Checkerboard copulas

Arguments

x

the data to be used

m

checkerboard parameters

pseudo

Boolean, defaults to FALSE. Set to TRUE if you are already providing pseudo data into the x argument.

Value

An instance of the cbCopula S4 class. The object represent the fitted copula and can be used through several methods to query classical (r/d/p/v)Copula methods, etc.

Details

The cbCopula class computes a checkerboard copula with a given checkerboard parameter $m$, as described by A. Cuberos, E. Masiello and V. Maume-Deschamps (2019). Assymptotics for this model are given by C. Genest, J. Neslehova and R. bruno (2017). The construction of this copula model is as follows :

Start from a dataset with $n$ i.i.d observation of a $d$-dimensional copula (or pseudo-observations), and a checkerboard parameter $m$,dividing $n$.

Consider the ensemble of multi-indexes $I = \{i = (i_1,..,i_d) \subset \{1,...,m \}^d\}$ which indexes the boxes :

$$B_{i} = \left]\frac{i-1}{m},\frac{i}{m}\right]$$

Let now $\lambda$ be the dimension-unspecific lebesgue measure on any power of $R$, that is :

$$\forall d \in N, \forall x,y \in R^p, \lambda(\left(x,y\right)) = \prod\limits_{p=1}^{d} (y_i - x_i)$$

Let furthermore $\mu$ and $\hat{\mu}$ be respectively the true copula measure of the sample at hand and the classical Deheuvels empirical copula, that is :

• For $n$ i.i.d observation of the copula of dimension $d$, let $\forall i \in \{1,...,d\}, \, R_i^1,...,R_i^d$ be the marginal ranks for the variable $i$.

• $\forall x \in I^d$ let $\hat{\mu}((0,x)) = \frac{1}{n} \sum\limits_{k=1}^n I_{R_1^k\le x_1,...,R_d^k\le x_d}$

The checkerboard copula, $C$, and the empirical checkerboard copula, $\hat{C}$, are then defined by the following :

$$\forall x \in (0,1)^d, C(x) = \sum\limits_{i\in I} {m^d \mu(B_{i}) \lambda((0,x)\cap B_{i})}$$

Where $m^d = \lambda(B_{i})$.

This copula is a special form of patchwork copulas, see F. Durante, J. Fernández Sánchez and C. Sempi (2013) and F. Durante, J. Fernández Sánchez, J. Quesada-Molina and M. Ubeda-Flores (2015). The estimator has the good property of always being a copula.

The checkerboard copula is a kind of patchwork copula that only uses independent copula as fill-in, only where there are values on the empirical data provided. To create such a copula, you should provide data and checkerboard parameters (depending on the dimension of the data).

References

cuberos2019cort

genest2017cort

durante2013cort

durante2015cort