, Volume 31, Issue 1, pp 99–113 | Cite as

Natural α-quantiles and conditional α-quantiles

  • D. Landers
  • L. Rogge


In general the α-quantile of a random variable is not uniquely determined. It is the aim of this paper to suggest a limit process for chosing exactly one α-quantile. Moreover a natural choice of a conditional α-quantile is suggested.


Conditional Median Convex Function Probability Space Natural Choice Regular Distribution 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ando, T., andL. Amemiya: Almost everywhere convergence of prediction sequences inL p (1<p<∞). Z. Wahrscheinlichkeitstheorie verw. Geb.4, 1965, 113–120.MathSciNetCrossRefMATHGoogle Scholar
  2. Freedman, D.: Markov Chains. San Francisco 1971.Google Scholar
  3. Hoch, H.: Diplomarbeit. Konstanz 1980.Google Scholar
  4. Landers, D., andL. Rogge: Best approximants inL ϕ-spaces. Z. Wahrscheinlichkeitstheorie verw. Geb.51, 1980, 215–237.MathSciNetCrossRefMATHGoogle Scholar
  5. —: Natural choice ofL 1-approximants. Journal of Approximation Theory33, 1981a, 268–280.MathSciNetCrossRefMATHGoogle Scholar
  6. —: Consistent estimation of the natural median. Statistics and Decision1, 1983, 269–284.MathSciNetMATHGoogle Scholar
  7. —: The Natural Median. Annals of Probability9, 1981b, 1041–1042.MathSciNetCrossRefMATHGoogle Scholar
  8. —: Isotonic approximation inL S. Journal of Approximation Theory31, 1981c, 199–223.MathSciNetCrossRefMATHGoogle Scholar
  9. Shintani, T., andT. Ando: Best approximants inL 1-space. Z. Wahrscheinlichkeitstheorie verw. Geb.33, 1975, 33–39.MathSciNetCrossRefMATHGoogle Scholar
  10. Tomkins, R.J.: On conditional medians. Ann. Prob.3, 1975, 375–379.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Physica-Verlag 1984

Authors and Affiliations

  • D. Landers
    • 1
  • L. Rogge
    • 2
  1. 1.Mathematisches Institut der Universität KölnKöln 41
  2. 2.Universität-Gesamthochschule DuisburgDuisburg

Personalised recommendations