Linearity of regression for overlapping order statistics

Adam Dołęgowski () and Jacek Wesołowski ()

Metrika: International Journal for Theoretical and Applied Statistics, 2015, vol. 78, issue 2, 205-218

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

Keywords: Order statistics; Overlapping samples; Linearity of regression; Characterization of probability distributions; Gamma distribution; Power distribution; Pareto distribution (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/s00184-014-0496-6

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