Advertisement

Journal of Mathematical Sciences

, Volume 220, Issue 6, pp 763–776 | Cite as

Asymptotics of the Optimal Confidence Region for Shift and Scale, Based on Two Order Statistics

  • A.Yu. Zaigraev
  • M. Alama-Bućko
Article

Using two order statistics, we construct the two-dimensional optimal strong confidence region for shift and scale parameters and study its asymptotics when the sample size tends to infinity. Also, we construct simpler confidence region asymptotically equivalent to the optimal one. The constructed confidence region is compared with similar confidence regions, based on sample mean and sample mean-square deviation.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    H. David, Order Statistics, Wiley, New York (1970).MATHGoogle Scholar
  2. 2.
    M. Alama-Bućko, A. V. Nagaev, and A. Zaigraev, “Asymptotic analysis of minimum volume confidence regions for location-scale families,” Appl. Math., 33, No. 1, 1–20 (2006).MathSciNetMATHGoogle Scholar
  3. 3.
    A. Czarnowska and A.V. Nagaev, “Confidence regions of minimal area for the scale-location parameter and their applications,” Appl. Math., 28, No. 2, 125–142 (2001).MathSciNetMATHGoogle Scholar
  4. 4.
    S. Jeyaratnam, “Minimum volume confidence regions,” Stat. Probab. Letters, No. 3, 307–308 (1985).Google Scholar
  5. 5.
    R.-D. Reiss, “On the accuracy of the normal approximation for quantiles,” Ann. Probab., 2, 741–744 (1974).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    R.-D. Reiss, “The asymptotic normality and asymptotic expansions for the joint distribution of several order statistics,” Colloq. Math. Soc. Janos Bolyai, 11, 297–340 (1975).MathSciNetMATHGoogle Scholar
  7. 7.
    R.-D. Reiss, “Asymptotic expansions for sample quantiles,” Ann. Probab., 4, No. 2, 249–258 (1976).MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    R.-D. Reiss, “Approximation of product measures with an application to order statistics,” Ann. Probab., 9, No. 2, 335–341 (1981).MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    R.-D. Reiss, Approximate Distributions of Order Statistics with Applications to Non-parametric Statistics, Springer-Verlag, New York (1989).CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Nikolay Kopernik UniversityTorunPoland
  2. 2.UTP University of Science and TechnologyBydgoszczPoland

Personalised recommendations