An Introduction to Stochastic Orders - download pdf or read online
By Felix Belzunce, Carolina Martinez Riquelme, Julio Mulero
An creation to Stochastic Orders discusses this strong software that may be utilized in evaluating probabilistic types in numerous components resembling reliability, survival research, hazards, finance, and economics. The e-book 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 learn.
In addition, clients will locate particular proofs of the most effects and functions to numerous probabilistic versions of curiosity in numerous fields, and discussions of primary homes of a number of stochastic orders, within the univariate and multivariate circumstances, besides purposes to probabilistic models.
- Introduces stochastic orders and its notation
- Discusses assorted orders of univariate stochastic orders
- Explains multivariate stochastic orders and their convex, chance ratio, and dispersive orders
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An creation to Stochastic Orders discusses this robust software that may be utilized in evaluating probabilistic types in several parts corresponding to reliability, survival research, hazards, finance, and economics. The booklet offers a basic historical past in this subject for college students and researchers who are looking to use it as a device for his or her examine.
Additional resources for An Introduction to Stochastic Orders
Next, we introduce some sufficient conditions for the mean residual life order that allow us to overcome this difficulty (see Ref.  for a discussion on the topic). 11. Let X and Y be two continuous random variables with survival functions F and G, respectively, and finite means such that E[X] ≤ E[Y]. If a value x0 ∈ R exists such that F(x)G(y) ≤ F(y)G(x), for all x < y ≤ x0 and F(x)G(y) ≥ F(y)G(x), for all x0 ≤ x < y, then X ≤mrl Y. Proof. From the assumption, we see that F(x)G(y) ≥ F(y)G(x), for all x0 ≤ x < y, and integrating with respect to y we get +∞ F(x) +∞ G(y) dy ≥ G(x) x F(y) dy, for all x > x0 .
There is a great variety of parametric families of copulas, and we refer the reader to Ref.  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].
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. Unless stated otherwise, the main references for the results provided in this chapter are Refs.
An Introduction to Stochastic Orders by Felix Belzunce, Carolina Martinez Riquelme, Julio Mulero