By Forman Sinnickson Acton
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Unless stated otherwise, the main references for the results provided in this chapter are Refs. [2, 3]. 2 THE USUAL STOCHASTIC ORDER As was mentioned in the introduction, the theory of stochastic orders arises because of the fact that comparisons based only on single measures are not very informative. 2, if the random variable X represents the random lifetime of a device, or an organism, the survival function F(x) is a function of interest in this context. If we have another random lifetime Y with survival function G(x), then it is of interest to study if one of the two survival functions lies above the other one.
Fn (pn )), for all (p1 , . . , pn ) ∈ [0, 1]n . There is a great variety of parametric families of copulas, and we refer the reader to Ref. [46] for examples and more details about copulas. 3 The multivariate dynamic hazard rate and mean residual life functions Next, multivariate extensions of the hazard rate and mean residual life functions are introduced. In the literature, several definitions can be found for these functions. The ones that we are going to consider are the multivariate dynamic versions introduced by Shaked and Shanthikumar [47, 48].
8) and the fact that E[φ(X2 (θ))] is increasing convex in θ. The case where E[φ(X1 (θ))] is increasing convex is similar. Additional results in this direction can be found in Ref. [59]. To finish, a result on the preservation under convolution of the increasing convex order is provided. As in the case of the stochastic order, the result follows by induction. 6). 13. Let X1 , . . , Xn and Y1 , . . , Yn be two sets of independent random variables. If Xi ≤icx Yi , for all i = 1, . . , n, then n n Xi ≤icx i=1 Yi .