# 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).

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.