Description Usage Arguments Details Value Author(s) References See Also Examples

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

1 | ```
BiCopKPlot(u1, u2, PLOT=TRUE, ...)
``` |

`u1,u2` |
Data vectors of equal length with values in [0,1]. |

`PLOT` |
Logical; whether the results are plotted. If |

`...` |
Additional plot arguments. |

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*.

`W.in` |
W-statistics (x-axis). |

`Hi.sort` |
H-statistics (y-axis). |

Natalia Belgorodski, Ulf Schepsmeier

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`

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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