BiCopLambda | R Documentation |
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 |
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 \lambda
-function is characteristic for each bivariate copula
family and defined by Kendall's distribution function K
:
\lambda(v,\theta) := v - K(v,\theta)
with
K(v,\theta) := P(C_{\theta}(U_1,U_2) \leq v),\ \ v\in [0,1].
For Archimedean copulas one has the following closed form expression
in terms of the generator function \varphi
of the copula
C_{\theta}
:
\lambda(v,\theta) = \frac{\varphi(v)}{\varphi '(v)},
where \varphi '
is the derivative of \varphi
. 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 \lambda
-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 \lambda
-function also shows the limits of
the \lambda
-function corresponding to Kendall's tau =0
and
Kendall's tau =1
(\lambda=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 \lambda
-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.
BiCopMetaContour()
, BiCopKPlot()
,
BiCopChiPlot()
, BiCop()
# 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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