Stochastic Modeling

Download An Introduction to Stochastic Orders by Felix Belzunce, Carolina Martinez Riquelme, Julio Mulero PDF

By Felix Belzunce, Carolina Martinez Riquelme, Julio Mulero

An advent to Stochastic Orders discusses this strong device that may be utilized in evaluating probabilistic versions in numerous components akin to reliability, survival research, hazards, finance, and economics. The publication offers a common historical past in this subject for college kids and researchers who are looking to use it as a device for his or her examine.

In addition, clients will locate specific proofs of the most effects and purposes to a number of probabilistic types of curiosity in different fields, and discussions of primary houses of a number of stochastic orders, within the univariate and multivariate circumstances, in addition to functions to probabilistic models.

  • Introduces stochastic orders and its notation
  • Discusses diverse orders of univariate stochastic orders
  • Explains multivariate stochastic orders and their convex, probability ratio, and dispersive orders

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Extra info for An Introduction to Stochastic Orders

Example text

To sum up, if α2 < α1 and α1 β1 ≤ α2 β2 , then X ≤icx Y, but X st Y or X st Y, by the previous comment. 6 shows a particular example of this situation. 5) (dashed line). 42 An Introduction to Stochastic Orders Next, some preservation results for the increasing convex order are provided. First, the result on the preservation under increasing convex transformations is stated. 10. Let X and Y be two random variables. Then, X ≤icx Y if, and only if, φ(X) ≤icx φ(Y), for all real valued increasing convex function φ.

Proof. Let us consider the random variables [X + Z| X = x] and [Y + Z|Y = x], for all x. From the independence hypothesis, they are equally distributed and it is not difficult to see that [X + Z|X = x] ≤hr [Y + Z|Y = x ], for all x ≤ x , since Z is IFR. 6. 1) leads to the following general result. 8. Let {(Xi , Yi )}ni=1 , be independent pairs of random variables. If Xi and Yi are IFR and Xi ≤hr Yi , for all i = 1, . . , n, then n n Xi ≤hr i=1 Yi . i=1 Proof. From the previous theorem, we see that X1 + X2 ≤hr X1 + Y2 ≤hr Y1 + Y2 .

Finally, some graphical procedures are given to compare two data sets. 9. An Introduction to Stochastic Orders. 00002-8 © 2016 Elsevier Ltd. All rights reserved. 1 Survival functions of X ∼ W(1, 2) (continuous line) and Y ∼ W( π /2, 1) (dashed line). In particular, several tables are included, which collect the sufficient conditions on the parameters of some family distributions so that the stochastic orders hold among two distributions belonging to the same model. Besides, some applications in reliability and risk theory are given.

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