rdm: Rearranged dependence measure

Description

This function estimates the asymmetric dependence between $X$ and $Y$ using the rearranged dependence measure $R_{μ} (X, Y)$ for different possible underlying measures $μ$ . A value of 0 characterizes independence of $X$ and $Y$ , while a value of 1 characterizes a functional relationship between $X$ and $Y$ , i.e. $Y = f (X)$ .

Usage

rdm(
  X,
  method = c("spearman", "kendall", "dss", "zeta1", "bkr", "all"),
  bandwidth_method = c("fixed", "cv", "cvsym"),
  bandwidth_parameter = 0.5,
  permutation = FALSE,
  npermutation = 1000,
  checkInput = FALSE
)

Value

The estimated value of the rearranged dependence measure

Arguments

X: A bivariate data.frame containing the observations. Each row contains one bivariate observation.
method: Options include "spearman", "kendall", "bkr", "dss", "chatterjee" and "zeta1".The option "all" returns the value for all aforementioned methods.
bandwidth_method: A character string indicating the use of either a cross-validation principle (square or non-square) or a fixed bandwidth (oftentimes called resolution).
bandwidth_parameter: A numerical vector which contains the necessary optional parameters for the exponent of the chosen bandwidth method. In case of N observations, the bandwidth_parameter $(s_{1}, s_{2})$ determines a lower bound $N^{s_{1}}$ and upper bound $N^{s_{2}}$ for the cross-validation methods or a single number s for the fixed bandwidth method resulting in $N^{s}$ . The parameters have to lie in $(0, 1 / 2)$ and fulfil $s_{1} < s_{2}$ .
permutation: Whether or not to perform a permutation test
npermutation: Number of repetitions of the permutation test
checkInput: Whether or not to perform validity checks of the input

Details

This function estimates $R_{μ} (X, Y)$ using the empirical checkerboard mass density $A$ . To arrive at $R_{μ} (X, Y)$ , $A$ is appropriately sorted and then evaluated for the underlying measure. The estimated $R_{μ}$ always takes values between 0 and 1 with

$R_{μ} (X, Y) = 0$ if and only if $X$ and $Y$ are independent.
$R_{μ} (X, Y) = 1$ if and only if $Y = f (X)$ for some measurable function $f$ .

Currently, the following underlying measures are implemented:

"spearman" Implements the concordance measure Spearman's $ρ$ (which is identical to the $L_{1}$ -Schweizer-Wolff-measure),
"kendall" Implements the concordance measure Kendall's $τ$ ,
"bkr" Implements the Blum–Kiefer–Rosenblatt $R$ , also known as the $L^{2}$ -Schweizer-Wolff-measure <doi:10.1214/aos/1176345528>,
"dss" Implements the Dette-Siburg-Stoimenov measure of complete dependence <doi:10.1111/j.1467-9469.2011.00767.x>, also known as Chatterjee's $ξ$ <doi:10.1080/01621459.2020.1758115>,
"zeta1" Implements the $ζ_{1}$ -measure of complete dependence established by W. Trutschnig <doi:10.1016/j.jmaa.2011.06.013>.

The estimation of the checkerboard mass density $A$ depends on the choice of the bandwidth for the checkerboard copula. For a detailed discussion of "cv" and "cvsym", see computeBandwidth.

Examples

Run this code

n <- 50
X <- cbind(runif(n), runif(n))
rdm(X, method="spearman", bandwidth_method="fixed", bandwidth_parameter=.3)
n <- 20
U <- runif(n)
rdm(cbind(U, U), method="spearman", bandwidth_method="cv", bandwidth_parameter=c(0.25, 0.5))