Robust confidence limits

  • Peter J. Huber


A location parameter is to be estimated from a sample of fixed size n, assuming that the shape of the true underlying distribution lies anywhere within ε of some given shape, e.g. the normal one. The metric in the space of distribution functions may be defined in various ways: total variation, Kolmogorov or Lévy distance. A minimax solution to this problem is described explicitly; it minimizes the maximum probability that the estimate exceeds, or falls below, the true value of the parameter by more than some fixed amount.


Distribution Function Total Variation Stochastic Process Probability Theory Confidence Limit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anscombe, F. J., and B. A. Barron: Treatment of outliers in samples of size three. J. Res. nat. Bur. Standards, Sect. B, 70B, 141–147 (1966).Google Scholar
  2. 2.
    Bickel, P. J.: On some robust estimates of location. Ann. math. Statistics 36, 847–858 (1965).Google Scholar
  3. 3.
    Birnbaum, A.: Confidence curves. J. Amer. statist. Assoc. 56, 246–249 (1961).Google Scholar
  4. 4.
    Czuber, E.: Theorie der Beobachtungsfehler. Leipzig: Teubner 1891.Google Scholar
  5. 5.
    Halmos, P. R.: The theory of unbiased estimation. Ann. math. Statistics 17, 34–43 (1946).Google Scholar
  6. 6.
    Huber, P. J.: Robust estimation of a location parameter. Ann. math. Statistics 35, 73–101 (1964).Google Scholar
  7. 7.
    Huber, P. J. A robust version of the probability ratio test. Ann. math. Statistics 36, 1753–1758 (1965).Google Scholar
  8. 8.
    Lehmann, E. L.: Testing statistical hypotheses. New York: Wiley 1959.Google Scholar
  9. 9.
    Lehmann, E. L., and H. Scheffé: Completeness, similar regions and unbiased estimation. Sankhyā 10, 305–340 (1950); 15, 219–236 (1955).Google Scholar
  10. 10.
    Locher, H.: Diplomarbeit ETH (unpublished) 1966.Google Scholar
  11. 11.
    Strassen, V.: Me\fehler und Information. Z. Wahrscheinlichkeitstheorie verw. Geb. 2, 273–305 (1964).Google Scholar
  12. 12.
    Tukey, J. W.: A survey of sampling from contaminated distributions. In: Contributions to probability and statistics. (Ed. Olkin.) Stanford Univ. Press 1960.Google Scholar

Copyright information

© Springer-Verlag 1968

Authors and Affiliations

  • Peter J. Huber
    • 1
  1. 1.Lehrstuhl f. math. StatistikEidgen. Techn. HochschuleZürich

Personalised recommendations