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Linearity of regression for overlapping order statistics

Author

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  • Adam Dołęgowski
  • Jacek Wesołowski

Abstract

We consider a problem of characterization of continuous distributions for which linearity of regression of overlapping order statistics, $$\mathbb {E}(X_{i:m}|X_{j:n})=aX_{j:n}+b$$ E ( X i : m | X j : n ) = a X j : n + b , $$m\le n$$ m ≤ n , holds. Due to a new representation of conditional expectation $$\mathbb {E}(X_{i:m}|X_{j:n})$$ E ( X i : m | X j : n ) in terms of conditional expectations $$\mathbb {E}(X_{l:n}|X_{j:n})$$ E ( X l : n | X j : n ) , $$l=i,\ldots ,n-m+i$$ l = i , … , n - m + i , we are able to use the already known approach based on the Rao-Shanbhag version of the Cauchy integrated functional equation. However this is possible only if $$j\le i$$ j ≤ i or $$j\ge n-m+i$$ j ≥ n - m + i . In the remaining cases the problem essentially is still open. Copyright The Author(s) 2015

Suggested Citation

  • Adam Dołęgowski & Jacek Wesołowski, 2015. "Linearity of regression for overlapping order statistics," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 78(2), pages 205-218, February.
    • Handle: RePEc:spr:metrik:v:78:y:2015:i:2:p:205-218
    DOI: 10.1007/s00184-014-0496-6
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    References listed on IDEAS

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    1. Ramesh Gupta & Mohammad Ahsanullah, 2004. "Some characterization results based on the conditional expectation of a function of non-adjacent order statistic (record value)," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 56(4), pages 721-732, December.
    2. Mariusz Bieniek & Dominik Szynal, 2003. "Characterizations of distributions via linearity of regression of generalized order statistics," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 58(3), pages 259-271, December.
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