Kendall's plot (K-plot) for bivariate copula data

Description

This function creates a Kendall's plot (K-plot) of given bivariate copula data.

Usage

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BiCopKPlot(u1, u2, PLOT=TRUE, ...)

Arguments

u1,u2

Data vectors of equal length with values in [0,1].

PLOT

Logical; whether the results are plotted. If PLOT = FALSE, the values W.in and Hi.sort are returned (see below; default: PLOT = TRUE).

...

Additional plot arguments.

Details

For observations u_{i,j}, i=1,...,N, j=1,2, the K-plot considers two quantities: First, the ordered values of the empirical bivariate distribution function H_i:=\hat{F}_{U_1U_2}(u_{i,1},u_{i,2}) and, second, W_{i:N}, which are the expected values of the order statistics from a random sample of size N of the random variable W=C(U_1,U_2) under the null hypothesis of independence between U_1 and U_2. W_{i:N} can be calculated as follows

W_{i:n}= N {N-1 \choose i-1} \int\limits_{0}^1 ω k_0(ω) ( K_0(ω) )^{i-1} ( 1-K_0(ω) )^{N-i} dω,

where

K_=(ω)=ω - ω log(ω)

and k_0() is the corresponding density.

K-plots can be seen as the bivariate copula equivalent to QQ-plots. If the points of a K-plot lie approximately on the diagonal y=x, then U_1 and U_2 are approximately independent. Any deviation from the diagonal line points towards dependence. In case of positive dependence, the points of the K-plot should be located above the diagonal line, and vice versa for negative dependence. The larger the deviation from the diagonal, the stronger is the degree of dependency. There is a perfect positive dependence if points ≤ft(W_{i:N},H_i\right) lie on the curve K_0(ω) located above the main diagonal. If points (W_{i:N},H_i) however lie on the x-axis, this indicates a perfect negative dependence between U_1 and U_2.

Value

W.in

W-statistics (x-axis).

Hi.sort

H-statistics (y-axis).

Author(s)

Natalia Belgorodski, Ulf Schepsmeier

References

Genest, C. and A. C. Favre (2007). Everything you always wanted to know about copula modeling but were afraid to ask. Journal of Hydrologic Engineering, 12 (4), 347-368.

See Also

BiCopMetaContour, BiCopChiPlot, BiCopLambda, BiCopGofKendall

Examples

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## Not run: 
# Gaussian and Clayton copulas
n = 500
tau = 0.5

# simulate from Gaussian copula
fam1 = 1	
theta1 = BiCopTau2Par(fam1,tau)
dat1 = BiCopSim(n,fam1,theta1)	

# simulate from Clayton copula
fam2 = 3
theta2 = BiCopTau2Par(fam2,tau)
dat2 = BiCopSim(n,fam2,theta2)

# create K-plots
dev.new(width=10,height=5)
par(mfrow=c(1,2))
BiCopKPlot(dat1[,1],dat1[,2],main="Gaussian copula")
BiCopKPlot(dat2[,1],dat2[,2],main="Clayton copula")

## End(Not run)

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