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Symmetry of a distribution via symmetry of order statistics

Author

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  • Balakrishnan, Narayanaswamy
  • Selvitella, Alessandro

Abstract

In this paper, we establish the following characterization of symmetric absolutely continuous distributions and symmetric discrete distributions. Suppose X1,…,Xn is a random sample from a distribution with pdf/pmf fX(x), and X1:n,…,Xn:n are the corresponding order statistics. Then, Xr:n=d−Xn−r+1:n for some r=1,…,n if and only if fX(x)=fX(−x). Here, =d means that the two random variables have the same distribution. In the discrete case, we assume the support to be finite.

Suggested Citation

  • Balakrishnan, Narayanaswamy & Selvitella, Alessandro, 2017. "Symmetry of a distribution via symmetry of order statistics," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 367-372.
    • Handle: RePEc:eee:stapro:v:129:y:2017:i:c:p:367-372
    DOI: 10.1016/j.spl.2017.06.023
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    Cited by:

    1. Jafar Ahmadi, 2021. "Characterization of continuous symmetric distributions using information measures of records," Statistical Papers, Springer, vol. 62(6), pages 2603-2626, December.
    2. Ahmadi, Jafar, 2020. "Characterization results for symmetric continuous distributions based on the properties of k-records and spacings," Statistics & Probability Letters, Elsevier, vol. 162(C).
    3. Withers, Christopher S. & Nadarajah, Saralees, 2019. "The distribution of the minimum of a positive sample," Statistics & Probability Letters, Elsevier, vol. 151(C), pages 89-96.

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