The almost sure representation of intermediate order statistics

  • Vernon Watts
Article

Summary

Let {Xn} be independent and identically distributed and let X kn (n) denote the kn-th order statistic for X1 ..., Xn, where kn→∞ but kn/n→0. A representation for X kn (n) in terms of the empirical distribution function is developed. The conditions include those under which X kn (n) is asymptotically normal.

Keywords

Distribution Function Stochastic Process Probability Theory Order Statistic Mathematical Biology 

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References

  1. 1.
    Bahadur, R.R.: A note on quantiles in large samples. Ann. Math. Statist. 37, 577–580 (1966)Google Scholar
  2. 2.
    Cheng, B.: The limiting distributions of order statistics. Chinese Math. 6, 84–104 (1965)Google Scholar
  3. 3.
    Csörgö, M., Révész, P.: Strong approximations of the quantile process. Ann. Statist. 6, 882–894 (1978)Google Scholar
  4. 4.
    Ghosh, J.K.: A new proof of the Bahadur representation of quantiles and an application. Ann. Math. Statist. 42, 1957–1961 (1971)Google Scholar
  5. 5.
    Kiefer, J.: On Bahadur's representation of sample quantiles. Ann. Math. Statist. 38, 1323–1342 (1967)Google Scholar
  6. 6.
    Kiefer, J.: Iterated logarithm analogues for sample quantiles when pn↓0. Proc. Sixth Berkeley Sympos. Math. Statist. Probab. 1, 227–244, (1972)Google Scholar
  7. 7.
    Sen, P.K.: Asymptotic normality of sample quantiles for m-dependent processes. Ann. Math. Statist. 39, 1724–1730 (1968)Google Scholar
  8. 8.
    Sen, P.K.: On the Bahadur representation of sample quantiles for sequences of ø-mixing random variables. J. Multivariate Anal. 2, 77–95 (1972)Google Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • Vernon Watts
    • 1
  1. 1.Durham Life Insurance CompanyRaleighUSA

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