CKT.kernel: 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",
  se = FALSE,
  confint = FALSE,
  level = 0.95,
  methodCV = "Kfolds",
  Kfolds = 5,
  nPairs = NULL,
  typeEstCKT = "wdm",
  progressBar = 1,
  warnNA = TRUE,
  observedX1 = NULL,
  observedX2 = NULL,
  observedZ = NULL
)

Value

an S3 object of class estimated_CKT_kernel with components including:

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.)
resultCV (only in case of cross-validation). This gives the output of the cross-validation function that is used, i.e. the output of either CKT.hCV.l1out or CKT.hCV.Kfolds.
se, and confint if requested.

Some methods (se, confint and plot) are available for such an object, see plot.estimated_CKT_kernel.

Arguments

X1

a vector of n observations of the first variable (or a 1-column matrix)

X2

a vector of n observations of the second variable (or a 1-column matrix)

Z

a vector of n observations of the conditioning variable, or a matrix with n rows of observations of the conditioning vector (in the case that several conditioning variables are given; in this case, each column corresponds to 1 conditioning variable). It can also be a data.frame, provided that all the entries are numeric.

newZ

the new data of observations of Z at which the conditional Kendall's tau should be estimated. It must have the same number of column as Z. It can be a vector (if NCOL(Z) == 1), a matrix or a data.frame, provided that all the entries are numeric.

h

the bandwidth used for kernel smoothing. If this is a vector, then cross-validation is used following the method given by argument methodCV to choose the best bandwidth before doing the estimation.

kernel.name

name of the kernel used for smoothing. Possible choices are "Gaussian" (Gaussian kernel) and "Epa" (Epanechnikov kernel).

se, confint

if TRUE, compute (asymptotic) standard errors and confidence intervals of conditional Kendall's tau. Standard errors are computed using Proposition 9 of (Derumigny & Fermanian, 2019).

level

the confidence level for the confidence intervals. By default, 95% confidence intervals are computed, i.e. level = 0.95.

methodCV

method used for the cross-validation. Possible choices are "leave-one-out" and "Kfolds".

Kfolds

number of subsamples used, if methodCV = "Kfolds".

nPairs

number of pairs used in the cross-validation criteria, if methodCV = "leave-one-out". Use nPairs = "all" to choose all pairs. The default is nPairs = 10 * n, where n is the sample size.

typeEstCKT

type of estimation of the conditional Kendall's tau. Possible choices are

1 and 3 produced biased estimators. 2 does not attain the full range $[-1,1]$. Therefore these 3 choices are not recommended for applications on real data.
4 is an improved version of 1,2,3 that has less bias and attains the full range $[-1,1]$.
"wdm" is the default version and produces the same results as 4 when they are no ties in the data.

progressBar

control the display of progress bars. Possible choices are:

0 no progress bar is displayed
1 a general progress bar is displayed
2 and larger values: a general progress bar is displayed, and additionally, a progressbar for each value of h is displayed to show the progress of the computation. This only applies when the bandwidth is chosen by cross-validation (i.e. when h is a vector).

warnNA

a Boolean to indicate whether warnings should be raised if NAs are produced. By default it is TRUE. If warnNA=FALSE, then no warning is raised even if NAs are produced. This is the case usually if either the bandwidth h is too small, or if there are already NAs in (some of) the inputs.

observedX1, observedX2, observedZ

old parameter names for X1, X2, Z. Support for this will be removed at a later version.

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.

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. tools:::Rd_expr_doi("10.1515/demo-2019-0016")

Examples

Run this code

# We simulate from a conditional copula
set.seed(1)
N = 100
# This is a small example for performance reason.
# For a better example, use:
# 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")

# Multivariate example
N = 100
# This is a small example for performance reason.
# For a better example, use:
# N = 1000
Z1 = rnorm(n = N, mean = 5, sd = 2)
Z2 = rnorm(n = N, mean = 5, sd = 2)
conditionalTau = -0.9 + 1.8 * pnorm(Z1 - Z2, mean = 2, sd = 2)
simCopula = VineCopula::BiCopSim(N = N , family = 1,
    par = VineCopula::BiCopTau2Par(1 , conditionalTau ))
X1 = qnorm(simCopula[,1])
X2 = qnorm(simCopula[,2])

Z = cbind(Z1, Z2)

newZ = expand.grid(Z1 = seq(2,8,by = 0.5),
                   Z2 = seq(2,8,by = 1))
estimatedCKT_kernel <- CKT.kernel(
   X1 = X1, X2 = X2, Z = Z,
   newZ = newZ, h = 1, kernel.name = "Epa")$estimatedCKT

if (requireNamespace("ggplot2", quietly = TRUE)) {
  df = rbind(
    data.frame(newZ, CKT = estimatedCKT_kernel,
               type = "estimated CKT") ,
    data.frame(newZ, CKT = -0.9 + 1.8 * pnorm(newZ$Z1 - newZ$Z2,
                                              mean = 2, sd = 2),
               type = "true CKT")
  )

  ggplot2::ggplot(df) +
  ggplot2::geom_tile(ggplot2::aes(x = Z1, y = Z2, fill = CKT)) +
  ggplot2::facet_grid(as.formula("~type"))
}

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