Likelihood ratio order of m-spacings for two samples

For two non-negative random variables X and Y, it is shown that if X is smaller than Y in the sense of the likelihood ratio order and either X or Y is of decreasing likelihood ratio, then the m-spacing V k : n ( m ) of an X-sample is smaller than W k : n ( m ) , the m-spacing of a Y-sample, in terms of the likelihood ratio order. It is also proved that a similar result for the upshifted likelihood ratio order is valid. Finally, it is proved that the decreasing failure rate in average property of a population X is preserved by sample spacing V k : n ( 1 ) .