Distribution functions invariant under residual-lifetime and length-biased sampling
- 62 Downloads
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.
KeywordsDistribution Function Hull Stochastic Process Probability Theory Convex Hull
Unable to display preview. Download preview PDF.
- 1.Barlow, R.E., Proschan, P.: Statistical Theory of Reliability and Life Testing. New York: Holt, Rinehart and Winston, Inc. 1975Google Scholar
- 2.Cox, D.R.: Renewal Theory. London: Methuen and Co. Ltd. 1962Google Scholar
- 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.Feller, W.: An Introduction to Probability Theory and Its Applications, Vol. II, 2nd ed. New York: John Wiley 1971Google Scholar
- 5.Harkness, W., Shantaram, R.: Convergence of a sequence of transformations of distribution functions. Pacific J. Math. 31, 403–415 (1969)Google Scholar
- 6.Heyde, C.C.: On a property of the lognormal distribution. J. R. Statist. Soc. B 25, 392–393 (1963)Google Scholar
- 7.Shantaram, R., Harkness, W.: On a certain class of limit distributions. Ann. Math. Statist. 43, 2067–2071 (1972)Google Scholar
- 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.Wilk, M.B., Gnanadesikan, R.: Probability plotting methods for the analysis of data. Biometrika 55, 1–17 (1968)Google Scholar