Type:	Package
Title:	Estimation and Inference for Conditional Copula Models
Version:	0.1.4.1
Description:	Provides functions for the estimation of conditional copulas models, various estimators of conditional Kendall's tau (proposed in Derumigny and Fermanian (2019a, 2019b, 2020) <doi:10.1515/demo-2019-0016>, <doi:10.1016/j.csda.2019.01.013>, <doi:10.1016/j.jmva.2020.104610>), and test procedures for the simplifying assumption (proposed in Derumigny and Fermanian (2017) <doi:10.1515/demo-2017-0011> and Derumigny, Fermanian and Min (2022) <doi:10.1002/cjs.11742>).
License:	GPL-3
Encoding:	UTF-8
RoxygenNote:	7.3.2
Suggests:	MASS, knitr, rmarkdown, DiagrammeR, ggplot2, mvtnorm, testthat (≥ 3.0.0)
Imports:	VineCopula, pbapply, glmnet, ordinalNet, tree, nnet, data.tree, statmod, wdm
VignetteBuilder:	knitr
BugReports:	https://github.com/AlexisDerumigny/CondCopulas/issues
URL:	https://github.com/AlexisDerumigny/CondCopulas
Config/testthat/edition:	3
NeedsCompilation:	no
Packaged:	2024-09-03 09:03:48 UTC; aderumigny
Author:	Alexis Derumigny [aut, cre], Jean-David Fermanian [ctb, ths], Aleksey Min [ctb], Rutger van der Spek [ctb]
Maintainer:	Alexis Derumigny <a.f.f.derumigny@tudelft.nl>
Repository:	CRAN
Date/Publication:	2024-09-03 12:40:06 UTC

Kendall's regression: choice of the penalization parameter by K-folds cross-validation

Description

In this model, three variables X_1, X_2 and Z are observed. We try to model the conditional Kendall's tau between X_1 and X_2 conditionally to Z=z, as follows:

\Lambda(\tau_{X_1, X_2 | Z = z}) = \sum_{i=1}^{p'} \beta_i \psi_i(z),

where \tau_{X_1, X_2 | Z = z} is the conditional Kendall's tau between X_1 and X_2 conditionally to Z=z, \Lambda is a function from ]-1, 1[] to R, (\beta_1, \dots, \beta_p) are unknown coefficients to be estimated and \psi_1, \dots, \psi_{p'}) are a dictionary of functions. To estimate beta, we used the penalized estimator which is defined as the minimizer of the following criteria

\frac{1}{2n'} \sum_{i=1}^{n'} [\Lambda(\hat\tau_{X_1, X_2 | Z = z}) - \sum_{j=1}^{p'} \beta_j \psi_j(z)]^2 + \lambda * |\beta|_1.

This function chooses the penalization parameter lambda by cross-validation.

Usage

CKT.KendallReg.LambdaCV(
  X1 = NULL,
  X2 = NULL,
  Z = NULL,
  ZToEstimate,
  designMatrixZ = cbind(ZToEstimate, ZToEstimate^2, ZToEstimate^3),
  typeEstCKT = 4,
  h_lambda,
  Lambda = identity,
  kernel.name = "Epa",
  Kfolds_lambda = 10,
  l_norm = 1,
  matrixSignsPairs = NULL,
  progressBars = "global",
  observedX1 = NULL,
  observedX2 = NULL,
  observedZ = NULL
)

Arguments

Value

A list with the following components

• lambdaCV: the chosen value of the penalization parameters lambda.

• vectorLambda: a vector containing the values of lambda that have been compared.

• vectorMSEMean: the estimated MSE for each value of lambda in vectorLambda

• vectorMSESD: the estimated standard deviation of the MSE for each lambda. It can be used to construct confidence intervals for estimates of the MSE given by vectorMSEMean.

References

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

See Also

the main fitting function CKT.kendallReg.fit.

Examples

# We simulate from a conditional copula
set.seed(1)
N = 400
Z = rnorm(n = N, mean = 5, sd = 2)
conditionalTau = -0.9 + 1.8 * pnorm(Z, mean = 5, sd = 2)
simCopula = VineCopula::BiCopSim(N=N , family = 1,
    par = VineCopula::BiCopTau2Par(1 , conditionalTau ))
X1 = qnorm(simCopula[,1])
X2 = qnorm(simCopula[,2])

newZ = seq(2, 10, by = 0.1)
result <- CKT.KendallReg.LambdaCV(X1 = X1, X2 = X2, Z = Z,
                                  ZToEstimate = newZ, h_lambda = 2)

plot(x = result$vectorLambda, y = result$vectorMSEMean,
     type = "l", log = "x")

Estimation of conditional Kendall's tau between two variables X1 and X2 given Z = z

Description

Let X_1 and X_2 be two random variables. The goal of this function is to estimate the conditional Kendall's tau (a dependence measure) between X_1 and X_2 given Z=z for a conditioning variable Z. Conditional Kendall's tau between X_1 and X_2 given Z=z is defined as:

P( (X_{1,1} - X_{2,1})(X_{1,2} - X_{2,2}) > 0 | Z_1 = Z_2 = z)

- P( (X_{1,1} - X_{2,1})(X_{1,2} - X_{2,2}) < 0 | Z_1 = Z_2 = z),

where (X_{1,1}, X_{1,2}, Z_1) and (X_{2,1}, X_{2,2}, Z_2) are two independent and identically distributed copies of (X_1, X_2, Z). In other words, conditional Kendall's tau is the difference between the probabilities of observing concordant and discordant pairs from the conditional law of

(X_1, X_2) | Z=z.

This function can use different estimators for conditional Kendall's tau, see the description of the parameter methodEstimation for a complete list of possibilities.

Usage

CKT.estimate(
  X1 = NULL, X2 = NULL, Z = NULL,
  newZ = Z, methodEstimation, h,
  listPhi = if(methodEstimation == "kendallReg")
               {list( function(x){return(x)}   ,
                      function(x){return(x^2)} ,
                      function(x){return(x^3)} )
               } else {list(identity)} ,
  ... ,
  observedX1 = NULL, observedX2 = NULL, observedZ = NULL )

Arguments

Value

the vector of estimated conditional Kendall's tau at each of the observations of newZ.

References

Derumigny, A., & Fermanian, J. D. (2019a). A classification point-of-view about conditional Kendall’s tau. Computational Statistics & Data Analysis, 135, 70-94. doi:10.1016/j.csda.2019.01.013

Derumigny, A., & Fermanian, J. D. (2019b). On kernel-based estimation of conditional Kendall’s tau: finite-distance bounds and asymptotic behavior. Dependence Modeling, 7(1), 292-321. doi:10.1515/demo-2019-0016

Derumigny, A., & Fermanian, J. D. (2020). On Kendall’s regression. Journal of Multivariate Analysis, 178, 104610. doi:10.1016/j.jmva.2020.104610

See Also

the specialized functions for estimating conditional Kendall's tau for each method: CKT.fit.tree, CKT.fit.randomForest, CKT.fit.GLM, CKT.fit.nNets, CKT.predict.kNN, CKT.fit.randomForest, CKT.kernel and CKT.kendallReg.fit.

See also the nonparametric estimator of conditional copula models estimateNPCondCopula, and the parametric estimators of conditional copula models estimateParCondCopula.

In the case where Z is discrete or in the case of discrete conditioning events, see bCond.treeCKT.

Examples

# We simulate from a conditional copula
set.seed(1)
N = 300
Z = rnorm(n = N, mean = 5, sd = 2)
conditionalTau = -0.9 + 1.8 * pnorm(Z, mean = 5, sd = 2)
simCopula = VineCopula::BiCopSim(N=N , family = 1,
    par = VineCopula::BiCopTau2Par(1 , conditionalTau ))
X1 = qnorm(simCopula[,1])
X2 = qnorm(simCopula[,2])

newZ = seq(2,10,by = 0.1)
h = 0.1
estimatedCKT_tree <- CKT.estimate(
  X1 = X1, X2 = X2, Z = Z,
  newZ = newZ,
  methodEstimation = "tree", h = h)

estimatedCKT_rf <- CKT.estimate(
  X1 = X1, X2 = X2, Z = Z,
  newZ = newZ,
  methodEstimation = "randomForest", h = h)

estimatedCKT_GLM <- CKT.estimate(
  X1 = X1, X2 = X2, Z = Z,
  newZ = newZ,
  methodEstimation = "logit", h = h,
  listPhi = list(function(x){return(x)}, function(x){return(x^2)},
                 function(x){return(x^3)}) )

estimatedCKT_kNN <- CKT.estimate(
  X1 = X1, X2 = X2, Z = Z,
  newZ = newZ,
  methodEstimation = "nearestNeighbors", h = h,
  number_nn = c(50,80, 100, 120,200),
  partition = 4
  )

estimatedCKT_nNet <- CKT.estimate(
  X1 = X1, X2 = X2, Z = Z,
  newZ = newZ,
  methodEstimation = "neuralNetwork", h = h,
  )

estimatedCKT_kernel <- CKT.estimate(
  X1 = X1, X2 = X2, Z = Z,
  newZ = newZ,
  methodEstimation = "kernel", h = h,
  )

estimatedCKT_kendallReg <- CKT.estimate(
   X1 = X1, X2 = X2, Z = Z,
   newZ = newZ,
   methodEstimation = "kendallReg", h = h)

# Comparison between true Kendall's tau (in black)
# and estimated Kendall's tau (in other colors)
trueConditionalTau = -0.9 + 1.8 * pnorm(newZ, mean = 5, sd = 2)
plot(newZ, trueConditionalTau , col="black",
   type = "l", ylim = c(-1, 1))
lines(newZ, estimatedCKT_tree, col = "red")
lines(newZ, estimatedCKT_rf, col = "blue")
lines(newZ, estimatedCKT_GLM, col = "green")
lines(newZ, estimatedCKT_kNN, col = "purple")
lines(newZ, estimatedCKT_nNet, col = "coral")
lines(newZ, estimatedCKT_kernel, col = "skyblue")
lines(newZ, estimatedCKT_kendallReg, col = "darkgreen")

Estimation of conditional Kendall's taus by penalized GLM

Description

The function CKT.fit.GLM fits a regression model for the conditional Kendall's tau \tau_{1,2|Z} between two variables X_1 and X_2 conditionally to some predictors Z. More precisely, this function fits the model

\tau_{1,2|Z} = 2 * \Lambda( \beta_0 + \beta_1 \phi_1(Z) + ... + \beta_p \phi_p(Z) )

for a link function \Lambda, and p real-valued functions \phi_1, ..., \phi_p. The function CKT.predict.GLM predicts the values of conditional Kendall's tau for some values of the conditioning variable Z.

Usage

CKT.fit.GLM(
  datasetPairs,
  designMatrix = datasetPairs[, 2:(ncol(datasetPairs) - 3), drop = FALSE],
  link = "logit",
  ...
)

CKT.predict.GLM(fit, newZ)

Arguments

Value

CKT.fit.GLM returns the fitted GLM, an object with S3 class ordinalNet.

CKT.predict.GLM returns a vector of (predicted) conditional Kendall's taus of the same size as the number of rows of the matrix newZ.

References

Derumigny, A., & Fermanian, J. D. (2019). A classification point-of-view about conditional Kendall’s tau. Computational Statistics & Data Analysis, 135, 70-94. (Algorithm 2) doi:10.1016/j.csda.2019.01.013

See Also

See also other estimators of conditional Kendall's tau: CKT.fit.tree, CKT.fit.randomForest, CKT.fit.nNets, CKT.predict.kNN, CKT.kernel, CKT.kendallReg.fit, and the more general wrapper CKT.estimate.

Examples

# We simulate from a conditional copula
set.seed(1)
N = 400
Z = rnorm(n = N, mean = 5, sd = 2)
conditionalTau = 2*plogis(-1 + 0.8*Z - 0.1*Z^2) - 1
simCopula = VineCopula::BiCopSim(N=N , family = 1,
    par = VineCopula::BiCopTau2Par(1 , conditionalTau ))
X1 = qnorm(simCopula[,1])
X2 = qnorm(simCopula[,2])

datasetP = datasetPairs(X1 = X1, X2 = X2, Z = Z, h = 0.07, cut = 0.9)
designMatrix = cbind(datasetP[,2], datasetP[,2]^2)
fitCKT_GLM <- CKT.fit.GLM(
  datasetPairs = datasetP, designMatrix = designMatrix,
  maxiterOut = 10, maxiterIn = 5)
print(coef(fitCKT_GLM))
# These are rather close to the true coefficients -1, 0.8, -0.1
# used to generate the data above.

newZ = seq(2,10,by = 0.1)
estimatedCKT_GLM = CKT.predict.GLM(
  fit = fitCKT_GLM, newZ = cbind(newZ, newZ^2))

# Comparison between true Kendall's tau (in red)
# and estimated Kendall's tau (in black)
trueConditionalTau = 2*plogis(-1 + 0.8*newZ - 0.1*newZ^2) - 1
plot(newZ, trueConditionalTau , col="red",
   type = "l", ylim = c(-1, 1))
lines(newZ, estimatedCKT_GLM)

Estimation of conditional Kendall's taus by model averaging of neural networks

Description

Let X_1 and X_2 be two random variables. The goal of this function is to estimate the conditional Kendall's tau (a dependence measure) between X_1 and X_2 given Z=z for a conditioning variable Z. Conditional Kendall's tau between X_1 and X_2 given Z=z is defined as:

P( (X_{1,1} - X_{2,1})(X_{1,2} - X_{2,2}) > 0 | Z_1 = Z_2 = z)

- P( (X_{1,1} - X_{2,1})(X_{1,2} - X_{2,2}) < 0 | Z_1 = Z_2 = z),

where (X_{1,1}, X_{1,2}, Z_1) and (X_{2,1}, X_{2,2}, Z_2) are two independent and identically distributed copies of (X_1, X_2, Z). In other words, conditional Kendall's tau is the difference between the probabilities of observing concordant and discordant pairs from the conditional law of

(X_1, X_2) | Z=z.

This function estimates conditional Kendall's tau using neural networks. This is possible by the relationship between estimation of conditional Kendall's tau and classification problems (see Derumigny and Fermanian (2019)): estimation of conditional Kendall's tau is equivalent to the prediction of concordance in the space of pairs of observations.

Usage

CKT.fit.nNets(
  datasetPairs,
  designMatrix = datasetPairs[, 2:(ncol(datasetPairs) - 3), drop = FALSE],
  vecSize = rep(3, times = 10),
  nObs_per_NN = 0.9 * nrow(designMatrix),
  verbose = 1
)

Arguments

Value

CKT.fit.nNets returns a list of the fitted neural networks

References

Derumigny, A., & Fermanian, J. D. (2019). A classification point-of-view about conditional Kendall’s tau. Computational Statistics & Data Analysis, 135, 70-94. (Algorithm 7) doi:10.1016/j.csda.2019.01.013

See Also

See also other estimators of conditional Kendall's tau: CKT.fit.tree, CKT.fit.randomForest, CKT.fit.GLM, CKT.predict.kNN, CKT.kernel, CKT.kendallReg.fit, and the more general wrapper CKT.estimate.

Examples

# We simulate from a conditional copula
set.seed(1)
N = 800
Z = rnorm(n = N, mean = 5, sd = 2)
conditionalTau = -0.9 + 1.8 * pnorm(Z, mean = 5, sd = 2)
simCopula = VineCopula::BiCopSim(N=N , family = 1,
    par = VineCopula::BiCopTau2Par(1 , conditionalTau ))
X1 = qnorm(simCopula[,1])
X2 = qnorm(simCopula[,2])

newZ = seq(2,10,by = 0.1)
datasetP = datasetPairs(X1 = X1, X2 = X2, Z = Z, h = 0.07, cut = 0.9)

fitCKT_nets <- CKT.fit.nNets(datasetPairs = datasetP)
estimatedCKT_nNets <- CKT.predict.nNets(
  fit = fitCKT_nets, newZ = matrix(newZ, ncol = 1))

# Comparison between true Kendall's tau (in black)
# and estimated Kendall's tau (in red)
trueConditionalTau = -0.9 + 1.8 * pnorm(newZ, mean = 5, sd = 2)
plot(newZ, trueConditionalTau , col="black",
   type = "l", ylim = c(-1, 1))
lines(newZ, estimatedCKT_nNets, col = "red")

Fit a Random Forest that can be used for the estimation of conditional Kendall's tau.

Description

Let X_1 and X_2 be two random variables. The goal of this function is to estimate the conditional Kendall's tau (a dependence measure) between X_1 and X_2 given Z=z for a conditioning variable Z. Conditional Kendall's tau between X_1 and X_2 given Z=z is defined as:

P( (X_{1,1} - X_{2,1})(X_{1,2} - X_{2,2}) > 0 | Z_1 = Z_2 = z)

- P( (X_{1,1} - X_{2,1})(X_{1,2} - X_{2,2}) < 0 | Z_1 = Z_2 = z),

where (X_{1,1}, X_{1,2}, Z_1) and (X_{2,1}, X_{2,2}, Z_2) are two independent and identically distributed copies of (X_1, X_2, Z). In other words, conditional Kendall's tau is the difference between the probabilities of observing concordant and discordant pairs from the conditional law of

(X_1, X_2) | Z=z.

These functions estimate and predict conditional Kendall's tau using a random forest. This is possible by the relationship between estimation of conditional Kendall's tau and classification problems (see Derumigny and Fermanian (2019)): estimation of conditional Kendall's tau is equivalent to the prediction of concordance in the space of pairs of observations.

Usage

CKT.fit.randomForest(
  datasetPairs,
  designMatrix = data.frame(x = datasetPairs[, 2:(ncol(datasetPairs) - 3)]),
  n,
  nTree = 10,
  mindev = 0.008,
  mincut = 0,
  nObs_per_Tree = ceiling(0.8 * n),
  nVar_per_Tree = ceiling(0.8 * (ncol(datasetPairs) - 4)),
  verbose = FALSE,
  nMaxDepthAllowed = 10
)

CKT.predict.randomForest(fit, newZ)

Arguments

Value

a list with two components

• list_tree a list of size nTree composed of all the fitted trees.

• list_variables a list of size nTree composed of the (predictor) variables for each tree.

CKT.predict.randomForest returns a vector of (predicted) conditional Kendall's taus of the same size as the number of rows of the newZ.

References

Derumigny, A., & Fermanian, J. D. (2019). A classification point-of-view about conditional Kendall’s tau. Computational Statistics & Data Analysis, 135, 70-94. (Algorithm 4) doi:10.1016/j.csda.2019.01.013

Examples

# We simulate from a conditional copula
set.seed(1)
N = 800
Z = rnorm(n = N, mean = 5, sd = 2)
conditionalTau = -0.9 + 1.8 * pnorm(Z, mean = 5, sd = 2)
simCopula = VineCopula::BiCopSim(N=N , family = 1,
    par = VineCopula::BiCopTau2Par(1 , conditionalTau ))
X1 = qnorm(simCopula[,1])
X2 = qnorm(simCopula[,2])

datasetP = datasetPairs(X1 = X1, X2 = X2, Z = Z, h = 0.07, cut = 0.9)
est_RF = CKT.fit.randomForest(datasetPairs = datasetP, n = N,
  mindev = 0.008)

newZ = seq(1,10,by = 0.1)
prediction = CKT.predict.randomForest(fit = est_RF,
   newZ = data.frame(x=newZ))
# Comparison between true Kendall's tau (in red)
# and estimated Kendall's tau (in black)
plot(newZ, prediction, type = "l", ylim = c(-1,1))
lines(newZ, -0.9 + 1.8 * pnorm(newZ, mean = 5, sd = 2), col="red")

Estimation of conditional Kendall's taus using a classification tree

Description

Let X_1 and X_2 be two random variables. The goal of this function is to estimate the conditional Kendall's tau (a dependence measure) between X_1 and X_2 given Z=z for a conditioning variable Z. Conditional Kendall's tau between X_1 and X_2 given Z=z is defined as:

P( (X_{1,1} - X_{2,1})(X_{1,2} - X_{2,2}) > 0 | Z_1 = Z_2 = z)

- P( (X_{1,1} - X_{2,1})(X_{1,2} - X_{2,2}) < 0 | Z_1 = Z_2 = z),

where (X_{1,1}, X_{1,2}, Z_1) and (X_{2,1}, X_{2,2}, Z_2) are two independent and identically distributed copies of (X_1, X_2, Z). In other words, conditional Kendall's tau is the difference between the probabilities of observing concordant and discordant pairs from the conditional law of

(X_1, X_2) | Z=z.

These functions estimate and predict conditional Kendall's tau using a classification tree. This is possible by the relationship between estimation of conditional Kendall's tau and classification problems (see Derumigny and Fermanian (2019)): estimation of conditional Kendall's tau is equivalent to the prediction of concordance in the space of pairs of observations.

Usage

CKT.fit.tree(datasetPairs, mindev = 0.008, mincut = 0)

CKT.predict.tree(fit, newZ)

Arguments

Value

CKT.fit.tree returns the fitted tree.

CKT.predict.tree returns a vector of (predicted) conditional Kendall's taus of the same size as the number of rows of newZ.

References

Derumigny, A., & Fermanian, J. D. (2019). A classification point-of-view about conditional Kendall’s tau. Computational Statistics & Data Analysis, 135, 70-94. (Section 3.2) doi:10.1016/j.csda.2019.01.013

See Also

See also other estimators of conditional Kendall's tau: CKT.fit.nNets, CKT.fit.randomForest, CKT.fit.GLM, CKT.predict.kNN, CKT.kernel, CKT.kendallReg.fit, and the more general wrapper CKT.estimate.

Examples

# We simulate from a conditional copula
set.seed(1)
N = 800
Z = rnorm(n = N, mean = 5, sd = 2)
conditionalTau = -0.9 + 1.8 * pnorm(Z, mean = 5, sd = 2)
simCopula = VineCopula::BiCopSim(N=N , family = 1,
    par = VineCopula::BiCopTau2Par(1 , conditionalTau ))
X1 = qnorm(simCopula[,1])
X2 = qnorm(simCopula[,2])

datasetP = datasetPairs(X1 = X1, X2 = X2, Z = Z, h = 0.07, cut = 0.9)
est_Tree = CKT.fit.tree(datasetPairs = datasetP, mindev = 0.008)
print(est_Tree)

newZ = seq(1,10,by = 0.1)
prediction = CKT.predict.tree(fit = est_Tree, newZ = data.frame(x=newZ))
# Comparison between true Kendall's tau (in red)
# and estimated Kendall's tau (in black)
plot(newZ, prediction, type = "l", ylim = c(-1,1))
lines(newZ, -0.9 + 1.8 * pnorm(newZ, mean = 5, sd = 2), col="red")

Choose the bandwidth for kernel estimation of conditional Kendall's tau using cross-validation

Description

Let X_1 and X_2 be two random variables. The goal here is to estimate the conditional Kendall's tau (a dependence measure) between X_1 and X_2 given Z=z for a conditioning variable Z. Conditional Kendall's tau between X_1 and X_2 given Z=z is defined as:

P( (X_{1,1} - X_{2,1})(X_{1,2} - X_{2,2}) > 0 | Z_1 = Z_2 = z)

- P( (X_{1,1} - X_{2,1})(X_{1,2} - X_{2,2}) < 0 | Z_1 = Z_2 = z),

where (X_{1,1}, X_{1,2}, Z_1) and (X_{2,1}, X_{2,2}, Z_2) are two independent and identically distributed copies of (X_1, X_2, Z). For this, a kernel-based estimator is used, as described in (Derumigny & Fermanian (2019)). These functions aims at finding the best bandwidth h among a given range_h by cross-validation. They use either:

• leave-one-out cross-validation: function CKT.hCV.l1out

• or K-folds cross-validation: function CKT.hCV.Kfolds

Usage

CKT.hCV.l1out(
  X1 = NULL,
  X2 = NULL,
  Z = NULL,
  range_h,
  matrixSignsPairs = NULL,
  nPairs = 10 * length(X1),
  typeEstCKT = "wdm",
  kernel.name = "Epa",
  progressBar = TRUE,
  verbose = FALSE,
  observedX1 = NULL,
  observedX2 = NULL,
  observedZ = NULL
)

CKT.hCV.Kfolds(
  X1,
  X2,
  Z,
  ZToEstimate,
  range_h,
  matrixSignsPairs = NULL,
  typeEstCKT = "wdm",
  kernel.name = "Epa",
  Kfolds = 5,
  progressBar = TRUE,
  verbose = FALSE,
  observedX1 = NULL,
  observedX2 = NULL,
  observedZ = NULL
)

Arguments

Value

Both functions return a list with two components:

• hCV: the chosen bandwidth

• scores: vector of the same length as range_h giving the value of the CV criteria for each of the h tested. Lower score indicates a better fit.

References

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. Page 296, Equation (4). doi:10.1515/demo-2019-0016

See Also

CKT.kernel for the corresponding estimator of conditional Kendall's tau by kernel smoothing.

Examples

# We simulate from a conditional copula
set.seed(1)
N = 200
Z = rnorm(n = N, mean = 5, sd = 2)
conditionalTau = -0.9 + 1.8 * pnorm(Z, mean = 5, sd = 2)
simCopula = VineCopula::BiCopSim(N=N , family = 1,
    par = VineCopula::BiCopTau2Par(1 , conditionalTau ))
X1 = qnorm(simCopula[,1])
X2 = qnorm(simCopula[,2])

newZ = seq(2,10,by = 0.1)
range_h = 3:10

resultCV <- CKT.hCV.l1out(X1 = X1, X2 = X2, Z = Z,
                          range_h = range_h, nPairs = 100)

resultCV <- CKT.hCV.Kfolds(X1 = X1, X2 = X2, Z = Z,
                           range_h = range_h, ZToEstimate = newZ)

plot(range_h, resultCV$scores, type = "b")

Fit Kendall's regression, a GLM-type model for conditional Kendall's tau

Description

The function CKT.kendallReg.fit fits a regression-type model for the conditional Kendall's tau between two variables X_1 and X_2 conditionally to some predictors Z. More precisely, it fits the model

\Lambda(\tau_{X_1, X_2 | Z = z}) = \sum_{j=1}^{p'} \beta_j \psi_j(z),

where \tau_{X_1, X_2 | Z = z} is the conditional Kendall's tau between X_1 and X_2 conditionally to Z=z, \Lambda is a function from ]-1, 1] to R, (\beta_1, \dots, \beta_p) are unknown coefficients to be estimated and \psi_1, \dots, \psi_{p'}) are a dictionary of functions. To estimate beta, we used the penalized estimator which is defined as the minimizer of the following criteria

\frac{1}{2n'} \sum_{i=1}^{n'} [\Lambda(\hat\tau_{X_1, X_2 | Z = z_i}) - \sum_{j=1}^{p'} \beta_j \psi_j(z_i)]^2 + \lambda * |\beta|_1,

where the z_i are a second sample (here denoted by ZToEstimate).

The function CKT.kendallReg.predict predicts the conditional Kendall's tau between two variables X_1 and X_2 given Z=z for some new values of z.

Usage

CKT.kendallReg.fit(
  X1 = NULL,
  X2 = NULL,
  Z = NULL,
  ZToEstimate,
  designMatrixZ = cbind(ZToEstimate, ZToEstimate^2, ZToEstimate^3),
  newZ = designMatrixZ,
  h_kernel,
  Lambda = identity,
  Lambda_inv = identity,
  lambda = NULL,
  Kfolds_lambda = 10,
  l_norm = 1,
  h_lambda = h_kernel,
  ...,
  observedX1 = NULL,
  observedX2 = NULL,
  observedZ = NULL
)

CKT.kendallReg.predict(fit, newZ, lambda = NULL, Lambda_inv = identity)

Arguments

Value

The function CKT.kendallReg.fit returns a list with the following components:

• estimatedCKT: the estimated CKT at the new data points newZ.

• fit: the fitted model, of S3 class glmnet (see glmnet::glmnet for more details).

• lambda: the value of the penalized parameter used. (i.e. either the one supplied by the user or the one determined by cross-validation)

CKT.kendallReg.predict returns the predicted values of conditional Kendall's tau.

References

Derumigny, A., & Fermanian, J. D. (2020). On Kendall’s regression. Journal of Multivariate Analysis, 178, 104610. doi:10.1016/j.jmva.2020.104610

See Also

See also other estimators of conditional Kendall's tau: CKT.fit.tree, CKT.fit.randomForest, CKT.fit.nNets, CKT.predict.kNN, CKT.kernel, CKT.fit.GLM, and the more general wrapper CKT.estimate.

See also the test of the simplifying assumption that a conditional copula does not depend on the value of the conditioning variable using the nullity of Kendall's regression coefficients: simpA.kendallReg.

Examples

# We simulate from a conditional copula
set.seed(1)
N = 400
Z = rnorm(n = N, mean = 5, sd = 2)
conditionalTau = -0.9 + 1.8 * pnorm(Z, mean = 5, sd = 2)
simCopula = VineCopula::BiCopSim(N=N , family = 1,
    par = VineCopula::BiCopTau2Par(1 , conditionalTau ))
X1 = qnorm(simCopula[,1])
X2 = qnorm(simCopula[,2])

newZ = seq(2, 10, by = 0.1)
estimatedCKT_kendallReg <- CKT.kendallReg.fit(
   X1 = X1, X2 = X2, Z = Z,
   ZToEstimate = newZ, h_kernel = 0.07)

coef(estimatedCKT_kendallReg$fit,
     s = estimatedCKT_kendallReg$lambda)

# Comparison between true Kendall's tau (in black)
# and estimated Kendall's tau (in red)
trueConditionalTau = -0.9 + 1.8 * pnorm(newZ, mean = 5, sd = 2)
plot(newZ, trueConditionalTau , col="black",
   type = "l", ylim = c(-1, 1))
lines(newZ, estimatedCKT_kendallReg$estimatedCKT, col = "red")

Estimation of conditional Kendall's tau using kernel smoothing

Description

Let X_1 and X_2 be two random variables. The goal of this function is to estimate the conditional Kendall's tau (a dependence measure) between X_1 and X_2 given Z=z for a conditioning variable Z. Conditional Kendall's tau between X_1 and X_2 given Z=z is defined as:

P( (X_{1,1} - X_{2,1})(X_{1,2} - X_{2,2}) > 0 | Z_1 = Z_2 = z)

- P( (X_{1,1} - X_{2,1})(X_{1,2} - X_{2,2}) < 0 | Z_1 = Z_2 = z),

where (X_{1,1}, X_{1,2}, Z_1) and (X_{2,1}, X_{2,2}, Z_2) are two independent and identically distributed copies of (X_1, X_2, Z). For this, a kernel-based estimator is used, as described in (Derumigny, & Fermanian (2019)).

Usage

CKT.kernel(
  X1 = NULL,
  X2 = NULL,
  Z = NULL,
  newZ,
  h,
  kernel.name = "Epa",
  methodCV = "Kfolds",
  Kfolds = 5,
  nPairs = 10 * length(observedX1),
  typeEstCKT = "wdm",
  progressBar = TRUE,
  observedX1 = NULL,
  observedX2 = NULL,
  observedZ = NULL
)

Arguments

Details

Choice of the bandwidth h. The choice of the bandwidth must be done carefully. In the univariate case, the default kernel (Epanechnikov kernel) has a support on [-1,1], so for a bandwidth h, estimation of conditional Kendall's tau at Z=z will only use points for which Z_i \in [z \pm h]. As usual in nonparametric estimation, h should not be too small (to avoid having a too large variance) and should not be large (to avoid having a too large bias).

We recommend that for each z for which the conditional Kendall's tau \tau_{X_1, X_2 | Z=z} is estimated, the set \{i: Z_i \in [z \pm h] \} should contain at least 20 points and not more than 30% of the points of the whole dataset. Note that for a consistent estimation, as the sample size n tends to the infinity, h should tend to 0 while the size of the set \{i: Z_i \in [z \pm h]\} should also tend to the infinity. Indeed the conditioning points should be closer and closer to the point of interest z (small h) and more and more numerous (h tending to 0 slowly enough).

In the multivariate case, similar recommendations can be made. Because of the curse of dimensionality, a larger sample will be necessary to reach the same level of precision as in the univariate case.

Value

a list with two components

• estimatedCKT the vector of size NROW(newZ) containing the values of the estimated conditional Kendall's tau.

• finalh the bandwidth h that was finally used for kernel smoothing (either the one specified by the user or the one chosen by cross-validation if multiple bandwidths were given.)

References

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. doi:10.1515/demo-2019-0016

See Also

CKT.estimate for other estimators of conditional Kendall's tau. CKTmatrix.kernel for a generalization of this function when the conditioned vector is of dimension d instead of dimension 2 here.

See CKT.hCV.l1out for manual selection of the bandwidth h by leave-one-out or K-folds cross-validation.

Examples

# We simulate from a conditional copula
set.seed(1)
N = 800
Z = rnorm(n = N, mean = 5, sd = 2)
conditionalTau = -0.9 + 1.8 * pnorm(Z, mean = 5, sd = 2)
simCopula = VineCopula::BiCopSim(N=N , family = 1,
    par = VineCopula::BiCopTau2Par(1 , conditionalTau ))
X1 = qnorm(simCopula[,1])
X2 = qnorm(simCopula[,2])

newZ = seq(2,10,by = 0.1)
estimatedCKT_kernel <- CKT.kernel(
   X1 = X1, X2 = X2, Z = Z,
   newZ = newZ, h = 0.1, kernel.name = "Epa")$estimatedCKT

# Comparison between true Kendall's tau (in black)
# and estimated Kendall's tau (in red)
trueConditionalTau = -0.9 + 1.8 * pnorm(newZ, mean = 5, sd = 2)
plot(newZ, trueConditionalTau , col = "black",
     type = "l", ylim = c(-1, 1))
lines(newZ, estimatedCKT_kernel, col = "red")

Prediction of conditional Kendall's tau using nearest neighbors

Description

Let X_1 and X_2 be two random variables. The goal of this function is to estimate the conditional Kendall's tau (a dependence measure) between X_1 and X_2 given Z=z for a conditioning variable Z. Conditional Kendall's tau between X_1 and X_2 given Z=z is defined as:

P( (X_{1,1} - X_{2,1})(X_{1,2} - X_{2,2}) > 0 | Z_1 = Z_2 = z)

- P( (X_{1,1} - X_{2,1})(X_{1,2} - X_{2,2}) < 0 | Z_1 = Z_2 = z),

where (X_{1,1}, X_{1,2}, Z_1) and (X_{2,1}, X_{2,2}, Z_2) are two independent and identically distributed copies of (X_1, X_2, Z). In other words, conditional Kendall's tau is the difference between the probabilities of observing concordant and discordant pairs from the conditional law of

(X_1, X_2) | Z=z.

This function estimates conditional Kendall's tau using a nearest neighbors. This is possible by the relationship between estimation of conditional Kendall's tau and classification problems (see Derumigny and Fermanian (2019)): estimation of conditional Kendall's tau is equivalent to the prediction of concordance in the space of pairs of observations.

Usage

CKT.predict.kNN(
  datasetPairs,
  designMatrix = datasetPairs[, 2:(ncol(datasetPairs) - 3), drop = FALSE],
  newZ,
  number_nn,
  weightsVariables = 1,
  normLp = 2,
  constantA = 1,
  partition = NULL,
  verbose = 1,
  lengthVerbose = 100,
  methodSort = "partial.sort"
)

Arguments

Value

a list with two components

• estimatedCKT the estimated conditional Kendall's tau, a vector of the same size as the number of rows in newZ;

• vect_k_chosen the locally selected number of nearest neighbors, a vector of the same size as the number of rows in newZ.

References

Derumigny, A., & Fermanian, J. D. (2019). A classification point-of-view about conditional Kendall’s tau. Computational Statistics & Data Analysis, 135, 70-94. (Algorithm 5) doi:10.1016/j.csda.2019.01.013

See Also

See also other estimators of conditional Kendall's tau: CKT.fit.tree, CKT.fit.randomForest, CKT.fit.nNets, CKT.fit.randomForest, CKT.fit.GLM, CKT.kernel, CKT.kendallReg.fit, and the more general wrapper CKT.estimate.

Examples

# We simulate from a conditional copula
set.seed(1)
N = 800
Z = rnorm(n = N, mean = 5, sd = 2)
conditionalTau = -0.9 + 1.8 * pnorm(Z, mean = 5, sd = 2)
simCopula = VineCopula::BiCopSim(N=N , family = 1,
    par = VineCopula::BiCopTau2Par(1 , conditionalTau ))
X1 = qnorm(simCopula[,1])
X2 = qnorm(simCopula[,2])

newZ = seq(2,10,by = 0.1)
datasetP = datasetPairs(X1 = X1, X2 = X2, Z = Z, h = 0.07, cut = 0.9)
estimatedCKT_knn <- CKT.predict.kNN(
  datasetPairs = datasetP,
  newZ = matrix(newZ,ncol = 1),
  number_nn = c(50,80, 100, 120,200),
  partition = 8)

# Comparison between true Kendall's tau (in black)
# and estimated Kendall's tau (in red)
trueConditionalTau = -0.9 + 1.8 * pnorm(newZ, mean = 5, sd = 2)
plot(newZ, trueConditionalTau , col="black",
   type = "l", ylim = c(-1, 1))
lines(newZ, estimatedCKT_knn$estimatedCKT, col = "red")

Predict the values of conditional Kendall's tau using Model Averaging of Neural Networks

Description

Predict the values of conditional Kendall's tau using Model Averaging of Neural Networks

Usage

CKT.predict.nNets(fit, newZ, aggregationMethod = "mean")

Arguments

Value

CKT.predict.nNets returns a vector of (predicted) conditional Kendall's taus of the same size as the number of rows of the matrix newZ.

Estimate the conditional Kendall's tau matrix at different conditioning points

Description

Assume that we are interested in a random vector (X, Z), where X is of dimension d > 2 and Z is of dimension 1. We want to estimate the dependence across the elements of the conditioned vector X given Z=z. This function takes in parameter observations of (X,Z) and returns kernel-based estimators of

\tau_{i,j | Z=zk}

which is the conditional Kendall's tau between X_i and X_j given to Z=zk, for every conditioning point zk in gridZ. If the conditional Kendall's tau matrix has a block structure, then improved estimation is possible by averaging over the kernel-based estimators of pairwise conditional Kendall's taus. Groups of variables composing the same blocks can be defined using the parameter blockStructure, and the averaging can be set on using the parameter averaging=all, or averaging=diag for faster estimation by averaging only over diagonal elements of each block.

Usage

CKTmatrix.kernel(
  dataMatrix,
  observedZ,
  gridZ,
  averaging = "no",
  blockStructure = NULL,
  h,
  kernel.name = "Epa",
  typeEstCKT = "wdm"
)

Arguments

Value

array with dimensions depending on averaging:

• If averaging = "no": it returns an array of dimensions (n, n, length(gridZ)), containing the estimated conditional Kendall's tau matrix given Z = z. Here, n is the number of rows in dataMatrix.

• If averaging = "all" or "diag": it returns an array of dimensions (length(blockStructure), length(blockStructure), length(gridZ)), containing the block estimates of the conditional Kendall's tau given Z = z with ones on the diagonal.

Author(s)

Rutger van der Spek, Alexis Derumigny

References

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.

See Also

CKT.kernel for kernel-based estimation of conditional Kendall's tau between two variables (i.e. the equivalent of this function when X is bivariate and d=2).

Examples

# Data simulation
n = 100
Z = runif(n)
d = 5
CKT_11 = 0.8
CKT_22 = 0.9
CKT_12 = 0.1 + 0.5 * cos(pi * Z)
data_X = matrix(nrow = n, ncol = d)
for (i in 1:n){
  CKT_matrix = matrix(data =
    c(  1      , CKT_11   , CKT_11   , CKT_12[i], CKT_12[i] ,
      CKT_11   ,   1      , CKT_11   , CKT_12[i], CKT_12[i] ,
      CKT_11   , CKT_11   ,    1     , CKT_12[i], CKT_12[i] ,
      CKT_12[i], CKT_12[i], CKT_12[i],   1      , CKT_22    ,
      CKT_12[i], CKT_12[i], CKT_12[i], CKT_22   ,   1
      ) ,
     nrow = 5, ncol = 5)
  sigma = sin(pi * CKT_matrix/2)
  data_X[i, ] = mvtnorm::rmvnorm(n = 1, sigma = sigma)
}
plot(as.data.frame.matrix(data_X))

# Estimation of CKT matrix
h = 1.06 * sd(Z) * n^{-1/5}
gridZ = c(0.2, 0.8)
estMatrixAll <- CKTmatrix.kernel(
  dataMatrix = data_X, observedZ = Z, gridZ = gridZ, h = h)
# Averaging estimator
estMatrixAve <- CKTmatrix.kernel(
  dataMatrix = data_X, observedZ = Z, gridZ = gridZ,
  averaging = "diag", blockStructure = list(1:3,4:5), h = h)

# The estimated CKT matrix conditionally to Z=0.2 is:
estMatrixAll[ , , 1]
# Using the averaging estimator,
# the estimated CKT between the first group (variables 1 to 3)
# and the second group (variables 4 and 5) is
estMatrixAve[1, 2, 1]

# True value (of CKT between variables in block 1 and 2 given Z = 0.2):
0.1 + 0.5 * cos(pi * 0.2)

Estimation of the conditional parameters of a parametric conditional copula with discrete conditioning events.

Description

By Sklar's theorem, any conditional distribution function can be written as

F_{1,2|A}(x_1, x_2) = c_{1,2|A}(F_{1|A}(x_1), F_{2,A}(x_2)),

where A is an event and c_{1,2|A} is a copula depending on the event A. In this function, we assume that we have a partition A_1,... A_p of the probability space, and that for each k=1,...,p, the conditional copula is parametric according to the following model

c_{1,2|Ak} = c_{\theta(Ak)},

for some parameter \theta(Ak) depending on the realized event Ak. This function uses canonical maximum likelihood to estimate \theta(Ak) and the corresponding copulas c_{1,2|Ak}.

Usage

bCond.estParamCopula(U1, U2, family, partition)

Arguments

Value

a list of size p containing the p conditional copulas

References

Derumigny, A., & Fermanian, J. D. (2017). About tests of the “simplifying” assumption for conditional copulas. Dependence Modeling, 5(1), 154-197. doi:10.1515/demo-2017-0011

Derumigny, A., & Fermanian, J. D. (2022) Conditional empirical copula processes and generalized dependence measures Electronic Journal of Statistics, 16(2), 5692-5719. doi:10.1214/22-EJS2075

See Also

bCond.pobs for the computation of (conditional) pseudo-observations in this framework.

bCond.simpA.param for a test of the simplifying assumption that all these conditional copulas are equal (assuming they all belong to the same parametric family). bCond.simpA.CKT for a test of the simplifying assumption that all these conditional copulas are equal, based on the equality of conditional Kendall's tau.

Examples

n = 800
Z = stats::runif(n = n)
CKT = 0.2 * as.numeric(Z <= 0.3) +
  0.5 * as.numeric(Z > 0.3 & Z <= 0.5) +
  - 0.8 * as.numeric(Z > 0.5)
simCopula = VineCopula::BiCopSim(N = n,
  par = VineCopula::BiCopTau2Par(CKT, family = 1), family = 1)
X1 = simCopula[,1]
X2 = simCopula[,2]
partition = cbind(Z <= 0.3, Z > 0.3 & Z <= 0.5, Z > 0.5)
condPseudoObs = bCond.pobs(X = cbind(X1, X2), partition = partition)

estimatedCondCopulas = bCond.estParamCopula(
  U1 = condPseudoObs[,1], U2 = condPseudoObs[,2],
  family = 1, partition = partition)
print(estimatedCondCopulas)
# Comparison with the true conditional parameters: 0.2, 0.5, -0.8.

Computing the pseudo-observations in case of discrete conditioning events

Description

Let A_1, ..., A_p be p events forming a partition of a probability space and X_1, ..., X_d be d random variables. Assume that we observe n i.i.d. replications of (X_1, ..., X_d), and that for each i=1, ..., d,

V_{i,j|A} = F_{X_j | A_k}(X_{i,j} | A_k),

we also know which of the A_k was realized. This function computes the pseudo-observations where k is such that the event A_k is realized for the i-th observation.

Usage

bCond.pobs(X, partition)

Arguments

Value

a matrix of size n * d containing the conditional pseudo-observations V_{i,j|A}.

References

Derumigny, A., & Fermanian, J. D. (2017). About tests of the “simplifying” assumption for conditional copulas. Dependence Modeling, 5(1), 154-197. doi:10.1515/demo-2017-0011

Derumigny, A., & Fermanian, J. D. (2022) Conditional empirical copula processes and generalized dependence measures Electronic Journal of Statistics, 16(2), 5692-5719. doi:10.1214/22-EJS2075

See Also

bCond.estParamCopula for the estimation of a (conditional) parametric copula model in this framework.

bCond.treeCKT that provides a binary tree based on conditional Kendall's tau and that can be used to derive relevant conditioning events.

Examples

n = 800
Z = stats::runif(n = n)
CKT = 0.2 * as.numeric(Z <= 0.3) +
  0.5 * as.numeric(Z > 0.3 & Z <= 0.5) +
  - 0.8 * as.numeric(Z > 0.5)
simCopula = VineCopula::BiCopSim(N = n,
  par = VineCopula::BiCopTau2Par(CKT, family = 1), family = 1)
X1 = simCopula[,1]
X2 = simCopula[,2]
partition = cbind(Z <= 0.3, Z > 0.3 & Z <= 0.5, Z > 0.5)
condPseudoObs = bCond.pobs(X = cbind(X1, X2),
                           partition = partition)

Function for testing the simplifying assumption with data-driven box-type conditioning events

Description

This function takes in parameter the matrix of (observations) of the conditioned variables and either matrixInd, a matrix of indicator variables describing which events occur for which observations

Usage

bCond.simpA.CKT(
  XI,
  XJ = NULL,
  matrixInd = NULL,
  minCut = 0,
  minProb = 0.01,
  minSize = minProb * nrow(XI),
  nPoints_xJ = 10,
  type.quantile = 7,
  verbose = 2,
  methodTree = "doSplit",
  propTree = 0.5,
  methodPvalue = "bootNP",
  nBootstrap = 100
)

Arguments

Value

a list with the following components

• p.value the estimated p-value.

• stat the test statistic.

• treeCKT the estimated tree if matrixInd is not provided.

• vec_statB the vector of bootstrapped statistics if methodPvalue is not covMatrix.

Author(s)

Alexis Derumigny, Jean-David Fermanian and Aleksey Min

References

Derumigny, A., Fermanian, J. D., & Min, A. (2022). Testing for equality between conditional copulas given discretized conditioning events. Canadian Journal of Statistics. doi:10.1002/cjs.11742

Derumigny, A., & Fermanian, J. D. (2022) Conditional empirical copula processes and generalized dependence measures Electronic Journal of Statistics, 16(2), 5692-5719. doi:10.1214/22-EJS2075

See Also

bCond.simpA.param for a test of this simplifying assumption in a parametric framework.

bCond.treeCKT provides the binary tree that is used in this function (if matrixInd is not provided).

Tests of the simplifying assumption for conditional copulas with a continuous conditioning variable:

• simpA.NP in a nonparametric setting

• simpA.param in a (semi)parametric setting, where the conditional copula belongs to a parametric family, but the conditional margins are estimated arbitrarily through kernel smoothing

• simpA.kendallReg: test based on the constancy of conditional Kendall's tau

Examples

set.seed(1)
n = 200
XJ = MASS::mvrnorm(n = n, mu = c(3,3), Sigma = rbind(c(1, 0.2), c(0.2, 1)))
XI = matrix(nrow = n, ncol = 2)
high_XJ1 = which(XJ[,1] > 4)
XI[high_XJ1, ]  = MASS::mvrnorm(n = length(high_XJ1), mu = c(10,10),
                                Sigma = rbind(c(1, 0.8), c(0.8, 1)))
XI[-high_XJ1, ] = MASS::mvrnorm(n = n - length(high_XJ1), mu = c(8,8),
                                Sigma = rbind(c(1, -0.2), c(-0.2, 1)))

result = bCond.simpA.CKT(XI = XI, XJ = XJ, minSize = 10, verbose = 2,
                         methodTree = "doSplit", nBootstrap = 4)
print(result$p.value)
result2 = bCond.simpA.CKT(XI = XI, XJ = XJ, minSize = 10, verbose = 2,
                          methodTree = "noSplit", nBootstrap = 4)
print(result2$p.value)

Test of the assumption that a conditional copulas does not vary through a list of discrete conditioning events

Description

Test of the assumption that a conditional copulas does not vary through a list of discrete conditioning events

Usage

bCond.simpA.param(
  X1,
  X2,
  partition,
  family,
  testStat = "T2c_tau",
  typeBoot = "boot.NP",
  nBootstrap = 100
)

Arguments

Value

a list containing

• true_stat: the value of the test statistic computed on the whole sample

• vect_statB: a vector of length nBootstrap containing the bootstrapped test statistics.

• p_val: the p-value of the test.

References

Derumigny, A., & Fermanian, J. D. (2017). About tests of the “simplifying” assumption for conditional copulas. Dependence Modeling, 5(1), 154-197. doi:10.1515/demo-2017-0011

Derumigny, A., & Fermanian, J. D. (2022) Conditional empirical copula processes and generalized dependence measures Electronic Journal of Statistics, 16(2), 5692-5719. doi:10.1214/22-EJS2075

See Also

bCond.estParamCopula for the estimation of a (conditional) parametric copula model in this framework.

bCond.simpA.CKT for a test of the simplifying assumption that all these conditional copulas are equal, based on the equality of conditional Kendall's tau (i.e. without any parametric assumption).

Tests of the simplifying assumption for conditional copulas with a continuous conditioning variable:

• simpA.NP in a nonparametric setting

• simpA.param in a (semi)parametric setting, where the conditional copula belongs to a parametric family, but the conditional margins are estimated arbitrarily through kernel smoothing

• simpA.kendallReg: test based on the constancy of conditional Kendall's tau

Examples

n = 800
Z = stats::runif(n = n)
CKT = 0.2 * as.numeric(Z <= 0.3) +
  0.5 * as.numeric(Z > 0.3 & Z <= 0.5) +
  + 0.3 * as.numeric(Z > 0.5)
family = 3
simCopula = VineCopula::BiCopSim(N = n,
  par = VineCopula::BiCopTau2Par(CKT, family = family), family = family)
X1 = simCopula[,1]
X2 = simCopula[,2]
partition = cbind(Z <= 0.3, Z > 0.3 & Z <= 0.5, Z > 0.5)

result = bCond.simpA.param(X1 = X1, X2 = X2, testStat = "T2c_tau",
  partition = partition, family = family, typeBoot = "boot.paramInd")
print(result$p_val)

n = 800
Z = stats::runif(n = n)
CKT = 0.1
family = 3
simCopula = VineCopula::BiCopSim(N = n,
  par = VineCopula::BiCopTau2Par(CKT, family = family), family = family)
X1 = simCopula[,1]
X2 = simCopula[,2]
partition = cbind(Z <= 0.3, Z > 0.3 & Z <= 0.5, Z > 0.5)

result = bCond.simpA.param(X1 = X1, X2 = X2,
  partition = partition, family = family, typeBoot = "boot.NP")
print(result$p_val)

Construct a binary tree for the modeling the conditional Kendall's tau

Description

This function takes in parameter two matrices of observations: the first one contains the observations of XI (the conditioned variables) and the second on contains the observations of XJ (the conditioning variables). The goal of this procedure is to find which of the variables in XJ have important influence on the dependence between the components of XI, (measured by the Kendall's tau).

Usage

bCond.treeCKT(
  XI,
  XJ,
  minCut = 0,
  minProb = 0.01,
  minSize = minProb * nrow(XI),
  nPoints_xJ = 10,
  type.quantile = 7,
  verbose = 2
)

Arguments

Details

The object return by this function is a binary tree. Each leaf of this tree correspond to one event (or, equivalently, one subset of R^{dim(XJ)}), and the conditional Kendall's tau conditionally to it.

Value

the estimated tree using the data 'XI, XJ'.

References

Derumigny, A., Fermanian, J. D., & Min, A. (2022). Testing for equality between conditional copulas given discretized conditioning events. Canadian Journal of Statistics. doi:10.1002/cjs.11742

See Also

bCond.simpA.CKT for a test of the simplifying assumption that all these conditional Kendall's tau are equal.

treeCKT2matrixInd for converting this tree to a matrix of indicators of each event. matrixInd2matrixCKT for getting the matrix of estimated conditional Kendall's taus for each event.

CKT.estimate for the estimation of pointwise conditional Kendall's tau, i.e. assuming a continuous conditioning variable Z.

Examples

set.seed(1)
n = 400
XJ = MASS::mvrnorm(n = n, mu = c(3,3), Sigma = rbind(c(1, 0.2), c(0.2, 1)))
XI = matrix(nrow = n, ncol = 2)
high_XJ1 = which(XJ[,1] > 4)
XI[high_XJ1, ]  = MASS::mvrnorm(n = length(high_XJ1), mu = c(10,10),
                                Sigma = rbind(c(1, 0.8), c(0.8, 1)))
XI[-high_XJ1, ] = MASS::mvrnorm(n = n - length(high_XJ1), mu = c(8,8),
                                Sigma = rbind(c(1, -0.2), c(-0.2, 1)))

result = bCond.treeCKT(XI = XI, XJ = XJ, minSize = 50, verbose = 2)
# Plotting the corresponding tree using the "DiagrammeR" package
if (requireNamespace("DiagrammeR", quietly = TRUE)){
  plot(result)
}

# Number of observations in the first two children
print(length(data.tree::GetAttribute(result$children[[1]], "condObs")))
print(length(data.tree::GetAttribute(result$children[[2]], "condObs")))

Computing the kernel matrix

Description

This function computes a matrix of dimensions (length(observedX3), length(newX3)), whose element at coordinate (i,j) is K_{h}(observedX3[i] - newX3[j] ), where K_h(x) := K(x/h) / h and K is the kernel.

Usage

computeKernelMatrix(observedX, newX, kernel, h)

Arguments

Value

a numeric matrix of dimensions (length(observedX), length(newX))

See Also

estimateCondCDF_matrix, estimateCondCDF_vec,

Examples

Y = MASS::mvrnorm(n = 100, mu = c(0,0), Sigma = cbind(c(1, 0.9), c(0.9, 1)))
matrixK = computeKernelMatrix(observedX = Y[,2], newX = c(0, 1, 2.5),
kernel = "Gaussian", h = 0.8)

# To have an estimator of the conditional expectation of Y1 given Y2 = 0, 1, 2.5
Y[,1] * matrixK[,1] / sum(matrixK[,1])
Y[,1] * matrixK[,2] / sum(matrixK[,2])
Y[,1] * matrixK[,3] / sum(matrixK[,3])

Compute the matrix of signs of pairs

Description

Compute a matrix giving the concordance or discordance of each pair of observations.

Usage

computeMatrixSignPairs(vectorX1, vectorX2, typeEstCKT = 4)

Arguments

Value

an n * n matrix with the signs of each pair of observations.

Examples

# We simulate from a conditional copula
N = 500
Z = rnorm(n = N, mean = 5, sd = 2)
conditionalTau = 0.9 * pnorm(Z, mean = 5, sd = 2)
simCopula = VineCopula::BiCopSim(N = N , family = 3,
    par = VineCopula::BiCopTau2Par(1 , conditionalTau) )
matrixPairs = computeMatrixSignPairs(vectorX1 = simCopula[,1],
                                     vectorX2 = simCopula[,2])

Converting to matrix of indicators / matrix of conditional Kendall's tau

Description

The function treeCKT2matrixInd takes as input a binary tree that has been returned by the function bCond.treeCKT. Since this tree describes a partition of the conditioning space, it can be interesting to get, for a given dataset, the matrix

1\{ X_{i,J} \in A_{j,J} \},

where each A_{j,J} corresponds to a conditioning subset. This is the so-called matrixInd. Finally, it can be interesting to get the matrix of

Usage

treeCKT2matrixInd(estimatedTree, newDataXJ = NULL)

matrixInd2matrixCKT(matrixInd, newDataXI)

treeCKT2matrixCKT(estimatedTree, newDataXI = NULL, newDataXJ = NULL)

Arguments

Value

• The function treeCKT2matrixInd returns a matrix of size N * m which component [i,j] is

  1\{ X_{i,J} \in A_{j,J} \}

  .

• The function matrixInd2matrixCKT and treeCKT2matrixCKT return a matrix of size |I| * (|I|-1) * m where each component corresponds to a conditional Kendall's tau between a pair of conditional variables conditionally to the conditioned variables in one of the boxes

See Also

bCond.treeCKT for the construction of such a binary tree.

Examples

set.seed(1)
n = 200
XJ = MASS::mvrnorm(n = n, mu = c(3,3), Sigma = rbind(c(1, 0.2), c(0.2, 1)))
XI = matrix(nrow = n, ncol = 2)
high_XJ1 = which(XJ[,1] > 4)
XI[high_XJ1, ]  = MASS::mvrnorm(n = length(high_XJ1), mu = c(10,10),
                                Sigma = rbind(c(1, 0.8), c(0.8, 1)))
XI[-high_XJ1, ] = MASS::mvrnorm(n = n - length(high_XJ1), mu = c(8,8),
                                Sigma = rbind(c(1, -0.2), c(-0.2, 1)))

result = bCond.treeCKT(XI = XI, XJ = XJ, minSize = 10, verbose = 2)

treeCKT2matrixInd(result)

matrixInd2matrixCKT(treeCKT2matrixInd(result), newDataXI = XI)

treeCKT2matrixCKT(result)

Construct a dataset of pairs of observations for the estimation of conditional Kendall's tau

Description

In (Derumigny, & Fermanian (2019)), it is described how the problem of estimating conditional Kendall's tau can be rewritten as a classification task for a dataset of pairs (of observations). This function computes such a dataset, that can be then used to estimate conditional Kendall's tau using one of the following functions: CKT.fit.tree, CKT.fit.randomForest, CKT.fit.GLM, CKT.fit.nNets, CKT.predict.kNN.

Usage

datasetPairs(
  X1,
  X2,
  Z,
  h,
  cut = 0.9,
  onlyConsecutivePairs = FALSE,
  nPairs = NULL
)

Arguments

Value

A matrix with (4+dimZ) columns and n*(n-1)/2 rows if onlyConsecutivePairs=FALSE and else (n/2) rows. It is structured in the following way:

• column 1 contains the information about the concordance of the pair (i,j) ;

• columns 2 to 1+dimZ contain the mean value of Z (the conditioning variables) ;

• column 2+dimZ contains the value of the kernel K_h(Z_j - Z_i) ;

• column 3+dimZ and 4+dimZ contain the corresponding values of i and j.

References

Derumigny, A., & Fermanian, J. D. (2019). A classification point-of-view about conditional Kendall’s tau. Computational Statistics & Data Analysis, 135, 70-94. (Algorithm 1 for all pairs and Algorithm 8 for the case of only consecutive pairs) doi:10.1016/j.csda.2019.01.013

See Also

the functions that require such a dataset of pairs to do the estimation of conditional Kendall's tau: CKT.fit.tree, CKT.fit.randomForest, CKT.fit.GLM, CKT.fit.nNets, CKT.predict.kNN, and CKT.fit.randomForest.

Examples

# We simulate from a conditional copula
N = 500
Z = rnorm(n = N, mean = 5, sd = 2)
conditionalTau = 0.9 * pnorm(Z, mean = 5, sd = 2)
simCopula = VineCopula::BiCopSim(N = N , family = 3,
    par = VineCopula::BiCopTau2Par(1 , conditionalTau) )
X1 = qnorm(simCopula[,1])
X2 = qnorm(simCopula[,2])

datasetP = datasetPairs(
X1 = X1, X2 = X2, Z = Z, h = 0.07, cut = 0.9)

Compute kernel-based conditional marginal (univariate) cdfs

Description

This function computes an estimate of the conditional (marginal) cdf of X1 given a conditioning variable X3.

Usage

estimateCondCDF_matrix(observedX1, newX1, matrixK3)

Arguments

Details

This function is supposed to be used with computeKernelMatrix. Assume that we observe a sample (X_{i,1}, X_{i,3}), i=1, \dots, n. We want to estimate the conditional cdf of X_1 given X_3 = x_3 at point x_1 using the following kernel-based estimator

\hat P(X_1 \le x_1 | X_3 = x_3) := \frac{\sum_{l=1}^n 1 \{X_{l,1} \leq x_1 \} K_h(X_{l,3} - x_3)} {\sum_{l=1}^n K_h(X_{l,3} - x_3)},

for every x_1 in newX1 and every x_3 in newX3. The matrixK3 should be a matrix of the values K_h(X_{l,3} - x_3) such as the one produced by computeKernelMatrix(observedX3, newX3, kernel, h).

Value

A matrix of dimensions (p1 = length(newX), p3 = length(matrixK3[,1])) of estimators \hat P(X_1 \leq x_1 | X_3 = x_3) for every possible choices of (x_1, x_3).

Examples

Y = MASS::mvrnorm(n = 100, mu = c(0,0), Sigma = cbind(c(1, 0.9), c(0.9, 1)))
newY1 = seq(-1, 1, by = 0.5)
newY2 = c(0, 1, 2)
matrixK = computeKernelMatrix(observedX = Y[,2], newX = newY2,
  kernel = "Gaussian", h = 0.8)
# In this matrix, there are the estimated conditionl cdf at points given by newY1
# conditionally to the points given by newY2.
matrixCondCDF = estimateCondCDF_matrix(observedX1 = Y[,1],
  newX1 = newY1, matrixK)
matrixCondCDF

Compute kernel-based conditional marginal (univariate) cdfs

Description

This function computes an estimate of the conditional (marginal) cdf of X1 given a conditioning variable X3. This function is supposed to be used with computeKernelMatrix. Assume that we observe a sample (X_{i,1}, X_{i,3}), i=1, \dots, n. We want to estimate the conditional cdf of X_1 given X_3 = x_3 at point x_1 using the following kernel-based estimator

\hat P(X_1 \leq x_1 | X_3 = x_3) := \frac{\sum_{l=1}^n 1 \{X_{l,1} \leq x_1 \} K_h(X_{l,3} - x_3)} {\sum_{l=1}^n K_h(X_{l,3} - x_3)},

for every couple (x_{j,1}, x_{j,3}) where x_{j,1} in newX1 and x_{j,3} in newX3. The matrixK3 should be a matrix of the values K_h(X_{l,3} - x_3) such as the one produced by computeKernelMatrix(observedX3, newX3, kernel, h).

Usage

estimateCondCDF_vec(observedX1, newX1, matrixK3)

Arguments

Value

It returns a vector of length newX1 of estimators \hat P(X_1 \leq x_1 | X_3 = x_3) for every couple (x_{j,1}, x_{j,3}).

Examples

Y = MASS::mvrnorm(n = 100, mu = c(0,0), Sigma = cbind(c(1, 0.9), c(0.9, 1)))
newY1 = seq(-1, 1, by = 0.5)
newY2 = newY1
matrixK = computeKernelMatrix(observedX = Y[,2], newX = newY2,
  kernel = "Gaussian", h = 0.8)
vecCondCDF = estimateCondCDF_vec(observedX1 = Y[,1],
  newX1 = newY1, matrixK)
vecCondCDF

Compute kernel-based conditional quantiles

Description

This function is supposed to be used with computeKernelMatrix. Assume that we observe a sample (X_{i,1}, X_{i,3}), i=1, \dots, n. We want to estimate the conditional quantiles of X_1 given X_3 = x_3 at point u_1 using the following kernel-based estimator

\hat Q(u_1 | X_3 = x_3) := \hat P^{(-1)}(u_1 \leq x_1 | X_3 = x_3),

where

\hat P(X_1 \leq x_1 | X_3 = x_3) := \frac{\sum_{l=1}^n 1 \{X_(l,1) \leq x_1 \} K_h(X_(l,3) - x_3)} {\sum_{l=1}^n K_h(X_(l,3) - x_3)},

for every u_1 in probsX1 and every x_3 in newX3. The matrixK3 should be a matrix of the values K_h(X_(l,3) - x_3) such as the one produced by computeKernelMatrix(observedX3, newX3, kernel, h).

Usage

estimateCondQuantiles(observedX1, probsX1, matrixK3)

Arguments

Value

A matrix of dimensions (p1,p2) whose (i,j) entry is \hat Q(u_1 | X_3 = x_3) with u_1 = probsX1[i] and x_3 = newX3[j], where newX3[j] is the vector that was used to construct matrixK3.

Examples

Y = MASS::mvrnorm(n = 100, mu = c(0,0), Sigma = cbind(c(1, 0.9), c(0.9, 1)))
matrixK = computeKernelMatrix(observedX = Y[,2] , newX = c(0, 1, 2.5),
  kernel = "Gaussian", h = 0.8)
matrixnp = estimateCondQuantiles(observedX1 = Y[,2],
  probsX1 = c(0.3, 0.5) , matrixK3 = matrixK)
matrixnp

Compute a kernel-based estimator of the conditional copula

Description

Assuming that we observe a sample (X_{i,1}, X_{i,2}, X_{i,3}), i=1, \dots, n, this function returns a array \hat C_{1,2|3}(u_1, u_2 | X_3 = x_3) for each choice of (u_1, u_2, x_3).

Usage

estimateNPCondCopula(
  X1 = NULL,
  X2 = NULL,
  X3 = NULL,
  U1_,
  U2_,
  newX3,
  kernel,
  h,
  observedX1 = NULL,
  observedX2 = NULL,
  observedX3 = NULL
)

Arguments

Value

An array of dimension (length(U1_, U2_, newX3)) whose element in position (i, j, k) is \hat C_{1,2|3}(u_1, u_2 | X_3 = x_3) where u_1 = U1_[i], u_2 = U2_[j] and x_3 = newX3[k]

References

Derumigny, A., & Fermanian, J. D. (2017). About tests of the “simplifying” assumption for conditional copulas. Dependence Modeling, 5(1), 154-197. doi:10.1515/demo-2017-0011

See Also

estimateParCondCopula for estimating a conditional copula in a parametric setting ( = where the conditional copula is assumed to belong to a parametric class). simpA.NP for a test that this conditional copula is constant with respect to the value x_3 of the conditioning variable.

Examples

# We simulate from a conditional copula
N = 500
X3 = rnorm(n = N, mean = 5, sd = 2)
conditionalTau = 0.9 * pnorm(X3, mean = 5, sd = 2)
simCopula = VineCopula::BiCopSim(N=N , family = 3,
    par = VineCopula::BiCopTau2Par(1 , conditionalTau ))
X1 = qnorm(simCopula[,1])
X2 = qnorm(simCopula[,2])

# We do the estimation
grid = c(0.2, 0.4, 0.6, 0.8)
arrayEst = estimateNPCondCopula(
  X1 = X1, X2 = X2, X3 = X3,
  U1_ = grid, U2_ = grid, newX3 = c(2, 5, 7),
  kernel = "Gaussian", h = 0.8)
arrayEst

Estimation of parametric conditional copulas

Description

The function estimateParCondCopula computes an estimate of the conditional parameters in a conditional parametric copula model, i.e.

C_{X_1, X_2 | X_3 = x_3} = C_{\theta(x_3)},

for some parametric family (C_\theta), some conditional parameter \theta(x_3), and a three-dimensional random vector (X_1, X_2, X_3). Remember that C_{X_1,X_2 | X_3 = x_3} denotes the conditional copula of X_1 and X_2 given X_3 = x_3.

The function estimateParCondCopula_ZIJ is an auxiliary function that is called when conditional pseudos-observations are already available when one wants to estimate a parametric conditional copula.

Usage

estimateParCondCopula(
  X1 = NULL,
  X2 = NULL,
  X3 = NULL,
  newX3,
  family,
  method = "mle",
  h,
  observedX1 = NULL,
  observedX2 = NULL,
  observedX3 = NULL
)

estimateParCondCopula_ZIJ(Z1_J, Z2_J, observedX3, newX3, family, method, h)

Arguments

Value

a vector of size length(newX3) containing the estimated conditional copula parameters for each value of newX3.

References

Derumigny, A., & Fermanian, J. D. (2017). About tests of the “simplifying” assumption for conditional copulas. Dependence Modeling, 5(1), 154-197. doi:10.1515/demo-2017-0011

See Also

estimateNPCondCopula for estimating a conditional copula in a nonparametric setting ( = without parametric assumption on the conditional copula). simpA.param for a test that this conditional copula is constant with respect to the value x_3 of the conditioning variable.

Examples

# We simulate from a conditional copula
N = 500

X3 = rnorm(n = N, mean = 5, sd = 2)
conditionalTau = 0.9 * pnorm(X3, mean = 5, sd = 2)
simCopula = VineCopula::BiCopSim(
    N=N , family = 1, par = VineCopula::BiCopTau2Par(1 , conditionalTau ))
X1 = qnorm(simCopula[,1])
X2 = qnorm(simCopula[,2])

gridnewX3 = seq(2, 8, by = 1)
conditionalTauNewX3 = 0.9 * pnorm(gridnewX3, mean = 5, sd = 2)

vecEstimatedThetas = estimateParCondCopula(
  X1 = X1, X2 = X2, X3 = X3,
  newX3 = gridnewX3, family = 1, h = 0.1)

# Estimated conditional parameters
vecEstimatedThetas
# True conditional parameters
VineCopula::BiCopTau2Par(1 , conditionalTauNewX3 )

# Estimated conditional Kendall's tau
VineCopula::BiCopPar2Tau(1 , vecEstimatedThetas )
# True conditional Kendall's tau
conditionalTauNewX3

Nonparametric testing of the simplifying assumption

Description

This function tests the “simplifying assumption” that a conditional copula

C_{1,2|3}(u_1, u_2 | X_3 = x_3)

does not depend on the value of the conditioning variable x_3 in a nonparametric setting, where the conditional copula is estimated by kernel smoothing.

Usage

simpA.NP(
  X1,
  X2,
  X3,
  testStat,
  typeBoot = "bootNP",
  h,
  nBootstrap = 100,
  kernel.name = "Epanechnikov",
  truncVal = h,
  numericalInt = list(kind = "legendre", nGrid = 10)
)

Arguments

Value

a list containing

• true_stat: the value of the test statistic computed on the whole sample

• vect_statB: a vector of length nBootstrap containing the bootstrapped test statistics.

• p_val: the p-value of the test.

References

Derumigny, A., & Fermanian, J. D. (2017). About tests of the “simplifying” assumption for conditional copulas. Dependence Modeling, 5(1), 154-197. doi:10.1515/demo-2017-0011

See Also

Other tests of the simplifying assumption:

• simpA.param in a (semi)parametric setting, where the conditional copula belongs to a parametric family, but the conditional margins are estimated arbitrarily through kernel smoothing

• simpA.kendallReg: test based on the constancy of conditional Kendall's tau

• the counterparts of these tests in the discrete conditioning setting: bCond.simpA.CKT (test based on conditional Kendall's tau) bCond.simpA.param (test assuming a parametric form for the conditional copula)

Examples

# We simulate from a conditional copula
set.seed(1)
N = 500
Z = rnorm(n = N, mean = 5, sd = 2)
conditionalTau = -0.9 + 1.8 * pnorm(Z, mean = 5, sd = 2)
simCopula = VineCopula::BiCopSim(N=N , family = 1,
    par = VineCopula::BiCopTau2Par(1 , conditionalTau ))
X1 = qnorm(simCopula[,1], mean = Z)
X2 = qnorm(simCopula[,2], mean = - Z)

result <- simpA.NP(
   X1 = X1, X2 = X2, X3 = Z,
   testStat = "I_chi", typeBoot = "boot.pseudoInd",
   h = 0.03, kernel.name = "Epanechnikov", nBootstrap = 10)

# In practice, it is recommended to use at least nBootstrap = 100
# with nBootstrap = 200 being a good choice.

print(result$p_val)

set.seed(1)
N = 500
Z = rnorm(n = N, mean = 5, sd = 2)
conditionalTau = 0.8
simCopula = VineCopula::BiCopSim(N=N , family = 1,
    par = VineCopula::BiCopTau2Par(1 , conditionalTau ))
X1 = qnorm(simCopula[,1], mean = Z)
X2 = qnorm(simCopula[,2], mean = - Z)

result <- simpA.NP(
   X1 = X1, X2 = X2, X3 = Z,
   testStat = "I_chi", typeBoot = "boot.pseudoInd",
   h = 0.08, kernel.name = "Epanechnikov", nBootstrap = 10)
print(result$p_val)

Test of the simplifying assumption using the constancy of conditional Kendall's tau

Description

This function computes Kendall's regression, a regression-like model for conditional Kendall's tau. More precisely, it fits the model

\Lambda(\tau_{X_1, X_2 | Z = z}) = \sum_{j=1}^{p'} \beta_j \psi_j(z),

where \tau_{X_1, X_2 | Z = z} is the conditional Kendall's tau between X_1 and X_2 conditionally to Z=z, \Lambda is a function from ]-1, 1] to R, (\beta_1, \dots, \beta_p) are unknown coefficients to be estimated and \psi_1, \dots, \psi_{p'}) are a dictionary of functions. Then, this function tests the assumption

\beta_2 = \beta_3 = ... = \beta_{p'} = 0,

where the coefficient corresponding to the intercept is removed.

Usage

simpA.kendallReg(
  X1,
  X2,
  Z,
  vectorZToEstimate = NULL,
  listPhi = list(z = function(z) {
     return(z)
 }),
  typeEstCKT = 4,
  h_kernel,
  Lambda = function(x) {
     return(x)
 },
  Lambda_deriv = function(x) {
     return(1)
 },
  Lambda_inv = function(x) {
     return(x)
 },
  lambda = NULL,
  h_lambda = h_kernel,
  Kfolds_lambda = 5,
  l_norm = 1
)

## S3 method for class 'simpA_kendallReg_test'
coef(object, ...)

## S3 method for class 'simpA_kendallReg_test'
vcov(object, ...)

## S3 method for class 'simpA_kendallReg_test'
print(x, ...)

## S3 method for class 'simpA_kendallReg_test'
plot(x, ylim = c(-1.5, 1.5), ...)

Arguments

Value

simpA.kendallReg returns an S3 object of class simpA_kendallReg_test, containing

• statWn: the value of the test statistic.

• p_val: the p-value of the test.

plot.simpA_kendallReg_test returns (invisibly) a matrix with columns z, est_CKT_NP, asympt_se_np, est_CKT_NP_q025, est_CKT_NP_q975, est_CKT_reg, asympt_se_reg, est_CKT_reg_q025, est_CKT_reg_q975. The first column correspond to the grid of values of z. The next 4 columns are the NP (kernel-based) estimator of conditional Kendall's tau, with its standard error, and lower/upper confidence bands. The last 4 columns are the equivalents for the estimator based on Kendall's regression.

plot.simpA_kendallReg_test plots the kernel-based estimator and its confidence band (in red), and the estimator based on Kendall's regression and its confidence band (in blue).

Usually the confidence band for Kendall's regression is much tighter than the pure non-parametric counterpart. This is because the parametric model is sparser and the corresponding estimator converges faster (even without penalization).

print.simpA_kendallReg_test has no return values and is only called for its side effects.

Function coef.simpA_kendallReg_test returns the matrix of coefficients with standard errors, z values and p-values.

Function vcov.simpA_kendallReg_test returns the (estimated) variance-covariance matrix of the estimated coefficients.

References

Derumigny, A., & Fermanian, J. D. (2020). On Kendall’s regression. Journal of Multivariate Analysis, 178, 104610. (page 7) doi:10.1016/j.jmva.2020.104610

See Also

The function to fit Kendall's regression: CKT.kendallReg.fit.

Other tests of the simplifying assumption:

• simpA.NP in a nonparametric setting

• simpA.param in a (semi)parametric setting, where the conditional copula belongs to a parametric family, but the conditional margins are estimated arbitrarily through kernel smoothing

• the counterparts of these tests in the discrete conditioning setting: bCond.simpA.CKT (test based on conditional Kendall's tau) bCond.simpA.param (test assuming a parametric form for the conditional copula)

Examples

# We simulate from a non-simplified conditional copula
set.seed(1)
N = 300
Z = runif(n = N, min = 0, max = 1)
conditionalTau = -0.9 + 1.8 * Z
simCopula = VineCopula::BiCopSim(N=N , family = 1,
    par = VineCopula::BiCopTau2Par(1 , conditionalTau ))
X1 = qnorm(simCopula[,1])
X2 = qnorm(simCopula[,2])

result = simpA.kendallReg(
  X1, X2, Z, h_kernel = 0.03,
  listPhi = list(z = function(z){return(z)} ) )
print(result)
plot(result)
# Obtain matrix of coefficients, std err, z values and p values
coef(result)
# Obtain variance-covariance matrix of the coefficients
vcov(result)

result_morePhi = simpA.kendallReg(
   X1, X2, Z, h_kernel = 0.03,
   listPhi = list(
     z = function(z){return(z)},
     cos10z = function(z){return(cos(10 * z))},
     sin10z = function(z){return(sin(10 * z))},
     `1(z <= 0.4)` = function(z){return(as.numeric(z <= 0.4))},
     `1(z <= 0.6)` = function(z){return(as.numeric(z <= 0.6))}) )
print(result_morePhi)
plot(result_morePhi)

# We simulate from a simplified conditional copula
set.seed(1)
N = 300
Z = runif(n = N, min = 0, max = 1)
conditionalTau = -0.3
simCopula = VineCopula::BiCopSim(N=N , family = 1,
    par = VineCopula::BiCopTau2Par(1 , conditionalTau ))
X1 = qnorm(simCopula[,1])
X2 = qnorm(simCopula[,2])

result = simpA.kendallReg(
   X1, X2, Z, h_kernel = 0.03,
   listPhi = list(
     z = function(z){return(z)},
     cos10z = function(z){return(cos(10 * z))},
     sin10z = function(z){return(sin(10 * z))},
     `1(z <= 0.4)` = function(z){return(as.numeric(z <= 0.4))},
     `1(z <= 0.6)` = function(z){return(as.numeric(z <= 0.6))}) )
print(result)
plot(result)

Semiparametric testing of the simplifying assumption

Description

This function tests the “simplifying assumption” that a conditional copula

C_{1,2|3}(u_1, u_2 | X_3 = x_3)

does not depend on the value of the conditioning variable x_3 in a semiparametric setting, where the conditional copula is of the form

C_{1,2|3}(u_1, u_2 | X_3 = x_3) = C_{\theta(x_3)}(u_1,u_2),

for all 0 <= u_1, u_2 <= 1 and all x_3. Here, (C_\theta) is a known family of copula and \theta(x_3) is an unknown conditional dependence parameter. In this setting, the simplifying assumption can be rewritten as “\theta(x_3) does not depend on x_3, i.e. is a constant function of x_3”.

Usage

simpA.param(
  X1,
  X2,
  X3,
  family,
  testStat = "T2c",
  typeBoot = "boot.NP",
  h,
  nBootstrap = 100,
  kernel.name = "Epanechnikov",
  truncVal = h,
  numericalInt = list(kind = "legendre", nGrid = 10)
)

Arguments

Value

a list containing

• true_stat: the value of the test statistic computed on the whole sample

• vect_statB: a vector of length nBootstrap containing the bootstrapped test statistics.

• p_val: the p-value of the test.

References

Derumigny, A., & Fermanian, J. D. (2017). About tests of the “simplifying” assumption for conditional copulas. Dependence Modeling, 5(1), 154-197. doi:10.1515/demo-2017-0011

See Also

Other tests of the simplifying assumption:

• simpA.NP in a nonparametric setting

• simpA.kendallReg: test based on the constancy of conditional Kendall's tau

• the counterparts of these tests in the discrete conditioning setting: bCond.simpA.CKT (test based on conditional Kendall's tau) bCond.simpA.param (test assuming a parametric form for the conditional copula)

Examples

# We simulate from a conditional copula
set.seed(1)
N = 500
Z = rnorm(n = N, mean = 5, sd = 2)
conditionalTau = -0.9 + 1.8 * pnorm(Z, mean = 5, sd = 2)
simCopula = VineCopula::BiCopSim(N=N , family = 1,
    par = VineCopula::BiCopTau2Par(1 , conditionalTau ))
X1 = qnorm(simCopula[,1], mean = Z)
X2 = qnorm(simCopula[,2], mean = - Z)

result <- simpA.param(
   X1 = X1, X2 = X2, X3 = Z, family = 1,
   h = 0.03, kernel.name = "Epanechnikov", nBootstrap = 5)
print(result$p_val)
# In practice, it is recommended to use at least nBootstrap = 100
# with nBootstrap = 200 being a good choice.


set.seed(1)
N = 500
Z = rnorm(n = N, mean = 5, sd = 2)
conditionalTau = 0.8
simCopula = VineCopula::BiCopSim(N=N , family = 1,
    par = VineCopula::BiCopTau2Par(1 , conditionalTau ))
X1 = qnorm(simCopula[,1], mean = Z)
X2 = qnorm(simCopula[,2], mean = - Z)

result <- simpA.param(
   X1 = X1, X2 = X2, X3 = Z, family = 1,
   h = 0.08, kernel.name = "Epanechnikov", nBootstrap = 5)
print(result$p_val)