Compare Mean Residual Life Order With Different Scale Parameters

Convolutions of independent random variables occur quite frequently in statistics, applied probability, operations research and other fields. Their distribution theory is quite complicated when the random variables are not identically distributed.In such situation, it is interesting if'we can compare two convolutions just compare their parameters.First, we give the distribution of a convolution of two Gamma distribution with the same shape parameter no less than 1 but different scale parameters. Then we show that two group of random variables with the same shape parameter no less than 1 but different scale parameters satisfy likelihood ratio order when their scale parameters satisfy majorizaton order. So in this condition, the convolutions satisfy mean residual life order. At last, we let the condition of the scale parameters be weaker that when they satisfy reciprocal majorization order, the convolutions still satisfy mean residual life order. At last, the result was extended to the n random variables'convolution.