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} = ≤ft]\frac{i-1}{m},\frac{i}{m}\right]

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

\forall d \in N, \forall x,y \in R^p, λ(≤ft(x,y\right)) = ∏\limits_{p=1}^{d} (y_i - x_i)

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

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) = ∑\limits_{i\in I} {m^d μ(B_{i}) λ((0,x)\cap B_{i})}

Where m^d = λ(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¡lehová JG, Rémillard B (2017-jul). “Asymptotic Behavior of the Empirical Multilinear Copula Process under Broad Conditions.” Journal of Multivariate Analysis, 159, 82–110.

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

Durante F, Fernández-Sánchez J, Quesada-Molina JJ, Úbeda-Flores M (2015-dec). “Convergence Results for Patchwork Copulas.” European Journal of Operational Research, 247, 525–531.


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