zetapm {depcoeff}

Zeta dependence coefficient of piecewise monotonicity

Description

zetapm is a function to evaluate the zeta dependence coefficients of piecewise monotonicity of two random variables x and y which is based on the copula. The regressor domain (domain of x) is split into two parts. The function searches for the optimal splitting point to obtain maximum depedence. The main part of the function is coded as C++ procedure

Usage

zetapm(x,y,amin=0.25,method="all",methodF=1,parp=1.5,parH=0.5)

Arguments

x, y

data vectors of the two variables whose dependence is analysed.

amin

minimum fraction of sample items to be used for one split region

method

vector of chosen special coefficients: Spearman...Spearman coefficient footrule...Spearman's footrule power...power coefficient Huber...Huber function coefficient, "all" refers to all coefficients

methodF

value 1,2 or 3 refers to several methods for computation of the distribution function values, 1 is the default value.

parp

parameter of the power function (default 1.5). The parameter has to be positive.

parH

parameter of the Huber function (default 0.5). Valid values for parH are between 0 and 1.

Details

Let X_{1},\ldots ,X_{n} be the sample of the X variable. Formulas for the estimators of values F(X_{i}) of the distribution function: methodF = 1 \rightarrow \hat{F}(X_{i})=\frac{1}{n}\textrm{rank}(X_{i}) methodF = 2 \rightarrow \hat{F}^{1}(X_{i})=\frac{1}{n+1}\textrm{rank}(X_{i}) methodF = 3 \rightarrow \hat{F}^{2}(X_{i})=\frac{1}{\sqrt{n^{2}-1}}\textrm{rank}(X_{i}) The values of the distribution function of Y are treated analogously.

Value

list of zeta dependence coefficients (plusminus coefficient and minusplus one) of piecewise monotonicity of two random variables containing the following elements or a subset of it in this order: Spearman coefficient, footrule, power coefficient, Huber function coefficient. position1 and position2 indicate the number of the sample items where the optimized split point is located

References

Eckhard Liebscher (2017). Copula-based dependence measures for piecewise monotonicity. Dependence Modeling 5 (2017), 198-220

Examples

library(MASS)
data<- gilgais
zetapm(data[,1],data[,2])