Linearity of regression for overlapping order statistics

We consider a problem of characterization of continuous distributions for which linearity of regression of overlapping order statistics, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\mathbb {E}(X_{i:m}|X_{j:n})=aX_{j:n}+b$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi mathvariant="double-struck">E</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow> <mi>i</mi> <mo>:</mo> <mi>m</mi> </mrow> </msub> <mo stretchy="false">|</mo> <msub> <mi>X</mi> <mrow> <mi>j</mi> <mo>:</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>a</mi> <msub> <mi>X</mi> <mrow> <mi>j</mi> <mo>:</mo> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mi>b</mi> </mrow> </math> </EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$m\le n$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>m</mi> <mo>≤</mo> <mi>n</mi> </mrow> </math> </EquationSource> </InlineEquation>, holds. Due to a new representation of conditional expectation <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$\mathbb {E}(X_{i:m}|X_{j:n})$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi mathvariant="double-struck">E</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow> <mi>i</mi> <mo>:</mo> <mi>m</mi> </mrow> </msub> <mo stretchy="false">|</mo> <msub> <mi>X</mi> <mrow> <mi>j</mi> <mo>:</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation> in terms of conditional expectations <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$\mathbb {E}(X_{l:n}|X_{j:n})$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi mathvariant="double-struck">E</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow> <mi>l</mi> <mo>:</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">|</mo> <msub> <mi>X</mi> <mrow> <mi>j</mi> <mo>:</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$l=i,\ldots ,n-m+i$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>l</mi> <mo>=</mo> <mi>i</mi> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>-</mo> <mi>m</mi> <mo>+</mo> <mi>i</mi> </mrow> </math> </EquationSource> </InlineEquation>, 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 <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$j\le i$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>j</mi> <mo>≤</mo> <mi>i</mi> </mrow> </math> </EquationSource> </InlineEquation> or <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$j\ge n-m+i$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>j</mi> <mo>≥</mo> <mi>n</mi> <mo>-</mo> <mi>m</mi> <mo>+</mo> <mi>i</mi> </mrow> </math> </EquationSource> </InlineEquation>. In the remaining cases the problem essentially is still open. Copyright The Author(s) 2015