# BiCopKPlot: 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

 1 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.

BiCopMetaContour, BiCopChiPlot, BiCopLambda, BiCopGofKendall

### Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 ## 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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