This function estimates the asymmetric dependence between \(X\) and \(Y\) using the rearranged dependence measure \(R_\mu(X, Y)\) for different possible underlying measures \(\mu\). 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. \(Y = f(X)\).
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
)The estimated value of the rearranged dependence measure
A bivariate data.frame containing the observations. Each row contains one bivariate observation.
Options include "spearman", "kendall", "bkr", "dss", "chatterjee" and "zeta1".The option "all" returns the value for all aforementioned methods.
A character string indicating the use of either a cross-validation principle (square or non-square) or a fixed bandwidth (oftentimes called resolution).
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 \(s_1 < s_2\).
Whether or not to perform a permutation test
Number of repetitions of the permutation test
Whether or not to perform validity checks of the input
This function estimates \(R_\mu(X, Y)\) using the empirical checkerboard mass density \(A\). To arrive at \(R_\mu(X, Y)\), \(A\) is appropriately sorted and then evaluated for the underlying measure. The estimated \(R_\mu\) always takes values between 0 and 1 with
\(R_\mu(X, Y) = 0\) if and only if \(X\) and \(Y\) are independent.
\(R_\mu(X, Y) = 1\) 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 \(\rho\) (which is identical to the \(L_1\)-Schweizer-Wolff-measure),
"kendall" Implements the concordance measure Kendall's \(\tau\),
"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 \(\xi\) <doi:10.1080/01621459.2020.1758115>,
"zeta1" Implements the \(\zeta_1\)-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.
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))
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