# Lambda-function (plot) for bivariate copula data

### Description

This function plots the lambda-function of given bivariate copula data.

### Usage

1 2 |

### Arguments

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

`family` |
An integer defining the bivariate copula family or indicating the empirical lambda-function: |

`par` |
Copula parameter; if the empirical lambda-function is chosen, |

`par2` |
Second copula parameter for t-, BB1, BB6, BB7 and BB8 copulas (default: |

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

`...` |
Additional plot arguments. |

### Value

`empLambda` |
If the empirical lambda-function is chosen and |

`theoLambda` |
If the theoretical lambda-function is chosen and |

### Note

The *λ*-function is characteristic for each bivariate copula family and defined by Kendall's distribution function *K*:

*
λ(v,θ) := v - K(v,θ)
*

with

*
K(v,θ) := P(C_{θ}(U_1,U_2) <= v), v \in [0,1].
*

For Archimedean copulas one has the following closed form expression in terms of the generator function *φ* of the copula *C_{θ}*:

*
λ(v,θ) = φ(v) / φ'(v),
*

where *φ'* is the derivative of *φ*.
For more details see Genest and Rivest (1993) or Schepsmeier (2010).

For the bivariate Gaussian and t-copula no closed form expression for the theoretical *λ*-function exists.
Therefore it is simulated based on samples of size 1000.
For all other implemented copula families there are closed form expressions available.

The plot of the theoretical *λ*-function also shows the limits of the *λ*-function corresponding to Kendall's tau *=0*
and Kendall's tau *=1* (*λ=0*).

For rotated bivariate copulas one has to transform the input arguments `u1`

and/or `u2`

.
In particular, for copulas rotated by 90 degrees `u1`

has to be set to `1-u1`

,
for 270 degrees `u2`

to `1-u2`

and for survival copulas `u1`

and `u2`

to `1-u1`

and `1-u2`

, respectively.
Then *λ*-functions for the corresponding non-rotated copula families can be considered.

### Author(s)

Ulf Schepsmeier

### References

Genest, C. and L.-P. Rivest (1993). Statistical inference procedures for bivariate Archimedean copulas. Journal of the American Statistical Association, 88 (423), 1034-1043.

Schepsmeier, U. (2010).
Maximum likelihood estimation of C-vine pair-copula constructions based on bivariate copulas from different families.
Diploma thesis, Technische Universitaet Muenchen.

http://mediatum.ub.tum.de/doc/1079296/1079296.pdf.

### See Also

`BiCopMetaContour`

, `BiCopKPlot`

, `BiCopChiPlot`

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 | ```
## Not run:
# Clayton and rotated Clayton copulas
n = 1000
tau = 0.5
# simulate from Clayton copula
fam = 3
theta = BiCopTau2Par(fam,tau)
dat = BiCopSim(n,fam,theta)
# create lambda-function plots
dev.new(width=16,height=5)
par(mfrow=c(1,3))
BiCopLambda(dat[,1],dat[,2]) # empirical lambda-function
BiCopLambda(family=fam,par=theta) # theoretical lambda-function
BiCopLambda(dat[,1],dat[,2],family=fam,par=theta) # both
# simulate from rotated Clayton copula (90 degrees)
fam = 23
theta = BiCopTau2Par(fam,-tau)
dat = BiCopSim(n,fam,theta)
# rotate the data to standard Clayton copula data
rot_dat = 1-dat[,1]
dev.new(width=16,height=5)
par(mfrow=c(1,3))
BiCopLambda(rot_dat,dat[,2]) # empirical lambda-function
BiCopLambda(family=3,par=-theta) # theoretical lambda-function
BiCopLambda(rot_dat,dat[,2],family=3,par=-theta) # both
## End(Not run)
``` |