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On the number of maxima in a discrete sample

Author

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  • Brands, J. J. A. M.
  • Steutel, F. W.
  • Wilms, R. J. G.

Abstract

Let Mn = max(N1,..., Nn), where N1, N2, ... are i.i.d., positive, integer-valued r.v.'s. We are interested in Kn, the number of values of j [epsilon] {1, 2, ..., n} for which Nj = Mn, especially for large values of n. There is strong evidence that Kn either tends to one or to infinity, or diverges in distribution as n tends to infinity. An interesting example of the latter type occurs when N1 has a geometric distribution. There is an application of results on Kn to the behaviour of the fractional parts of sample maxima from non-integer populations.

Suggested Citation

  • Brands, J. J. A. M. & Steutel, F. W. & Wilms, R. J. G., 1994. "On the number of maxima in a discrete sample," Statistics & Probability Letters, Elsevier, vol. 20(3), pages 209-217, June.
    • Handle: RePEc:eee:stapro:v:20:y:1994:i:3:p:209-217
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    Citations

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    Cited by:

    1. Li, Yun, 1999. "A note on the number of records near the maximum," Statistics & Probability Letters, Elsevier, vol. 43(2), pages 153-158, June.
    2. Baryshnikov, Yuliy & Eisenberg, Bennett & Stengle, Gilbert, 1995. "A necessary and sufficient condition for the existence of the limiting probability of a tie for first place," Statistics & Probability Letters, Elsevier, vol. 23(3), pages 203-209, May.
    3. Archibald, Margaret & Knopfmacher, Arnold, 2009. "The average position of the dth maximum in a sample of geometric random variables," Statistics & Probability Letters, Elsevier, vol. 79(7), pages 864-872, April.
    4. Jasiński, Krzysztof, 2016. "Asymptotic normality of numbers of observations near order statistics from stationary processes," Statistics & Probability Letters, Elsevier, vol. 119(C), pages 259-263.
    5. Qi, Yongcheng & Wilms, R. J. G., 1997. "The limit behavior of maxima modulo one and the number of maxima," Statistics & Probability Letters, Elsevier, vol. 32(4), pages 357-366, April.
    6. Qi, Yongcheng, 1997. "A note on the number of maxima in a discrete sample," Statistics & Probability Letters, Elsevier, vol. 33(4), pages 373-377, May.
    7. Eisenberg, Bennett, 2008. "On the expectation of the maximum of IID geometric random variables," Statistics & Probability Letters, Elsevier, vol. 78(2), pages 135-143, February.
    8. Eisenberg, Bennett, 2009. "The number of players tied for the record," Statistics & Probability Letters, Elsevier, vol. 79(3), pages 283-288, February.
    9. Qi, Y. & Wilms, R. J. G., 1997. "The limit behavior of maxima modulo one and the number of maxima," Statistics & Probability Letters, Elsevier, vol. 34(1), pages 75-84, May.
    10. Khmaladze, E. & Nadareishvili, M. & Nikabadze, A., 1997. "Asymptotic behaviour of a number of repeated records," Statistics & Probability Letters, Elsevier, vol. 35(1), pages 49-58, August.
    11. 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.
    12. Lin, Jin-Guan & Huang, Chao & Zhuang, Qing-Yun & Zhu, Li-Ping, 2010. "Estimating generalized state density of near-extreme events and its applications in analyzing stock data," Insurance: Mathematics and Economics, Elsevier, vol. 47(1), pages 13-20, August.

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