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Spacings around an order statistic

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  • H. Nagaraja
  • Karthik Bharath
  • Fangyuan Zhang

Abstract

We determine the joint limiting distribution of adjacent spacings around a central, intermediate, or an extreme order statistic $$X_{k:n}$$ X k : n of a random sample of size $$n$$ n from a continuous distribution $$F$$ F . For central and intermediate cases, normalized spacings in the left and right neighborhoods are asymptotically i.i.d. exponential random variables. The associated independent Poisson arrival processes are independent of $$X_{k:n}$$ X k : n . For an extreme $$X_{k:n}$$ X k : n , the asymptotic independence property of spacings fails for $$F$$ F in the domain of attraction of Fréchet and Weibull ( $$\alpha \ne 1$$ α ≠ 1 ) distributions. This work also provides additional insight into the limiting distribution for the number of observations around $$X_{k:n}$$ X k : n for all three cases. Copyright The Institute of Statistical Mathematics, Tokyo 2015

Suggested Citation

  • H. Nagaraja & Karthik Bharath & Fangyuan Zhang, 2015. "Spacings around an order statistic," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(3), pages 515-540, June.
    • Handle: RePEc:spr:aistmt:v:67:y:2015:i:3:p:515-540
    DOI: 10.1007/s10463-014-0466-9
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    References listed on IDEAS

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    1. Michael Falk, 1989. "A note on uniform asymptotic normality of intermediate order statistics," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 41(1), pages 19-29, March.
    2. Pakes, Anthony G. & Li, Yun, 1998. "Limit laws for the number of near maxima via the Poisson approximation," Statistics & Probability Letters, Elsevier, vol. 40(4), pages 395-401, November.
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    Cited by:

    1. Arvydas Astrauskas, 2023. "Some Bounds for the Expectations of Functions on Order Statistics and Their Applications," Journal of Theoretical Probability, Springer, vol. 36(2), pages 1116-1147, June.
    2. Loertscher, Simon & Marx, Leslie M., 2020. "Asymptotically optimal prior-free clock auctions," Journal of Economic Theory, Elsevier, vol. 187(C).
    3. Chaitra H. Nagaraja & Haikady N. Nagaraja, 2020. "Distribution‐free Approximate Methods for Constructing Confidence Intervals for Quantiles," International Statistical Review, International Statistical Institute, vol. 88(1), pages 75-100, April.
    4. Loertscher, Simon & Mezzetti, Claudio, 2019. "The deficit on each trade in a Vickrey double auction is at least as large as the Walrasian price gap," Journal of Mathematical Economics, Elsevier, vol. 84(C), pages 101-106.
    5. Anna Dembińska, 2017. "An ergodic theorem for proportions of observations that fall into random sets determined by sample quantiles," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 80(3), pages 319-332, April.

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