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

By Felix Belzunce, Carolina Martinez Riquelme, Julio Mulero

An advent 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 ebook offers a basic heritage 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 certain proofs of the most effects and purposes to numerous probabilistic types of curiosity in different fields, and discussions of basic houses of numerous stochastic orders, within the univariate and multivariate situations, besides functions to probabilistic models.

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

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Taking into account this comment, most of the results for the hazard rate order can be translated to the reversed hazard rate order. 3 of the hazard rate order, it is natural to consider the following conditions as possible candidates to keep comparing random quantities in terms of their residual lives +∞ +∞ F(u) du x y +∞ G(u) du ≥ +∞ F(u) du y G(u) du, x for all x ≤ y, and [X − x|X > x] ≤icx [Y − x|Y > x], for all x such that F(x), G(x) > 0. 14) where m and l denote the mean residual life functions of X and Y, respectively.

Hence, S− (g−f ) ≤ 2 with the sign sequence +, −, + when the equality holds. 7, if E[X] ≤ E[Y] and g(x) g(x) lim , lim > 1, x→l f(x) x→u f(x) where l and u denote the left and right extremes of the common supports, respectively, then X ≤icx Y, but X st Y or X st Y. Otherwise, the previous theorem implies X ≤st Y, since S− (g − f ) = 1 with the sign sequence −, +. The following example shows the simplicity and usefulness of this result to compare, for instance, two gamma distributions. 9. Let X ∼ G(α1 , β1 ) and Y ∼ G(α2 , β2 ).

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. [2, 3]. 2 THE USUAL STOCHASTIC ORDER As was mentioned in the introduction, the theory of stochastic orders arises because of the fact that comparisons based only on single measures are not very informative. 2, if the random variable X represents the random lifetime of a device, or an organism, the survival function F(x) is a function of interest in this context.

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