By Felix Belzunce, Carolina Martinez Riquelme, Julio Mulero
An creation to Stochastic Orders discusses this robust instrument that may be utilized in evaluating probabilistic versions in several parts resembling reliability, survival research, dangers, finance, and economics. The ebook presents a common history 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 particular proofs of the most effects and purposes to numerous probabilistic versions of curiosity in different fields, and discussions of primary houses of a number of stochastic orders, within the univariate and multivariate circumstances, besides functions to probabilistic models.
- Introduces stochastic orders and its notation
- Discusses varied orders of univariate stochastic orders
- Explains multivariate stochastic orders and their convex, probability ratio, and dispersive orders
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Additional info for An Introduction to Stochastic Orders
We assume a2 , a1 > 1 in order to have finite means. 2, if k2 ≤ k1 and a2 > a1 , we see that S− (G(x) − F(x)) = 1, for all x ≥ k1 , with the sign sequence −, +. In addition, if E[X] = a1 k1 a2 k2 ≤ = E[Y], a1 − 1 a2 − 1 we get X ≤icx Y from the previous theorem. We observe that the conditions k2 ≤ k1 and a1 k1 /(a1 − 1) ≤ a2 k2 /(a2 − 1), when the equalities do not hold at the same time, imply a2 < a1 . Therefore, to sum up, if k2 ≤ k1 and a1 k1 /(a1 − 1) ≤ a2 k2 /(a2 − 1), when the equalities do not hold at the same time, then X ≤icx Y, but X ≤st Y or X ≥st Y.
In actuarial science, copulas are used to model dependent mortality and losses [36–38]. In finance, copulas are used in asset allocation, credit scoring, derivative pricing, and risk management [39–41]. In biomedical studies, copulas are used to model correlated event times and competing risks [42, 43]. In engineering, copulas are used in multivariate process control and hydrological modeling [44, 45]. Let us give the formal definition. A copula C : [0, 1]n → [0, 1] is a cumulative distribution function with uniformly distributed marginal functions on [0, 1].
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]. For a recent review, the reader can look at Shaked and Shanthikumar . These functions require the “history” notion, which will be recalled next. Let us consider a random vector X = (X1 , . . , Xn ) where Xi represents the lifetimes of n units, therefore the components are assumed to be nonnegative.