On stochastic orders for sums of independent random variables
Journal of Multivariate Analysis
Stochastic comparisons of order statistics from gamma distributions
Journal of Multivariate Analysis
Journal of Multivariate Analysis
Mean residual life order of convolutions of heterogeneous exponential random variables
Journal of Multivariate Analysis
On the right spread order of convolutions of heterogeneous exponential random variables
Journal of Multivariate Analysis
On hazard rate ordering of the sums of heterogeneous geometric random variables
Journal of Multivariate Analysis
Ordering convolutions of heterogeneous exponential and geometric distributions revisited
Probability in the Engineering and Informational Sciences
Some new results on convolutions of heterogeneous gamma random variables
Journal of Multivariate Analysis
Hazard rate comparison of parallel systems with heterogeneous gamma components
Journal of Multivariate Analysis
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Convolutions of random variables which are either exponential or geometric are studied with respect to majorization of parameter vectors and the likelihood ratio ordering (=^l^r) of random variables. Let X"@l, ..., X"@l"""n be independent exponential random variables with respective hazards @l"i (means 1/@l"i), i = 1 ..., n. Then if @l = (@l"1, ..., @l"n) =^m (@l"1"', ..., @l"n"') = @l', it follows that @S"i" "=" "1^nX"@l =^l^r @S"i" "=" "1^nX"@l"'"1. Similarly if X"p"1, ..., X"p"n are independent geometric random variables with respective parameters p"1, ..., p"n, then p = (p"1, ..., p"n) =^m(p'"1, ..., p'"n) = p' or log p = (log p"1, ..., log p"n) = ^m (log p"1, ..., log p"n) = log p' implies @S"i" "=" "1^nX"p"l = ^l^r @S"i" "=" "1^nX"P"'"""1. Applications of these results are given yielding convenient upper bounds for the hazard rate function of convolutions of exponential (geometric) random variables in terms of those of gamma (negative binomial) distributions. Other applications are also given for a server model, the range of a sample of i.i.d. exponential random variables, and the duration of a multistate component performing in excess of a given level.