Asymptotic Results for m-th Exponential Spacings

Abstract

In this work, we discuss m-th exponential spacings △k:m:n obtained from order statistics. We study limit results for such spacings when the sample size n tends to infinity and the indices k and m are either fixed or also tend to infinity. We also investigate asymptotic properties of largest exponential m-th spacing.

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References

  1. Ahsanullah, M (1984). A characterization of the exponential distribution by higher order gap. Metrika 31, 323–326.

    MathSciNet  Article  Google Scholar 

  2. Arnold, BC, Balakrishnan, N and Nagaraja, HN (1992). A first course in order statistics. Wiley, New York.

    MATH  Google Scholar 

  3. Balakrishnan, NK and Koutras, MV (2002). Runs and scans with applications. Wiley, New York.

    MATH  Google Scholar 

  4. Balakrishnan, N and Stepanov, A (2010). Generalization of Borel-Cantelli lemma. Math. Sci. 35, 61–62.

    MathSciNet  MATH  Google Scholar 

  5. Barndorff-Nielsen, O (1961). On the rate of growth of the partial maxima of a sequence of independent identically distributed random variables. Math. Scand. 9, 383–394.

    MathSciNet  Article  Google Scholar 

  6. Beirlant, J and Zuijlen, M (1985). The empirical distribution function and strong laws for functions of order statistics of uniform spacings. J. Multivar. Anal.16, 300–317.

    MathSciNet  Article  Google Scholar 

  7. Berred, A and Stepanov, A (2020). Asymptotic results for lower exponential spacings. Commun. Stat. – Theory Methods 49, 1730–1741.

    MathSciNet  Article  Google Scholar 

  8. Cressie, N (1979). An optimal statistic based on higher order gaps. Biometrika 66, 619–627.

    MathSciNet  Article  Google Scholar 

  9. David, HA and Nagaraja, HN (2003). Order statistics, 3rd edn. Wiley, Hoboken.

    Book  Google Scholar 

  10. Del Pino, GE (1979). On the asymptotic distribution of k-spacings with applications to goodness-of-fit tests. Ann. Statist. 7, 1058–1065.

    MathSciNet  Article  Google Scholar 

  11. Devroye, L (1984). The largest exponential spacing. Utilitas Mathematica25, 303–313.

    MathSciNet  MATH  Google Scholar 

  12. Ederer, F, Meyers, MH and Mantel, N (1964). A statistical problem in space and time: Do leukemia cases come in clusters? Biometrics 20, 626–636.

    Article  Google Scholar 

  13. Glaz, J and Balakrishnan, N (eds.) (1999). JScan statistics and applications, Birkhauser, Boston.

  14. Hall, PG (1984). Limit theorems for sums of general functions of m-spacings. Math. Proc. Cambridge Philos. Soc. 96, 517–532.

    MathSciNet  Article  Google Scholar 

  15. Naus, JI (1966). Some probabilities, expectations and variances for the side of largest clusters and smallest intervals. J. Amer. Statist. Ass. 61, 1191–1199.

    MathSciNet  Article  Google Scholar 

  16. Nevzorov, V (2001). Records: Mathematical theory american mathematical society. Providence, Rhode Island.

    Google Scholar 

  17. Pyke, R (1965). Spacings (with discussions). J. R. Stat. Soc. Series B27, 395–449.

    MathSciNet  MATH  Google Scholar 

  18. Riffi, MI (2017). Distributions of gamma m-spacings. IUG J. Nat. Stud. 4, 01–06.

    Google Scholar 

  19. Riffi, MI (2018). Characterizing the exponential distribution by m-spacings. J. Scientif. Eng. Res. 5, 211–214.

    Google Scholar 

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Acknowledgments

The last author’s work was partially supported by RFBR grant N 18-01-00393.

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Correspondence to Narayanaswamy Balakrishnan.

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Balakrishnan, N., Stepanov, A. & Nevzorov, V.B. Asymptotic Results for m-th Exponential Spacings. Sankhya A (2021). https://doi.org/10.1007/s13171-021-00259-y

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Keywords and phrases.

  • Order statistics
  • spacings
  • exponential distribution
  • limit laws.

AMS (2000) subject classification.

  • 60G70
  • 62G30