Based on samples from a bivariate copula, the copula density is estimated. The user can choose between different methods. If no bandwidth is provided by the user, it will be set by a methodspecific automatic selection procedure. The related (d/p/r)kdecop functions evaluate the density and cdf or simulate synthetic data, respectively.
kdecop(udata, bw = NA, mult = 1, method = "TLL2nn", knots = 30, renorm.iter = 3L, info = TRUE)
udata 


bw  bandwidth specification; if 
mult  bandwidth multiplier, has to be positive; useful for making estimates more/less smooth manually. 
method 

knots  integer; number of knots in each dimension for the spline approximation. 
renorm.iter  integer; number of iterations for the renormalization procedure (see Details). 
info  logical; if 
The function kdecop
returns an
object of class kdecopula
that contains all information necessary for
evaluation of the estimator. If no bandwidth was provided in the function
call, the automatically selected value can be found in the variable
object$bw
. If info=TRUE
, also the following will be available
under object$info
:
Estimator evaluated in sample points
Log likelihood
Effective number of parameters
Akaike information criterion
Biascorrected version of Akaike information criterion
Bayesian information criterion.
We use a Gaussian product kernel function for all methods
except the beta kernel and Bernstein estimators. For details on bandwidth
selection for a specific method, see: bw_t
,
bw_tll
, bw_tll_nn
, bw_tt_pi
,
bw_tt_cv
, bw_mr
, bw_beta
,
bw_bern
.
Kernel estimates are usually no proper copula densities. In particular, the
estimated marginal densities are not uniform. We mitigate this issue by
a renormalization procedure. The number of iterations of the
renormalization algorithm can be specified with the renorm.iter
argument. Typically, a very small number of iterations is sufficient.
The implementation of the tapered transformation estimator ("TTPI"/"TTCV") was kindly provided by Kuangyu Wen.
Geenens, G., Charpentier, A., and Paindaveine, D. (2017). Probit transformation for nonparametric kernel estimation of the copula density. Bernoulli, 23(3), 18481873. Wen, K. and Wu, X. (2015). TransformationKernel Estimation of the Copula Density, Working paper, http://agecon2.tamu.edu/people/faculty/wuximing/agecon2/public/copula.pdf Gijbels, I. and Mielniczuk, J. (1990). Estimating the density of a copula function. Communications in Statistics  Theory and Methods, 19(2):445464. Charpentier, A., Fermanian, J.D., and Scaillet, O. (2006). The estimation of copulas: Theory and practice. In Rank, J., editor, Copulas: From theory to application in finance. Risk Books. Weiss, G. and Scheffer, M. (2012). Smooth Nonparametric Bernstein Vine Copulas. arXiv:1210.2043 Nagler, T. (2014). Kernel Methods for Vine Copula Estimation. Master's Thesis, Technische Universitaet Muenchen, https://mediatum.ub.tum.de/node?id=1231221
kdecopula
,
plot.kdecopula
,
predict.kdecopula
,
fitted.kdecopula
,
simulate.kdecopula
,
dkdecop
,
pkdecop
,
rkdecop
## load data and transform with empirical cdf data(wdbc) udat < apply(wdbc[, 1], 2, function(x) rank(x) / (length(x) + 1)) ## estimation of copula density of variables 5 and 6 fit < kdecop(udat[, 5:6]) summary(fit)#> Kernel copula density estimate (tau = 0.49) #>  #> Variables: mean smoothness  mean compactness #> Observations: 569 #> Method: Transformation local likelihood, logquadratic (nearestneighbor, 'TLL2nn') #> Bandwidth: alpha = 0.5689854 #> B = matrix(c(0.71, 0.71, 0.71, 0.71), 2, 2) #>  #> logLik: 190.97 AIC: 357 cAIC: 356.4 BIC: 302.85 #> Effective number of parameters: 12.47plot(fit)contour(fit)#> [1] 1.28916#> [1] 0.09779868