Lambda-Function (Plot) for Bivariate Copula Data
This function plots/returns the lambda-function of given bivariate copula data.
BiCopLambda( u1 = NULL, u2 = NULL, family = "emp", par = 0, par2 = 0, PLOT = TRUE, obj = NULL, ... )
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 |
obj |
|
... |
Additional plot arguments. |
If the family and parameter specification is stored in a BiCop()
object obj, the alternative versions
BiCopLambda(obj, PLOT = TRUE, ...)
and
BiCopLambda((u1, u2, obj, PLOT = TRUE, ...)
can be used.
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 Student-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.
https://mediatum.ub.tum.de/?id=1079296.
# simulate from Clayton copula cop <- BiCop(3, tau = 0.5) dat <- BiCopSim(1000, cop) # create lambda-function plots op <- par(mfrow = c(1, 3)) BiCopLambda(dat[, 1], dat[, 2]) # empirical lambda-function BiCopLambda(cop) # theoretical lambda-function BiCopLambda(dat[, 1], dat[, 2], cop) # both par(op)
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