The dilation order, the dispersion order, and orderings of residual lives

One purpose of this paper is to study the relationship of the dilation order ([less-than-or-equals, slant]dil) to two other stochastic orders: the mean residual life order ([less-than-or-equals, slant]mrl) and the increasing convex order ([less-than-or-equals, slant]icx). Regarding these orders, it is already known that X [less-than-or-equals, slant]mrlY => X [less-than-or-equals, slant]icxY. In this paper we show that for non-negative random variables we actually have X [less-than-or-equals, slant]mrlY => X [less-than-or-equals, slant]dilY => X [less-than-or-equals, slant]icxY (the first implication holds under the assumption that at least one of the two underlying random variables satisfies some aging property). Thus, we refine the result of Theorem 3.A.13 in Shaked and Shanthikumar (1994). Another purpose of this paper is to identify conditions under which all the residual lives, that are associated with two random variables X and Y, are ordered according to the dilation or the dispersion orders. Some of these results extend parts (a) and (b) of Theorem 2.B.13 in Shaked and Shanthikumar (1994).