depchi: Estimating extremal dependence coefficientes

In IndTestPP: Tests of Independence and Analysis of Dependence Between Point Processes in Time

Description

This function estimates and plots the extremal dependence functions χ(u) and \bar χ(u) against a grid of values in [0,1] to analyse the extremal dependence of two variables.

Usage

depchi(X, Y, thresval = c(0:99)/100, tit = "", indgraph = TRUE, bothest = TRUE, 
       xlegend = "topleft",mfrow=c(1,2),...)

Arguments

X	Numeric vector. Values of the first variable.
Y	Numeric vector. Values of the second variable.
thresval	Numeric vector. Grid values where the functions χ(u) and \bar χ(u) are evaluated.
tit	Character string. A title for the plots.
indgraph	Logical flag. If it is TRUE, plots are shown in separate windows. If it is FALSE, the layout in mfrow is used.
bothest	Logical flag. If it is TRUE, two estimated coefficientes (for X given Y and for Y given X) are displayed in the same plot. Otherwise, only the coefficient for Y given X is plotted.
xlegend	Optional. Label "topleft","bottomright", etc. Position where the legend on the graph will be located.
mfrow	Optional. A vector of the form c(2, 1) or c(1,2). It gives the layout to draw the two figures in the function.
...	Further arguments to be passed to the function plot.

Details

The extremal dependence between two variables X and Y is the tendency for one variable to be large, given that the other one is large. The extremal dependence coefficients χ and \bar χ are defined as χ= \lim_{u \to 1} χ(u) where χ(u)= P(U>u |V>u) and (U,V) are the transformed uniform marginals of the variables X and Y.

\bar χ= \lim_{u \to 1} \bar χ(u) where \bar χ(u)= 2log P(U>u)/log P(U>u, V>u)-1.

The function plots χ(u) and \bar χ(u). These graphs can be used to estimate \hat χ and \widehat{\bar χ}, the limits of the functions. In the χ (u) plot, the expected behaviour under independence of X and Y is also plotted.

χ is on the scale [0, 1], with the set (0, 1] corresponding to asymptotic dependence, and the measure \bar χ falls within the range [-1, 1], with the set [-1, 1) corresponding to asymptotic independence. See Coles et al. (1999) for more details on the definition and interpretation of these indexes.

Value

A list with elements

chiX	Estimated χ (u) function for Y given X evaluated at the threshold grid.
chiY	Estimated χ (u) function for X given Y evaluated at the threshold grid.
chiBX	Estimated \bar χ (u) function for Y given X evaluated at the threshold grid.
chiBY	Estimated \bar χ (u) function for X given Y evaluated at the threshold grid.
PX	Estimation of the probabilities P(U<thresval).
PY	Estimation of the probabilities P(V<thresval).
PXY	Estimation of the probabilities P[(U<thresval)\&(V<thresval)].
thresval	Input argument.

References

Coles, S., Heffernan, J. and Tawn, J. (1999) Dependence measures for extreme value analysis. Extremes, 2, 339-365.

See Also

CountingCor, BinPer

Examples

data(TxBHZ)

aux<-depchi(X=TxBHZ$TxZ,Y=TxBHZ$TxH, thresval = c(0:99)/100, 
	tit = "Tx Zaragoza and Tx Huesca", xlegend = "bottom",indgraph="FALSE")

IndTestPP documentation built on Aug. 29, 2020, 1:06 a.m.