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We study the random variables of radial asymmetry based on copulas. We research on the structure of random variables which radial asymmetry degree isand get the exact best-possible bounds of random variables which radial asymmetry degree is equal to. Then we expand to general case. We propose an essential condition of radial asymmetry degree is and study the structure of copula. We provide a broad bounds of copula that the radial asymmetry degree is .     

Intervals of 1-Lipschitz aggregation operators, quasi-copulas, and copulas with given affine section

Best lower and upper bounds for 1-Lipschitz aggregation operators with a given affine section are given. These are used to determine best bounds for quasi-copulas and copulas with a given affine section. However, in general there is no greatest copula with a given non-decreasing affine section. These results are used to study (quasi-)copulas with arbitrary affine sections.     

On the construction of copulas and quasi-copulas with given diagonal sections

We study a method, which we call a copula (or quasi-copula) diagonal splice, for creating new functions by joining portions of two copulas (or quasi-copulas) with a common diagonal section. The diagonal splice of two quasi-copulas is always a quasi-copula, and we find a necessary and sufficient condition for the diagonal splice of two copulas to be a copula. Applications of this method include the construction of absolutely continuous asymmetric copulas with a prescribed diagonal section, and determining the best-possible upper bound on the set of copulas with a particular type of diagonal section. Several examples illustrate our results.     

Intervals of 1-Lipschitz aggregation operators,quasi-copulas,and copulas with given affine section

Best lower and upper bounds for 1-Lipschitz aggregation operators with a given affine section are given. These are used to determine best bounds for quasi-copulas and copulas with a given affine section. However, in general there is no greatest copula with a given non-decreasing affine section. These results are used to study (quasi-)copulas with arbitrary affine sections. A significant part of this work was done during a visit of the second author at the Johannes Kepler University, Linz (Austria). The second author was supported by the grant VEGA 1/3012/06 and by the Science and Technology Assistance Agency (Contract No. APVT-20-003204). Both authors would like to thank the anonymous referee whose comments (including the two copulas C ₁ and C ₂ given in the Conclusion) not only solved the originally stated open problem in the negative, but also allowed them to formulate two more interesting open problems.     

A Comparison of Bounds on Three Sets of Copulas with Given Degree of Non-exchangeability

We establish best-possible supremum bounds of copulas with the degree of non-exchangeabilityt=3/4, t=3/5 and t=3/6=1/2, and study the structures of these sets of copulas. The volumes between the upper and lower bounds are calculated to illustrate that the supremum bounds are specific practical and effective in narrowing the Fr\'{e}chet-Hoeffding bounds.     

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??We establish best-possible supremum bounds of copulas with the degree of non-exchangeabilityt=3/4, t=3/5 and t=3/6=1/2, and study the structures of these sets of copulas. The volumes between the upper and lower bounds are calculated to illustrate that the supremum bounds are specific practical and effective in narrowing the Fr\'{e}chet-Hoeffding bounds.     

Dominating and large induced trees in regular graphs

Recently Chen et al. [Tree domination in graphs, Ars Combin. 73 (2004) 193-203] asked for characterizations of the class of graphs and the class of regular graphs that have an induced dominating tree, i.e. for which there exists a dominating set that induces a tree.We give a somewhat negative answer to their question by proving that the corresponding decision problems are NP-complete. Furthermore, we prove essentially best-possible lower bounds on the maximum order of induced trees in connected cacti of maximum degree 3 and connected cubic graphs.Finally, we give a forbidden induced subgraph condition for the existence of induced dominating trees.     

Best-possible bounds on sets of bivariate distribution functions

The fundamental best-possible bounds inequality for bivariate distribution functions with given margins is the Fréchet–Hoeffding inequality: If H denotes the joint distribution function of random variables X and Y whose margins are F and G, respectively, then max(0,F(x)+G(y)−1)H(x,y)min(F(x),G(y)) for all x,y in [−∞,∞]. In this paper we employ copulas and quasi-copulas to find similar best-possible bounds on arbitrary sets of bivariate distribution functions with given margins. As an application, we discuss bounds for a bivariate distribution function H with given margins F and G when the values of H are known at quartiles of X and Y.     

The lattice-theoretic structure of sets of bivariate copulas and quasi-copulas

In this Note we show that the set of quasi-copulas is a complete lattice, which is order-isomorphic to the Dedekind–MacNeille completion of the set of copulas. Consequently, any set of copulas sharing a particular statistical property is guaranteed to have pointwise best-possible bounds within the set of quasi-copulas. To cite this article: R.B. Nelsen, M. Úbeda Flores, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Best-possible bounds for the distribution of a sum — a problem of Kolmogorov

Summary Recently, in answer to a question of Kolmogorov, G.D. Makarov obtained best-possible bounds for the distribution function of the sumX+Y of two random variables,X andY, whose individual distribution functions,F _X andF _Y, are fixed. We show that these bounds follow directly from an inequality which has been known for some time. The techniques we employ, which are based on copulas and their properties, yield an insightful proof of the fact that these bounds are best-possible, settle the question of equality, and are computationally manageable. Furthermore, they extend to binary operations other than addition and to higher dimensions.     

Upper comonotonicity and convex upper bounds for sums of random variables

It is well-known that if a random vector with given marginal distributions is comonotonic, it has the largest sum with respect to convex order. However, replacing the (unknown) copula by the comonotonic copula will in most cases not reflect reality well. For instance, in an insurance context we may have partial information about the dependence structure of different risks in the lower tail. In this paper, we extend the aforementioned result, using the concept of upper comonotonicity, to the case where the dependence structure of a random vector in the lower tail is already known. Since upper comonotonic random vectors have comonotonic behavior in the upper tail, we are able to extend several well-known results of comonotonicity to upper comonotonicity. As an application, we construct different increasing convex upper bounds for sums of random variables and compare these bounds in terms of increasing convex order.     

基于灵敏性分析的Q-MSGSID的优化及应用

本文针对绝对关联度、综合关联度以及相对关联度的取值范围存在的不足,首先,设置了控制因子λ以及空间中的距离d,以此来调节关联度值的范围,建立了新模型。其次,研究了它的一些性质,并在理论上证明了新模型满足灰色关联公理。另外,提出了新模型的准优值所满足的几个原则,并结合灵敏性分析原理给出了准优值的算法步骤。最后,通过实例研究,验证了新模型所得结果不但能够使关联度的值扩充到(0,1]这一更大的范围,而且提高了区分度和分辨效果。     

Bivariate option pricing with copulas

The adoption of copula functions is suggested in order to price bivariate contingent claims. Copulas enable the marginal distributions extracted from vertical spreads in the options markets to be imbedded in a multivariate pricing kernel. It is proved that such a kernel is a copula function, and that its super-replication strategy is represented by the Fréchet bounds. Applications provided include prices for binary digital options, options on the minimum and options to exchange one asset for another. For each of these products, no-arbitrage pricing bounds, as well as values consistent with the independence of the underlying assets are provided. As a final reference value, a copula function calibrated on historical data is used.     

Worst VaR scenarios with given marginals and measures of association

This paper studies the problem of finding best-possible upper bounds on the Value-at-Risk for a function of two random variables when the marginal distributions are known and additional nonparametric information on the dependence structure, such as the value of a measure of association, is available. The same problem for the Tail-Value-at-Risk is also briefly discussed.     

Reducing model risk via positive and negative dependence assumptions

We give analytical bounds on the Value-at-Risk and on convex risk measures for a portfolio of random variables with fixed marginal distributions under an additional positive dependence structure. We show that assuming positive dependence information in our model leads to reduced dependence uncertainty spreads compared to the case where only marginals information is known. In more detail, we show that in our model the assumption of a positive dependence structure improves the best-possible lower estimate of a risk measure, while leaving unchanged its worst-possible upper risk bounds. In a similar way, we derive for convex risk measures that the assumption of a negative dependence structure leads to improved upper bounds for the risk while it does not help to increase the lower risk bounds in an essential way. As a result we find that additional assumptions on the dependence structure may result in essentially improved risk bounds.     

Reality-Based Algebras,Generalized Camina-Frobenius Pairs,and the Nonexistence of Degree Maps

The notion of a generalized Camina-Frobenius pair is extended to reality-based algebras, and a construction that characterizes such pairs is given. Zero-product sets are defined, and a best-possible upper bound on their size is proved and related to Camina-Frobenius pairs. It is shown that there exist commutative reality-based algebras with zero-product sets and, hence, no degree map, of every dimension at least 4. All such 4-dimensional algebras are constructed explicitly.     

Branch and bound algorithm with applications to robust stability

We discuss a branch and bound algorithm for global optimization of NP-hard problems related to robust stability. This includes computing the distance to instability of a system with uncertain parameters, computing the minimum stability degree of a system over a given set of uncertain parameters, and computing the worst case \(H_\infty \) norm over a given parameter range. The success of our method hinges (1) on the use of an efficient local optimization technique to compute lower bounds fast and reliably, (2) a method with reduced conservatism to compute upper bounds, and (3) the way these elements are favorably combined in the algorithm.     

Worst case risk measurement: Back to the future?

This paper studies the problem of finding best-possible upper bounds on a rich class of risk measures, expressible as integrals with respect to measures, under incomplete probabilistic information. Both univariate and multivariate risk measurement problems are considered. The extremal probability distributions, generating the worst case scenarios, are also identified.The problem of worst case risk measurement has been studied extensively by Etienne De Vijlder and his co-authors, within the framework of finite-dimensional convex analysis. This paper revisits and extends some of their results.     

A refinement of multivariate value set bounds

Over finite fields, if the image of a polynomial map is not the entire field, then its cardinality can be bounded above by a significantly smaller value. Earlier results bound the cardinality of the value set using the degree of the polynomial, but more recent results make use of the powers of all monomials.In this paper, we explore the geometric properties of the Newton polytope and show how they allow for tighter upper bounds on the cardinality of the multivariate value set. We then explore a method which allows for even stronger upper bounds, regardless of whether one uses the multivariate degree or the Newton polytope to bound the value set. Effectively, this provides improvement of a degree matrix-based result given by Zan and Cao, making our new bound the strongest upper bound thus far.     

Two algorithms for the Student-Project Allocation problem

We study the Student-Project Allocation problem (SPA), a generalisation of the classical Hospitals/Residents problem (HR). An instance of SPA involves a set of students, projects and lecturers. Each project is offered by a unique lecturer, and both projects and lecturers have capacity constraints. Students have preferences over projects, whilst lecturers have preferences over students. We present two optimal linear-time algorithms for allocating students to projects, subject to the preference and capacity constraints. In particular, each algorithm finds a stable matching of students to projects. Here, the concept of stability generalises the stability definition in the HR context. The stable matching produced by the first algorithm is simultaneously best-possible for all students, whilst the one produced by the second algorithm is simultaneously best-possible for all lecturers. We also prove some structural results concerning the set of stable matchings in a given instance of SPA. The SPA problem model that we consider is very general and has applications to a range of different contexts besides student-project allocation.