Let X1,X2,Θ and Θ′ be independent non-negative random variables. The residual life of Xi at random time Θ, that is, XiΘ=Xi−Θ∣XiΘ is considered. Some sufficient conditions which lead to the likelihood ratio ordering, the failure rate ordering, the reverse failure rate ordering and the...

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>

Let X1,X2, and X3 be independent random variables with absolutely continuous distributions having the common support [0,∞). We show that if X1≤hr[mrl,lr]X3 and X2≤hr[mrl,lr]X3, then max{X1,X2}≤hr[mrl,lr]max{X1,X3}. We also show that if X2≤rh[lr]X1 and X2≤rh[lr]X3, then...

The purpose of this paper is to study new notions of stochastic comparisons and aging classes based on the Laplace transform order of residual lives. We give relationships to other stochastic orders and aging classes given previously. Finally we study some applications to shock models.