Abstract
This chapter is devoted to a short review of the basic concepts of the theory of dependence functions or copula functions, as they are more usually called. In particular our objective is to define the volume of a copula function: this is one of the basic ingredients of the aggregation algorithm that will be discussed in Part II.
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Notes
- 1.
We define the generalized inverse\(F^{[-1]}:(0,1)\rightarrow \mathbb {R}\) as \( F^{[-1]}(w)=\inf \left \{l\in \mathbb {R} :F(l)\geq w\right \}, \) that coincides with the standard inverse function when this exists.
- 2.
As for example in the Archimedean family, the assumption to work with strict generators is sufficient to assure the existence of the tail dependence coefficients (see Nelsen 25).
- 3.
These features define a famous class of functionals, i.e., the family of Laplace transforms.
- 4.
We talk about copula’s parameter because in this procedure we work with hierarchical copulas and so with Archimedean ones that are characterized by only one parameter.
- 5.
This threshold is fixed coherently with the problem at hand.
- 6.
The coordinates of the box S are computed as \(\mathbf {u}=col(\tilde {\mathbf {u}}'),\mathbf {v}=col(\tilde {\mathbf {v}}')\), where the function \(col:[0,1]^{\max \{m_s, s=1,\ldots ,k\}}\times [0,1]^k\rightarrow [0,1]^n\) sets orderly all the columns of a matrix into the same column.
- 7.
We refer to D(i, k) as the number of the ways in which one can distribute the integer i into k groups taking into account the order into the groups (o.c.ds for short). If we also take into account the cardinalities of the groups, we refer to D c(i, k) as the number of the ordered compatible (for groups’ cardinalities) combinatorial distributions.
- 8.
We assume to know the estimations of the real parameters within and between the groups which may be estimated by an econometric methodology, such as the maximum likelihood method, possibly simplifying the problem by considering the limiting distributions proposed for Archimedean copulas with granularity adjustment, and then compensated to account for the homogeneous approximation as suggested in Definition 2.4.2.
- 9.
We must recall the representation of the survival copula function proposed in Corollary 4.3.2.
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Bernardi, E., Romagnoli, S. (2021). Copula Function and C-Volume. In: Counting Statistics for Dependent Random Events. Springer, Cham. https://doi.org/10.1007/978-3-030-64250-1_2
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