Spacings around an order statistic

Abstract

We determine the joint limiting distribution of adjacent spacings around a central, intermediate, or an extreme order statistic \(X_{k:n}\) of a random sample of size \(n\) from a continuous distribution \(F\). 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 \(X_{k:n}\). For an extreme \(X_{k:n}\), the asymptotic independence property of spacings fails for \(F\) in the domain of attraction of Fréchet and Weibull (\(\alpha \ne 1\)) distributions. This work also provides additional insight into the limiting distribution for the number of observations around \(X_{k:n}\) for all three cases.

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References

  1. Balakrishnan, N., Stepanov, A. (2005). A note on the number of observations near an order statistic. Journal of Statistical Planning and Inference, 134, 1–14.

  2. Chanda, K. C. (1975). Some comments on the asymptotic probability laws of sample quantiles. Calcutta Statistical Association Bulletin, 24, 123–126.

  3. Daley, D. J., Vere-Jones, D. (2003). An introduction to the theory of point processes. Elementary theory and methods, Vol. I, (2nd ed.) New York: Springer.

  4. David, H. A., Nagaraja, H. N. (2003). Order Statistics (3rd ed.). Hoboken: Wiley.

  5. de Haan, L. (1970). On regular variation and its application to the weak convergence of sample extremes. Mathematical centre tracts 32. Amsterdam: Mathematisch Centrum.

  6. Dembińska, A., Balakrishnan, N. (2010). On the asymptotic independence of numbers of observations near order statistics. Statistics, 44, 517–528.

  7. Dembińska, A., Stephanov, A., Wesołowski, J. (2007). How many observations fall in a neighborhood of an order statistic? Communications in Statistics -Theory and Methods, 36, 851–867.

  8. Embrechts, P., Klüppelberg, C., Mikosch, T. (1997). Modelling extremal events. Berlin: Springer.

  9. Falk, M. (1989). A note of uniform asymptotic normality of intermediate order statistics. Annals of Institute of Statistical Mathematics, 41, 19–29.

  10. Ghosh, J. K. (1971). A new of proof of Bahadur’s representation of quantiles and an application. Annals of Mathematical Statistics, 42, 1957–1961.

  11. Ghosh, M., Sukhatme, S. (1981). On Bahadur’s representation of quantiles in nonregular cases. Communications in Statistics -Theory and Methods, 10, 269–282.

  12. Gnedenko, B. V. (1943). Sur la distribution limite du terme maximum d’une série altéatoire. Annals of Mathematics, 44, 423–453.

  13. Hall, P. (1978). Representations and limit theorems for extreme value distributions. Journal of Applied Probability, 15, 639–644.

  14. Harshova, E., Hüsler, J. (2000). On the number of near-maxima. Rendiconti del Circolo Mathematico di Palermo, Serie II, Supplementi, 65, 121–136.

  15. Nagaraja, H. N. (1982). Record values and extreme value distributions. Journal of Applied Probability, 19, 233–239.

  16. Pakes, A. G. (2009). Number of observations near order statistics. Australian and New Zealand Journal of Statistics, 51(4), 375–395.

  17. Pakes, A. G., Li, Y. (1998). Limit laws for the number of near maxima via the Poisson approximation. Statistics and Probability Letters, 40, 395–401.

  18. Pakes, A. G., Steutel, F. (1997). On the number of records near the maximum. Australian Journal of Statistics, 39, 179–192.

  19. Pyke, R. (1965). Spacings. Journal of Royal Statistical Society, Series B, 27, 395–436.

  20. Reiss, R.-D. (1989). Approximate distributions of order statistics. New York: Springer.

  21. Siddiqui, M. M. (1960). Distribution of quantiles in samples from a bivariate population. Journal of Research, National Bureau of Standards, 64B, 145–150.

  22. Smirnov, N.V. (1952). Limit distributions for the terms of a variational series. American Mathematical Translations, Series 1, no. 67. Original published in 1949.

  23. Smirnov, N. V. (1967). Some remarks on limit laws for order statistics. Theory of Probability and Its Applications, 12, 82–143.

  24. Teugels, J. L. (2001). On order statistics close to the maixmum. In M. de Gunst, C. Klaassen, A. van der Vaart (Eds.), State of the art in probability and statistics (pp. 568–581). Beachwood: Institute of Mathematical Statistics.

  25. von Mises, R. (1936). La distribution de la plus grande de \(n\) valeurs. Review of Mathematical Union Interbalkanique, 1, 141–160.

  26. Weissman, I. (1978). Estimation of parameters and larger quantiles based on the \(k\) largest observations. Journal of American Statistical Association, 73, 812–815.

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Acknowledgments

The authors thank two anonymous reviewers for a close reading of the manuscript and many helpful suggestions.

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Correspondence to H. N. Nagaraja.

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Nagaraja, H.N., Bharath, K. & Zhang, F. Spacings around an order statistic. Ann Inst Stat Math 67, 515–540 (2015). https://doi.org/10.1007/s10463-014-0466-9

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Keywords

  • Spacings
  • Uniform distribution
  • Central order statistics
  • Intermediate order statistics
  • Extremes
  • Poisson process