Stochastic Orders and Applications

Abstract

X ::;maj y. 11. X = IIy, for some doubly stochastic matrix II.iii. I:i=l g(Xi) ::; I:i=l g(y;) for all 9 convex.Another important paper in the history of stochastic orderings is due to Karamata (1932), who provided an analysis of concepts that will then be known as dilation and second degree stochastic dominance, both of which can be considered as a generalization of majorization.After World War II more results have been discovered and new areas of research have been developed. Many results now originate from applications: for instance Blackwell's 1. J <p dPI ::; J <p dP2 y<p convex.2. There exists a Markov kernel M(·,·) such that J x M(y,dx) = y and \fA E A.3. There exist two random variables Xl, X 2 on a probability space (0" F, P), with values in (X, A) such that £(XI) = PI, £(X2) = P2 and E(X2IXd = Xl P-a.s ..It is interesting to notice that the properly said stochastic order (namely the simplest and most intuitive form of stochastic ordering) was studied only later. introduced the order for distributions on lR n in connection with some inferential problems. The idea has been subsequently generalized and several characterizations have been proved. The most general and thorough analysis of the stochastic order can be found in Kamae, Krengel and O'Brien (1977) whose scheme we follow here. Given two probability measures PI ,P2 on a partially ordered Polish space (X,A), PI ::;st P2 iff any of the following equivalent conditions holds. \f <p increasing. 11. PI(A) ::; P2 (A) for all upper sets A.Ill. There exists a Markov kernel M(·,·) such that J M(y, dx) ~ y and \fA EA.IV. There exist two random variables Xl, X 2 on a probability space (0" F, P) with values in (X,A) such that £(XI) = PI, £(X2) = P2 and Xl::; X 2 P-a.s ..The generality of the setting allows the above definitions to be used for comparing (the law of) real valued random variables, random vectors, stochastic process, etc ..The stochastic order, the dilation order, and their variations are the orderings that have generated the most work both in theoretical and applied sense. Several other orderings have been defined and studied, motivated by different areas of applications.Queueing theory has provided many applications for the comparison of stochastic processes. See Stoyan (1977Stoyan ( , 1983. In reliability theory, the need often arises to compare the lifelengths of some complex systems (or of some components of the systems). The original studies (e.g., were rarely related to stochastic IntroductionSome of the most interesting articles concern the optimal composition of a portfolio under different assumptions on the distribution of the assets and the attitude towards risk of the investors. Another interesting area of applications is the measurement of inequality. Kolm (1969) and Atkinson (1970) gave a coherent mathematical form to some basic principles for inequality measurement and its characterizations. It is worth remembering that the first contributions concerning the comparison of income distributions date back to 'Lorenz (1905), whose co...

Nonparametric Tests for Positive Quadrant Dependence

“…the review papers by Kroll and Levy (1980) and Levy (1992), the classified bibliography by Mosler and Scarsini (1993) and the book by Shaked and Shanthikumar (1994). For a rigorous treatment of k−icv , see Rolski (1976) and Fishburn (1976 …”

Multidimensional First-Order Dominance Comparisons of Population Wellbeing

“…For real world applications, computationally efficient algorithms for checking dominance are required (Range and Østerdal 2013). Mosler and Scarsini (1991) and Dyckerhoff and Mosler (1997) show that, appealing to definition (A), checking FOD corresponds to determining if a certain linear programme has a feasible solution. The first empirical implementation of this approach was provided by Arndt et al (2012) in a study of child poverty in Mozambique and Vietnam.…”

Ordinal Welfare Comparisons with Multiple Discrete Indicators: A First Order Dominance Approach and Application to Child Poverty

“…Preston (1974) and Hansel and Troallic (1978) assert that an algorithm for finding the maximum flow in a properly defined network can be used to determine dominance. More usefully, for the multivariate discrete case, Mosler and Scarsini (1991) and Dyckerhoff and Mosler (1997) show from definition (A) that FOD corresponds to a linear program that has a feasible solution. Hence, FOD can be verified using a linear programming package.…”