激波模型中Copula的非交换性
Non-exchangeability of copulas arising from shock models
摘要：
当为我们的数据选择正确的copula时，关键是要区分最好地描述它的族。在这方面，一个更好的选择copula可以通过信息的（非）对称性的数据。自20世纪30年代以来，作为概率概念的可交换性（仅次于独立性）已被研究，自20世纪50年代以来，Copula已被研究，甚至从应用的观点来看，最重要的一类Copula，即由Shock模型引起的Copula。Marshall的Copulas，从20世纪60年代开始研究。然而，Copula的不可交换性直到2006年才被提出，并从那时起得到了深入的研究。本文的主要贡献之一是给出了一类Copula函数族的极大不对称函数。我们计算这个函数的主要家庭的冲击为基础的copula，即马歇尔，最大最小和反射最大最小（简称RMM）copula和其他一些重要的家庭。我们计算了最重要的非对称性测度$μ_\infty$对于Marshall copula族和maxmin copula族的精确界，它们都等于$\frac{4}{27}\ (\approx 0.148)$ .我们应该将这个界与它们所属的PQD Copula类的界（$3-2\sqrt{2}\ \approx 0.172)$）和所有Copula的一般界（$\frac13$）进行比较。此外，我们给出了RMM copula的相同的不对称性测度$3-2\sqrt{2}$的精确界，与NQD copula的相同界相比，它们属于$\sqrt{5}-2\ (\approx 0.236)$。我们的主要结果之一也是在一个给定的模型，在最大的不对称性措施界达到冲击的统计解释。这些解释的三个家庭研究的例子，应该是有帮助的从业者选择模型时，他们的数据说明。
英文摘要：
When choosing the right copula for our data a key point is to distinguish the family that describes it at the best. In this respect, a better choice of the copulas could be obtained through the information about the (non)symmetry of the data. Exchangeability as a probability concept (first next to independence) has been studied since 1930's, copulas have been studied since 1950's, and even the most important class of copulas from the point of view of applications, i.e. the ones arising from shock models s.a. Marshall's copulas, have been studied since 1960's. However, the point of non-exchangeability of copulas was brought up only in 2006 and has been intensively studied ever since. One of the main contributions of this paper is the maximal asymmetry function for a family of copulas. We compute this function for the major families of shock-based copulas, i.e. Marshall, maxmin and reflected maxmin (RMM for short) copulas and also for some other important families. We compute the sharp bound of asymmetry measure $μ_\infty$, the most important of the asymmetry measures, for the family of Marshall copulas and the family of maxmin copulas, which both equal to $\frac{4}{27}\ (\approx 0.148)$. One should compare this bound to the one for the class of PQD copulas to which they belong, which is $3-2\sqrt{2}\ \approx 0.172)$, and to the general bound for all copulas that is $\frac13$. Furthermore, we give the sharp bound of the same asymmetry measure for RMM copulas which is $3-2\sqrt{2}$, compared to the same bound for NQD copulas, where they belong, which is $\sqrt{5}-2\ (\approx 0.236)$. One of our main results is also the statistical interpretation of shocks in a given model at which the maximal asymmetry measure bound is attained. These interpretations for the three families studied are illustrated by examples that should be helpful to practitioners when choosing the model for their data.
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