cbCopula-Class {cort} | R Documentation |

cbCopula contructor

```
cbCopula(x, m = rep(nrow(x), ncol(x)), pseudo = FALSE)
```

`x` |
the data to be used |

`m` |
checkerboard parameters |

`pseudo` |
Boolean, defaults to |

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).

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.

Cuberos A, Masiello E, Maume-Deschamps V (2019-mar).
“Copulas Checker-Type Approximations: Application to Quantiles Estimation of Sums of Dependent Random Variables.”
*Communications in Statistics - Theory and Methods*, 1–19.

Genest C, NeÅ¡lehovÃ¡ JG, RÃ©millard B (2017-jul).
“Asymptotic Behavior of the Empirical Multilinear Copula Process under Broad Conditions.”
*Journal of Multivariate Analysis*, **159**, 82–110.

Durante F, FernÃ¡ndez SÃ¡nchez J, Sempi C (2013-nov).
“Multivariate Patchwork Copulas: A Unified Approach with Applications to Partial Comonotonicity.”
*InsuranceMathematics and Economics*, **53**, 897–905.

Durante F, FernÃ¡ndez-SÃ¡nchez J, Quesada-Molina JJ, Ãšbeda-Flores M (2015-dec).
“Convergence Results for Patchwork Copulas.”
*European Journal of Operational Research*, **247**, 525–531.

[Package *cort* version 0.3.2 Index]