<Para ID="Par1">In order to take into account any possible dependence between alternatives in optimization problems, bivariate characterizations of some well-know univariate stochastic orders have been defined and studied by Shanthikumar and Yao (Adv Appl Probab 23:642–659, <CitationRef CitationID="CR15">1991</CitationRef>). These characterizations gave...</citationref></para>

In this paper we study convolution residuals, that is, if <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$X_1,X_2,\ldots ,X_n$$</EquationSource> </InlineEquation> are independent random variables, we study the distributions, and the properties, of the sums <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$\sum _{i=1}^lX_i-t$$</EquationSource> </InlineEquation> given that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$\sum _{i=1}^kX_it$$</EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$t\in \mathbb R $$</EquationSource> </InlineEquation>, and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$1\le k\le l\le n$$</EquationSource> </InlineEquation>....</equationsource></inlineequation></equationsource></inlineequation></equationsource></inlineequation></equationsource></inlineequation></equationsource></inlineequation>

In this paper we study a family of stochastic orders of random variables defined via the comparison of their percentile residual life functions. Some interpretations of these stochastic orders are given, and various properties of them are derived. The relationships to other stochastic orders are...