cbCopula-Class {cort} R Documentation

## Checkerboard copulas

### Description

cbCopula contructor

### Usage

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


### 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.

### 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).

### 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.

### References

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]