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The empirical distribution function and strong laws for functions of order statistics of uniform spacings

Author

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  • Beirlant, Jan
  • van Zuijlen, M. C. A.

Abstract

Let N points be arbitrarily chosen on the circle with unit circumference, and order them clockwise. The uniform mth order spacings are then defined as the clockwise distances between any pair of points having m - 1 other points in between. A Glivenko-Cantelli theorem and nonlinear almost sure bounds for the empirical distribution function based on these uniform spacings are derived. The parameter m is allowed to increase with N to infinity. Applications to linear combinations of functions of mth order spacings are given.

Suggested Citation

  • Beirlant, Jan & van Zuijlen, M. C. A., 1985. "The empirical distribution function and strong laws for functions of order statistics of uniform spacings," Journal of Multivariate Analysis, Elsevier, vol. 16(3), pages 300-317, June.
    • Handle: RePEc:eee:jmvana:v:16:y:1985:i:3:p:300-317
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    Cited by:

    1. Balakrishnan, N. & Nevzorov, V.B. & Stepanov, A., 2023. "On normal spacings," Statistics & Probability Letters, Elsevier, vol. 193(C).
    2. Narayanaswamy Balakrishnan & Alexei Stepanov & Valery Borisovich Nevzorov, 2023. "Asymptotic Results for m-th Exponential Spacings," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(1), pages 468-477, February.
    3. Nelly Litvak & Maria Vlasiou, 2010. "A survey on performance analysis of warehouse carousel systems," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 64(4), pages 401-447, November.
    4. Nevzorov, Valery B. & Stepanov, Alexei, 2024. "On uppermost discrete spacing," Statistics & Probability Letters, Elsevier, vol. 208(C).

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