Stochastic Orders in Reliability and Risk: In Honor of by Haijun Li, Xiaohu Li

By Haijun Li, Xiaohu Li

Stochastic Orders in Reliability and chance Management consists of nineteen contributions at the thought of stochastic orders, stochastic comparability of order facts, stochastic orders in reliability and possibility research, and functions. those review/exploratory chapters current contemporary and present study on stochastic orders pronounced on the foreign Workshop on Stochastic Orders in Reliability and chance administration, or SORR2011, which happened within the urban inn, Xiamen, China, from June 27 to June 29, 2011. The conference’s talks and invited contributions additionally signify the get together of Professor Moshe Shaked, who has made entire, primary contributions to the idea of stochastic orders and its functions 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 energetic learn on stochastic orders and multivariate dependence, and exemplifies shut collaborations among students operating in numerous fields. The Xiamen Workshop and this quantity search to restore the neighborhood workshop culture on stochastic orders and dependence and develop examine collaboration, whereas honoring the paintings of a uncommon pupil.

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Additional info for Stochastic Orders in Reliability and Risk: In Honor of Professor Moshe Shaked (Lecture Notes in Statistics)

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171 172 195 344 346 347 348 xxxi Part I Theory of Stochastic Orders Chapter 1 A Global Dependence Stochastic Order Based on the Presence of Noise Moshe Shaked, Miguel A. Sordo, and Alfonso Su´arez-Llorens Abstract: Two basic ideas that give rise to global dependence stochastic orders were introduced and studied in Shaked et al. (Methodology and Computing in Applied Probability 14:617–648, 2012). Here these are reviewed, and two new ideas that give rise to new global dependence orders are then brought out and discussed.

First let us intuitively Eq. 8), but now we consider d(U ) and d( ˜ ˜ examine the case when U and Y are “close to independence” (as before, ˜ and Y˜ are “close to total dependence”). Then, this means that X ˜ = u tells us “almost nothing” about Y˜ ; that is, again, the event U ˜U ˜ ˜ ) is large. given U , the uncertainty about Y˜ is large. As a result, d( ˜ and Y˜ are independent we have that In the extreme case when U ˜ ˜U ˜ ˜ ) = Var[Y˜ ]; we have noticed d(U ) is as large as possible, that is, d( ˜ ˜ (although earlier that in this case Y then is totally dependent on X the dependence need not be one-to-one).

8) we have G(X, U ) = X · FZ−1 (U ). Denote the marginal distributions of X and Y by FX and FY , respectively. A straightforward computation shows that Var(Y ) = E(X 2 ). Now, recall the notation X and Y from Eq. 11). Explicitly, st let X be a random variable such that X = X, and define Y by Y = FY−1 (FX (X )). Formally, as in Eq. 4), let U be a uniform (0,1) random variable that is independent of X , and let G be a function such that Y = G (X , U ). s. FY−1 (FX (x)) is actually independent of u.

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