limits of sequences of iterated overshoot distribution functions. Stochastic Processes and Their Applications 1, 307–316. Amsterdam: North-Holland 1973Google Scholar
  9. 9.
    Wilk, M.B., Gnanadesikan, R.: Probability plotting methods for the analysis of data. Biometrika 55, 1–17 (1968)Google Scholar

Copyright information

© Springer-Verlag 1981

Authors and Affiliations

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

Personalised recommendations