CDVine (version 1.4)

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

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.

Value

• W.in
  W-statistics (x-axis).
• Hi.sort
  H-statistics (y-axis).

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 $$W_{i:n}= N {N-1 \choose i-1} \int\limits_{0}^1 \omega k_0(\omega) ( K_0(\omega) )^{i-1} ( 1-K_0(\omega) )^{N-i} d\omega,$$ where $$K_0(\omega)=\omega - \omega \log(\omega), \nonumber$$ and $k_0(\cdot)$ 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 $\left(W_{i:N},H_i\right)$ lie on the curve $K_0(\omega)$ located above the main diagonal. If points $\left(W_{i:N},H_i\right)$ however lie on the x-axis, this indicates a perfect negative dependence between $U_1$ and $U_2$.

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

# 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")