By Haijun Li, Xiaohu Li

Stochastic Orders in Reliability and danger Management consists of nineteen contributions at the thought of stochastic orders, stochastic comparability of order information, stochastic orders in reliability and possibility research, and purposes. those review/exploratory chapters current contemporary and present learn on stochastic orders mentioned on the overseas Workshop on Stochastic Orders in Reliability and danger administration, or SORR2011, which happened within the urban resort, Xiamen, China, from June 27 to June 29, 2011. The conference’s talks and invited contributions additionally characterize the get together of Professor Moshe Shaked, who has made accomplished, primary contributions to the idea of stochastic orders and its purposes in reliability, queueing modeling, operations learn, economics and chance research. This quantity is in honor of Professor Moshe Shaked. The paintings awarded during this quantity represents lively study on stochastic orders and multivariate dependence, and exemplifies shut collaborations among students operating in several fields. The Xiamen Workshop and this quantity search to restore the group workshop culture on stochastic orders and dependence and enhance study collaboration, whereas honoring the paintings of a exceptional student.

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The order ≤GDO3 -cx satisfies property (O1). We do not know whether ≤GDO3 -cx satisfies property (O2). 1 we conjecture that it does not. However, we have the following result. 2. The order ≤GDO3 -cx satisfies property (O3). Proof : Let (X, Y ) be a random vector. Furthermore, let φ be a oneto-one measurable function, and consider (φ(X), Y ). Let G(x, u) be defined as in Eq. 2), and define Gφ by Gφ (x, u) = G(φ−1 (x), u). Let U be a uniform random variable on [0, 1], which is independent of st X. Since, by Eq.

4. M. SHAKED, M. A. SORDO, AND A. SUAREZ-LLORENS 19 and hence, E[Y ⊥ U ⊥ ] = h(U ⊥ ). 4) Thus E[E[Y ⊥ U ⊥ ]] = E[h(U ⊥ )] = E[Y ⊥ ] = E[Y ] = E[E[Y U ]], where the first equality follows from Eq. 4) and the second equality follows from Eq. 3). Thus, the two variables in Eq. 2) have the same expected value. Now, let φ be a convex function. Then, E[φ(E[Y U ])] ≤ E[E[φ(Y ) U ]] = E[φ(Y )] = E[φ(Y ⊥ )] = E[φ(E[Y ⊥ U ⊥ ])], where the inequality follows from Jensen’s Inequality. In order to prove the second inequality in Eq.

SORDO, AND A. SUAREZ-LLORENS 9 (O1) The order ≤GDO is reflexive and transitive. st ˜ Y˜ ), and if (V, W ) and (V˜ , W ˜ ) satisfy V = (O2) If (X, Y ) ≤GDO (X, st st ˜ st st = Y , as well as E[W V ] = E[Y X] and V˜ = X and W = W st ˜ , then (V, W ) ≤GDO (V˜ , W ˜ ). ˜ V˜ = E Y˜ X E W ˜ Y˜ ) then (φ(X), l(Y )) ≤GDO (φ(X), ˜ l(Y˜ )), (O3) If (X, Y ) ≤GDO (X, where φ is a one-to-one measurable function, and l is a linear function. (O4) Let (X, Y ) be any random vector, then (X, Y ) ≥GDO (X ⊥ , Y ⊥ ).

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