zetaci {depcoeff}

Zeta coefficient of piecewise monotonicity with split domain

Description

The function zetaci evaluates the coefficient of piecewise monotonicity of variables x and y where the x-domain is split into a fixed number of intervals.

Usage

zetaci(x,y,a,method="Spearman",methodF=1,parH=0.5,parp=1.5)

Arguments

x, y

data vectors of the two variables whose dependence is analysed.

a

vector of fractions a_{i},0<a_{i}<a_{i+1}<1 for the splitting. A fraction of a_{1},a_{2}-a_{1},a_{3}-a{2}... of data points are in the corresponding split region. The number of split regions is equal to the length of a plus 1.

method

value (default "Spearman")

methodF

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

parH

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

parp

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

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 of piecewise monotonicity of two random variables containing the following elements: Spearman...Spearman coefficient footrule...Spearman's footrule power...power coefficient Huber...Huber function coefficient

References

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

Examples

library(MASS)
data<- gilgais
zetaci(data[, 1], data[, 2], a=c(0.25, 0.5, 0.75))