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Large Deviations for Proportions of Observations Which Fall in Random Sets Determined by Order Statistics

Author

Listed:
  • Enkelejd Hashorva

    (Université de Lausanne)

  • Claudio Macci

    (Università di Roma Tor Vergata)

  • Barbara Pacchiarotti

    (Università di Roma Tor Vergata)

Abstract

Let {X n :n ≥ 1} be independent random variables with common distribution function F and consider $K_{h:n}(D)=\sum_{j=1}^n1_{\{X_j-X_{h:n}\in D\}}$ , where h ∈ {1,...,n}, X 1:k ≤ ⋯ ≤ X k:k are the order statistics of the sample X 1,...,X k and D is some suitable Borel set of the real line. In this paper we prove that, if F is continuous and strictly increasing in the essential support of the distribution and if $\lim_{n\to\infty}\frac{h_n}{n}=\lambda$ for some λ ∈ [0,1], then $\{K_{h_n:n}(D)/n:n\geq 1\}$ satisfies the large deviation principle. As a by product we derive the large deviation principle for order statistics $\{X_{h_n:n}:n\geq 1\}$ . We also present results for the special case of Bernoulli distributed random variables with mean p ∈ (0,1), and we see that the large deviation principle holds only for p ≥ 1/2. We discuss further almost sure convergence of $\{K_{h_n:n}(D)/n:n\geq 1\}$ and some related quantities.

Suggested Citation

  • Enkelejd Hashorva & Claudio Macci & Barbara Pacchiarotti, 2013. "Large Deviations for Proportions of Observations Which Fall in Random Sets Determined by Order Statistics," Methodology and Computing in Applied Probability, Springer, vol. 15(4), pages 875-896, December.
    • Handle: RePEc:spr:metcap:v:15:y:2013:i:4:d:10.1007_s11009-012-9290-y
    DOI: 10.1007/s11009-012-9290-y
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    References listed on IDEAS

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    1. Claudio Macci, 2010. "Large deviations for estimators of some threshold parameters," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 19(1), pages 63-77, March.
    2. Hashorva, Enkelejd, 2003. "On the number of near-maximum insurance claim under dependence," Insurance: Mathematics and Economics, Elsevier, vol. 32(1), pages 37-49, February.
    3. Smid, B. & Stam, A. J., 1975. "Convergence in distribution of quotients of order statistics," Stochastic Processes and their Applications, Elsevier, vol. 3(3), pages 287-292, July.
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    Cited by:

    1. Augustynowicz, Aneta, 2020. "Asymptotic behavior of proportions of observations falling to random regions determined by central order statistics," Statistics & Probability Letters, Elsevier, vol. 162(C).
    2. 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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