simpA.kendallReg: 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 simpA_kendallReg_test
se(object, type, ...)
# S3 method for simpA_kendallReg_test
coef(object, ...)
# S3 method for simpA_kendallReg_test
vcov(object, ...)
# S3 method for simpA_kendallReg_test
print(x, ...)
# S3 method for simpA_kendallReg_test
plot(x, ylim = c(-1.5, 1.5), ...)

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.

Function se.simpA_kendallReg_test returns the standard errors of the estimated coefficients or of the estimated conditional Kendall's taus.

Arguments

X1

vector of observations of the first conditioned variable

X2

vector of observations of the second conditioned variable

Z

vector of observations of the conditioning variable

vectorZToEstimate

vector containing the points $Z'_i$ to be used at which the conditional Kendall's tau should be estimated.

listPhi

the list of transformations $phi$ to be used.

typeEstCKT

the type of estimation of the kernel-based estimation of conditional Kendall's tau.

h_kernel

the bandwidth used for the kernel-based estimations.

Lambda

the function to be applied on conditional Kendall's tau. By default, the identity function is used.

Lambda_deriv

the derivative of the function Lambda.

Lambda_inv

the inverse function of Lambda.

lambda

the penalization parameter used for Kendall's regression. By default, cross-validation is used to find the best value of lambda if length(listPhi) > 1. Otherwise lambda = 0 is used.

h_lambda

bandwidth used for the smooth cross-validation in order to get a value for lambda.

Kfolds_lambda

the number of subsets used for the cross-validation in order to get a value for lambda.

l_norm

type of norm used for selection of the optimal lambda by cross-validation. l_norm=1 corresponds to the sum of absolute values of differences between predicted and estimated conditional Kendall's tau while l_norm=2 corresponds to the sum of squares of differences.

object, x

an S3 object of class simpA_kendallReg_test.

type

type of standard error to be returned. Possible values are:

"coefficients": standard errors for estimated coefficients
"kernel-based CKT": standard errors for the kernel-based estimated values of conditional Kendall's taus
"regression-based CKT": standard errors for the regression-based estimated values of conditional Kendall's taus

...

other arguments, unused

ylim

graphical parameter, see plot

References

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

Examples

Run this code


# \donttest{
# 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)
# Obtain standard errors of the kernel-based estimates of CKT
se(result, type = "kernel-based CKT")
# Obtain standard errors of the regression-based estimates of CKT
se(result, type = "regression-based CKT")

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)
# }