gofKernel {gofCopula} | R Documentation |
2 dimensional gof test of Scaillet (2007)
Description
gofKernel
tests a 2 dimensional dataset with the Scaillet test for a
copula. The possible copulae are "normal"
, "t"
,
"clayton"
, "gumbel"
, "frank"
, "joe"
,
"amh"
, "galambos"
, "huslerReiss"
, "tawn"
,
"tev"
, "fgm"
and "plackett"
. The parameter
estimation is performed with pseudo maximum likelihood method. In case the
estimation fails, inversion of Kendall's tau is used. The approximate
p-values are computed with a parametric bootstrap, which computation can be
accelerated by enabling in-build parallel computation.
Usage
gofKernel(
copula = c("normal", "t", "clayton", "gumbel", "frank", "joe", "amh", "galambos",
"huslerReiss", "tawn", "tev", "fgm", "plackett"),
x,
param = 0.5,
param.est = TRUE,
df = 4,
df.est = TRUE,
margins = "ranks",
flip = 0,
M = 1000,
MJ = 100,
dispstr = "ex",
delta.J = 0.5,
nodes.Integration = 12,
lower = NULL,
upper = NULL,
seed.active = NULL,
processes = 1
)
Arguments
copula |
The copula to test for. Possible are |
x |
A matrix containing the data with rows being observations and columns being variables. |
param |
The parameter to be used. |
param.est |
Shall be either |
df |
Degrees of freedom, if not meant to be estimated. Only necessary
if tested for |
df.est |
Indicates if |
margins |
Specifies which estimation method for the margins shall be
used. The default is |
flip |
The control parameter to flip the copula by 90, 180, 270 degrees clockwise. Only applicable for bivariate copula. Default is 0 and possible inputs are 0, 90, 180, 270 and NULL. |
M |
Number of bootstrapping loops. |
MJ |
Size of bootstrapping sample. |
dispstr |
A character string specifying the type of the symmetric
positive definite matrix characterizing the elliptical copula. Implemented
structures are "ex" for exchangeable and "un" for unstructured, see package
|
delta.J |
Scaling parameter for the matrix of smoothing parameters. |
nodes.Integration |
Number of knots of the bivariate Gauss-Legendre quadrature. |
lower |
Lower bound for the maximum likelihood estimation of the copula
parameter. The constraint is also active in the bootstrapping procedure. The
constraint is not active when a switch to inversion of Kendall's tau is
necessary. Default |
upper |
Upper bound for the maximum likelihood estimation of the copula
parameter. The constraint is also active in the bootstrapping procedure. The
constraint is not active when a switch to inversion of Kendall's tau is
necessary. Default |
seed.active |
Has to be either an integer or a vector of M+1 integers.
If an integer, then the seeds for the bootstrapping procedure will be
simulated. If M+1 seeds are provided, then these seeds are used in the
bootstrapping procedure. Defaults to |
processes |
The number of parallel processes which are performed to speed up the bootstrapping. Shouldn't be higher than the number of logical processors. Please see the details. |
Details
The Scaillet test is a kernel-based goodness-of-fit test with a fixed
smoothing parameter. For the copula density , the corresponding kernel estimator is given by
where for
;
are the pseudo observations,
and
for which a bivariate
quadratic kernel is used, as in Scaillet (2007). The matrix of smoothing
parameters is
with
the sample covariance
matrix. The test statistic is then given by
where denotes the convolution operator and
is
a weight function, see Zhang et al. (2015). The bivariate Gauss-Legendre
quadrature method is used to compute the integral in the test statistic
numerically, see Scaillet (2007).
The approximate p-value is computed by the formula
For small values of M
, initializing the parallelisation via
processes
does not make sense. The registration of the parallel
processes increases the computation time. Please consider to enable
parallelisation just for high values of M
.
Value
An object of the class
gofCOP with the components
method |
a character which informs about the performed analysis |
copula |
the copula tested for |
margins |
the method used to estimate the margin distribution. |
param.margins |
the parameters of
the estimated margin distributions. Only applicable if the margins were not
specified as |
theta |
dependence parameters of the copulae |
df |
the degrees of freedem of the copula. Only applicable for t-copula. |
res.tests |
a matrix with the p-values and test statistics of the hybrid and the individual tests |
References
Zhang, S., Okhrin, O., Zhou, Q., and Song, P.. Goodness-of-fit
Test For Specification of Semiparametric Copula Dependence Models.
Journal of Econometrics, 193, 2016, pp. 215-233
doi: 10.1016/j.jeconom.2016.02.017
Scaillet, O.
(2007). Kernel based goodness-of-fit tests for copulas with fixed smoothing
parameters. Journal of Multivariate Analysis, 98:533-543
Examples
data(IndexReturns2D)
gofKernel("normal", IndexReturns2D, M = 5, MJ = 5)