Abstract
We consider a problem of characterization of continuous distributions for which linearity of regression of overlapping order statistics, \(\mathbb {E}(X_{i:m}|X_{j:n})=aX_{j:n}+b\), \(m\le n\), holds. Due to a new representation of conditional expectation \(\mathbb {E}(X_{i:m}|X_{j:n})\) in terms of conditional expectations \(\mathbb {E}(X_{l:n}|X_{j:n})\), \(l=i,\ldots ,n-m+i\), 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 \(j\le i\) or \(j\ge n-m+i\). In the remaining cases the problem essentially is still open.
Highlights
Consider a sequence (Xk)k≥1 of independent identically distributed continuous random variables
The case of adjacent order statistics was completed in Ferguson (1967) who considered the case m = n, i = j + 1 with no restriction on a and characterized three families of distributions: exponential for a = 1, Pareto for a > 1 and power for 0 < a < 1
We are ready to state and to prove our main result which is a characterization of both the triplets of distributions described in Sect. 3 by linearity of regression of order statistics from overlapping samples
Summary
Consider a sequence (Xk)k≥1 of independent identically distributed continuous random variables. Wesołowski where a, b are some real constants, and we want to describe the family of parent distribution for which (1) holds. It goes back to Fisz (1958) who considered the case m = n = i = 2, j = 1, a = 1 and characterized the exponential distribution.
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