# 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 |

`...` |
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

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)
``` |