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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. 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.
    2. 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.
    3. 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.
    4. Loertscher, Simon & Marx, Leslie M., 2020. "Asymptotically optimal prior-free clock auctions," Journal of Economic Theory, Elsevier, vol. 187(C).
    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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