Stochastic orderings with respect to a capacity and an application to a financial optimization problem

CHAPTER I - VECTOR MEASURES

Order continuous of monotone set function and convergence of measurable functions sequence

Multivariate Concave and Convex Stochastic Dominance

Stochastic dominance permits a partial ordering of alternatives (probability distributions on consequences) based only on partial information about a decision maker’s utility function. Univariate stochastic dominance has been widely studied and applied, with general agreement on classes of utility functions for dominance of different degrees. Extensions to the multivariate case have received less attention and have used different classes of utility functions, some of which require strong assumptions about utility. The multivariate concave stochastic dominance we investigate is a natural extension of the stochastic order typically used in the univariate case and is consistent with a basic preference assumption. The corresponding utility functions are multivariate risk averse, and reversing the preference assumption allows us to investigate stochastic dominance for utility functions that are multivariate risk seeking. We provide insight into these two contrasting forms of stochastic dominance, develop some criteria to compare probability distributions (hence alternatives) via multivariate stochastic dominance, and illustrate how this dominance could be used in practice to identify inferior alternatives. Connections between our approach and dominance using different stochastic orders are discussed.

The Lebesgue-Nikodym Theorem states that for a Lebesgue measure an additive set function which is -absolutely continuous is the integral of a Lebegsue integrable a measurable function ; that is, for all measurable sets. Such a property is not shared by vector valued set functions. We introduce a suitable definition of the integral that will extend the above property to the vector valued case in its full generality. We also discuss a further extension of the Fundamental Theorem of Calculus for additive set functions with values in an infinite dimensional normed space.

Submodular set functions, matroids and the greedy algorithm: Tight worst-case bounds and some generalizations of the Rado-Edmonds theorem

In this paper, we present order invariance theoretical results for weighted quasi-arithmetic means of a monotonic series of numbers. The quasi-arithmetic mean, or Kolmogorov–Nagumo mean, generalizes the classical mean and appears in many disciplines, from information theory to physics, from economics to traffic flow. Stochastic orders are defined on weights (or equivalently, discrete probability distributions). They were introduced to study risk in economics and decision theory, and recently have found utility in Monte Carlo techniques and in image processing. We show in this paper that, if two distributions of weights are ordered under first stochastic order, then for any monotonic series of numbers their weighted quasi-arithmetic means share the same order. This means for instance that arithmetic and harmonic mean for two different distributions of weights always have to be aligned if the weights are stochastically ordered, this is, either both means increase or both decrease. We explore the invariance properties when convex (concave) functions define both the quasi-arithmetic mean and the series of numbers, we show its relationship with increasing concave order and increasing convex order, and we observe the important role played by a new defined mirror property of stochastic orders. We also give some applications to entropy and cross-entropy and present an example of multiple importance sampling Monte Carlo technique that illustrates the usefulness and transversality of our approach. Invariance theorems are useful when a system is represented by a set of quasi-arithmetic means and we want to change the distribution of weights so that all means evolve in the same direction.

In this paper, a convenient and parsimonious family of the quantile functions is proposed for modeling time-to-event data. Detailed mathematical arguments are presented for deriving the explicit expressions of the quantile density, quantile hazard and quantile mean remaining life functions. Their functional properties are also explored. Some important members of the family are provided. A special case, the so-called additive power logarithmic quantile family, is discussed with detailed properties, parameter estimation and application to time-to-event data. Measures of skewness and kurtosis for the proposed quantile function are derived. Important statistical measures useful for reliability analysis are also provided with their explicit expressions. Stochastic ordering, tail weight and order statistics are also discussed. L-moments are explicitly derived for the proposed quantile function. Estimation is approached using the likelihood function and percentiles. A real data set consists of times between occurrence of the coal-mining accidents is analyzed for illustration purposes.

Stochastic orders using quantile-based reliability functions

The Application of Conflict Measure to Estimating Incoherence of Analyst's Forecasts about the Cost of Shares of Russian Companies

Extended Abstract. The simplest and the most common way of comparing two random variables is through their means and variances. It may happen that in some cases the median of X is larger than that of Y, while the mean of X is smaller than the mean of Y. However, this confusion will not arise if the random variables are stochastically ordered. Similarly, the same may happen if one would like to compare the variability of X with that of Y based only on numerical measures like standard deviation etc. Besides, these characteristics of distributions might not exist in some cases. In most cases one can express various forms of knowledge about the underlying distributions in terms of their survival functions, hazard rate functions, mean residual functions, quantile functions and other suitable functions of probability distributions. These methods are much more informative than those based only on few numerical characteristics of distributions. Comparisons of random variables based on such functions usually establish partial orders among them. We call them as stochastic orders.

We introduce a new hierarchy over monotone set functions, that we refer to as MPH (Maximum over Positive Hypergraphs). Levels of the hierarchy correspond to the degree of complementarity in a given function. The highest level of the hierarchy, MPH-m (where m is the total number of items) captures all monotone functions. The lowest level, MPH-1, captures all monotone submodular functions, and more generally, the class of functions known as XOS. Every monotone function that has a positive hypergraph representation of rank k (in the sense defined by Abraham, Babaioff, Dughmi and Roughgarden [EC 2012]) is in MPH-k. Every monotone function that has supermodular degree k (in the sense defined by Feige and Izsak [ITCS 2013]) is in MPH-(k+1). In both cases, the converse direction does not hold, even in an approximate sense. We present additional results that demonstrate the expressiveness power of MPH-k.One can obtain good approximation ratios for some natural optimization problems, provided that functions are required to lie in low levels of the MPH hierarchy. We present two such applications. One shows that the maximum welfare problem can be approximated within a ratio of k+1 if all players hold valuation functions in MPH-k. The other is an upper bound of 2k on the price of anarchy of simultaneous first price auctions.

Misleading signals (MS) are likely to happen while using a simultaneous scheme to control the mean vector (μ) and the covariance matrix (Σ) of a bivariate process. They correspond to valid signals that lead to a misinterpretation of a shift in μ (resp. Σ) as a shift in Σ (resp. μ). Following previous work, focused on the quantitative assessment of the probabilities of misleading signals (PMS) in simultaneous schemes for bivariate processes, we now make use of stochastic ordering to qualitatively assess the impact of changes in μ and Σ in those probabilities.

Our purpose is to answer the question of Leader [1, p. 2341, by producing an example of a sigma algebra a of subsets of a set X and a universally a-integrable function f (i.e., ffdu exists for every countably additive set function u on a) which is not measurable with respect to a. We shall show that the sigma algebra a of Borel subsets of the interval [0, 1] admits nonmeasurable universally a-integrable functions (R. E. Zink suggested that this Borel algebra might yield an example). Lusin has shown [2] that, subject to the continuum hypothesis, there exists an uncountable subset V of X = [0, 1] such that each perfect nowhere dense subset G of X contains at most an enumerable set of points of V. Let H be a subset of V that is not a Borel set. We shall now show that the characteristic function x(H) of H is universally sigma integrable. To this end suppose u is a countably additive set function on a-. Then [5] u = c +j where c is continuous and j= Ijk where jk is a two valued jump function, the range of k is at most enumerable, and the variation of j is the sum of the variations of the jk'S; moreover, since the variation of u is the variation of j plus the variation c of c, each of j, c, and c is countably additive on a. Corresponding to each jk there is [4, Theorem 27.1] a point Xk such that jk(E) =jk(En [Xk ]) for every Borel subset E of X. Hence fX(H)dj= Ejk(HCn[Xk]) and it suffices to show that there exists a Borel subset B of X such that HCB and c(B) = 0 (then fX(H)dc= 0). There exists [3 ] a set K = U i Fi, F1 closed for i > 1, of the first category in X and a set L such that c(L) = 0 and KUL = X. For each positive integer i there exists a perfect set Gi and an at most enumerable set Hi such that Fi = Gi U Hi. Thus H n K = HC(Ui Fi) = Hf (Ui (GiUkHi)) = Ui (Hrl (GiU-Hi)) CUi((HnGi) UHi) is at most enumerable since every HrlGi (Fi is a closed subset of X and K is of the first category in X; hence Fi is nowhere dense in X which implies that G, is a perfect nowhere dense subset of X) and every H, is at most enumerable. Since e is countably additive on o and c vanishes on one point sets, c vanishes on countable sets. Hence H=Hr)(KUL) C(HnK)UL and c(HnK)UL) <?(HnK) +Z(L) =0.

It is a well known fact that non-additive measures, i.e., monotone set functions, and corresponding integrals have been successfully applied in many different areas, both theoretical and practical. Due to the course of research, this problem has diversified into two directions. One direction covers real-valued set functions, while the second one leads towards the set-valued case. The first option can be considered as the more classical one and it covers some well known notions, such as the Choquet integral, the Sugeno integral, etc. On the other hand, the set-valued form has imposed itself as being worth studying since, while working with uncertainty, instead of the actual values, sets (intervals) are quite often being used. Lately, decision making theory has emerged as the area of special interest in terms of research of possible applications of non-additive measures. The benefits of the use of non-additive measures and corresponding integrals lie in the flexibility that is provided precisely by the monotonicity of non-additive measures and which is essential for modelling the Decision Maker's behavior. Therefore, the focus of this presentation is on some interesting aspects of non-additive measures and decision making theory.