BiCopLambda {CDVine} | R Documentation |
This function plots the lambda-function of given bivariate copula data.
BiCopLambda(u1=NULL, u2=NULL, family="emp", par=0, par2=0, PLOT=TRUE, ...)
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. |
empLambda |
If the empirical lambda-function is chosen and |
theoLambda |
If the theoretical lambda-function is chosen and |
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.
Ulf Schepsmeier
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.
BiCopMetaContour
, BiCopKPlot
, BiCopChiPlot
## 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)