Characterization of distributions with the length-bias scaling property

Marcos Lopez-Garcia (Instituto de Matematicas, UNAM)

Abstract


This paper characterizes the density functions of absolutely continuous positive random variables with finite expectation whose respective distribution functions satisfy the so-called length-bias scaling property.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 186-191

Publication Date: May 3, 2009

DOI: 10.1214/ECP.v14-1458

References

  1. J. Bertoin, P. Biane and M. Yor. Poissonian exponential functionals, q-series, q-integrals, and the moment problem for log-normal distributions. Seminar on Stochastic Analysis, Random Fields and Applications IV, 45-56. Progr. Probab., 58, Birkhauser, Basel, 2004. Math. Review 2005h:60143
  2. D. R. Cox. Renewal theory. Methuen & Co. Ltd., London. John Wiley & Sons, Inc., New York 1962. ix+142 pp. Math. Review 27 #3030
  3. T. S. Chihara. A characterization and a class of distributions functions for the Stieltjes-Wigert polynomials. Canadian Math. Bull. 13 (1970), 529-532. Math. Review 43 #6480
  4. R. Gomez, M. Lopez-Garcia. A family of heat functions as solutions of indeterminate moment problems. Int. J. Math. Math. Sci., vol. 2007, Article ID 41526, 11 pages, doi:10.1155/2007/41526. Math. Review 2008j:44003
  5. R. C. Gupta, S. N. U. A. Kirmani. The role of weighted distributions in stochastic modeling. Comm. Statist. Theory Methods 19 (1990), no. 9, 3147-3162. Math. Review 92e:62180
  6. V. Leyva, A. Sanhueza, J. M. Angulo. A length-biased version of the Birnbaum-Saunders distribution with application in water quality. Stoch. Environ Res Risk Asses (2009) 23, 299-307, doi:10.1007/s00477-008-0215-9.
  7. M. Lopez-Garcia. Characterization of solutions to the log-normal moment problem, to appear in Theory of Probability and its Applications.
  8. S. Y. Oncel, M. Ahsanullah, F. A. Aliev and F. Aygun. Switching record and order statistics via random contractions. Statist. Probab. Lett. 73 (2005), no. 3, 207-217. Math. Review 2006i:62051
  9. A. G. Pakes, R. Khattree. Length-biasing, characterization of laws, and the moment problem. Austral. J. Statist. 34 (1992), 307-322. Math. Review 94a:60018
  10. A. G. Pakes. Length biasing and laws equivalent to the log-normal. J. Math. Anal. Appl. 197 (1996), 825-854. Math. Review 97f:60034
  11. A. G. Pakes. Structure of Stieltjes classes of moment-equivalent probability laws. J. Math. Anal. Appl. 326 (2007), 1268-1290. Math. Review 2008e:60034
  12. A. G. Pakes, J. Navarro. Distributional characterizations through scaling relations. Aust. N. Z. J. Stat. 49 (2007), no. 2, 115-135. Math. Review 2009a:62054
  13. C. R. Rao, D. N. Shanbhag. Choquet-Deny type functional equations with applications to stochastic models. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Ltd., Chichester (1994). xii+290 pp. Math. Review 97d:60020
  14. Y. Vardi, L. A. Shepp, B. F. Logan. Distribution functions invariant under residual-lifetime and length-biased sampling, Wahrscheinlichkeitstheorie verw. Gebiete 56 (1981), 415-426. Electronic Communications in Probability. Math. Review 83c:62167
  15. D. V. Widder. The Heat Equation. Pure and Applied Mathematics, Vol. 67. Academic Press, New York-London (1975). xiv+267 pp. Math. Review 57 #6840
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.