Estimation and Inference for Conditional Copula Models.

How to install

The release version on CRAN:

install.packages("CondCopulas")

The development version from GitHub, using the devtools package:

# install.packages("devtools")
devtools::install_github("AlexisDerumigny/CondCopulas")

If you have any questions or suggestions, feel free to open an issue.

Conditional copulas with pointwise conditioning

In this first part, we are interesting in the inference of the conditional copula of a random vector $X$ given the pointwise conditioning $Z = z$, where $Z$ is another random vector and $z$ is a fixed value.

Tests of the simplifying assumption

These functions perform a test of the "simplifying assumption" that the conditional copula $C_{X | Z = z}$ does not depend on the value of $z$.

• simpA.NP: in a purely nonparametric framework

• simpA.param: assuming that the conditional copula belongs to a parametric family of copulas for all values of the conditioning variable

• simpA.kendallReg: test of the simplifying assumption based on the constancy of the conditional Kendall's tau assuming that it satisfies a regression-like equation

Estimation of conditional copulas (using kernel smoothing)

These functions estimate the conditional copula $C_{X | Z = z}$ in different frameworks.

• estimateNPCondCopula: nonparametric estimation of conditional copulas.

• estimateParCondCopula: parametric estimation of conditional copulas.

• estimateParCondCopula_ZIJ: parametric estimation of conditional copulas using (already computed) conditional pseudo-observations.

Estimation of conditional Kendall's tau (CKT)

In this part, we assume that the dimension of $X$ is $2$, i.e. $X = (X_1, X_2)$. Instead of estimating the conditional copula $C_{X | Z = z}$ which is an infinite-dimensional object for every value of $z$, it is possible to estimate the conditional Kendall's tau (CKT) $\tau_{1,2|Z=z}$ which is a real number in $[-1, 1]$ for every value of $z$.

To estimate the conditional Kendall's tau, the package provides a general wrapper function:

• CKT.estimate: that can be used for any method of estimating conditional Kendall's tau. Each of these methods is detailed below and has its own function.

Kernel-based estimation of conditional Kendall's tau

• CKT.kernel: use kernel smoothing to estimate the conditional Kendall's tau. The bandwidth can be given by the user or determined by cross-validation.

Kendall's regression

• CKT.kendallReg.fit: fit Kendall's regression, a regression-like method for the estimation of conditional Kendall's tau.

• CKT.kendallReg.predict: predict the conditional Kendall's tau given new values $z$ of the covariates.

Classification-based estimation of conditional Kendall's tau

• using tree:

  • CKT.fit.tree: for fitting a tree-based model for the conditional Kendall's tau
  • CKT.predict.tree: for prediction of new conditional Kendall's taus

• using random forests:

  • CKT.fit.randomForest: for fitting a random forest-based model for the conditional Kendall's tau
  • CKT.predict.randomForest: for prediction of new conditional Kendall's taus

• using nearest neighbors:

  • CKT.predict.kNN: for several numbers of nearest neighbors

• using neural networks:

  • CKT.fit.nNets: for fitting a neural networks-based model for the conditional Kendall's tau
  • CKT.predict.nNets: for prediction of new conditional Kendall's taus

• using GLM:

  • CKT.fit.GLM: for fitting a GLM-like model for the conditional Kendall's tau
  • CKT.predict.GLM: for prediction of new conditional Kendall's taus

Advanced functions for manual hyperparameter choices

• CKT.hCV.Kfolds: for K-fold cross-validation choice of the bandwidth for kernel smoothing

• CKT.hCV.l1out: for leave-one-out cross-validation choice of the bandwidth for kernel smoothing

• CKT.KendallReg.LambdaCV : cross-validated choice of the penalization parameter lambda

• CKT.adaptkNN: for a (local) aggregation of the number of nearest neighbors based on Lepski's method

Conditional copulas with discrete conditioning by Borel sets

In this second part, we are interesting in the inference of the conditional copula of a random vector $X$ given the discrete conditioning $Z \in A$, where $Z$ is another random vector and $A$ is a Borel subset of possible values of $Z$.

Test of the hypothesis that the conditioning Borel subset has no influence on the conditional copula

These functions perform a test of the hypothesis that the conditional copula $C_{X | Z \in A}$ does not depend on the value of $A$ for different choices of the conditioning set $A$.

• bCond.simpA.param : test of this hypothesis, assuming that the copula belongs to a parametric family

• bCond.simpA.CKT: test of the hypothesis that conditional Kendall's tau are equal over all the different conditioning subsets.

Estimation

• bCond.pobs : computation of the conditional pseudo-observations $F_{1|A(i)}(X_{i,1} | A(i))$ and $F_{2|A(i)}(X_{i,2} | A(i))$ for every $i=1, \dots, n$.

• bCond.estParamCopula : estimation of a conditional parametric copula, i.e. for every set $A$, a conditional parameter $\theta(A)$ is estimated.

Data-driven choice of conditioning subsets

• bCond.treeCKT: construction of binary tree whose leaves corresponds to the most relevant conditioning subsets (in the sense of maximizing the difference between estimated conditional Kendall's taus).

References

Derumigny, A., & Fermanian, J. D. (2017). About tests of the “simplifying” assumption for conditional copulas. Dependence Modeling, 5(1), 154-197. pdf

Derumigny, A., & Fermanian, J. D. (2019). A classification point-of-view about conditional Kendall’s tau. Computational Statistics & Data Analysis, 135, 70-94. pdf

Derumigny, A., & Fermanian, J. D. (2019). On kernel-based estimation of conditional Kendall’s tau: finite-distance bounds and asymptotic behavior. Dependence Modeling, 7(1), 292-321. pdf

Derumigny, A., & Fermanian, J. D. (2020). On Kendall’s regression. Journal of Multivariate Analysis, 178, 104610. pdf

Derumigny, A., & Fermanian, J. D. (2022). Conditional empirical copula processes and generalized dependence measures. Electronic Journal of Statistics, 16(2), 5692-5719. pdf

Derumigny, A., Fermanian, J. D., & Min, A. (2022). Testing for equality between conditional copulas given discretized conditioning events. Canadian Journal of Statistics. pdf

van der Spek, R., & Derumigny, A. (2022). Fast estimation of Kendall’s Tau and conditional Kendall’s Tau matrices under structural assumptions. arXiv:2204.03285.