gofArchmSnC {gofCopula} | R Documentation |
The ArchmSnC test based on the Rosenblatt transformation for archimedean copula
Description
gofArchmSnC
contains the SnC gof test with a Rosenblatt
transformation for archimedean copulae, described in Hering and Hofert (2015).
The test follows the RosenblattChisq test as described in Genest (2009)
and Hofert (2014), and compares the empirical copula against a parametric
estimate of the copula derived under the null hypothesis. The margins can be
estimated by a bunch of distributions and the time which is necessary for
the estimation can be given. The approximate p-values are computed with a
parametric bootstrap, which computation can be accelerated by enabling
in-build parallel computation. The gof statistics are computed with the
function gofTstat
from the package copula.
It is possible to insert datasets of all dimensions above 1
(except for the "amh"
copula) and the possible copulae are
"clayton"
, "gumbel"
, "frank"
, "joe"
and
"amh"
. The parameter estimation is performed with pseudo maximum
likelihood method. In case the estimation fails, inversion of Kendall's tau
is used.
Usage
gofArchmSnC(
copula = c("clayton", "gumbel", "frank", "joe", "amh"),
x,
param = 0.5,
param.est = TRUE,
margins = "ranks",
flip = 0,
M = 1000,
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 copula parameter to use, if it shall not be estimated. |
param.est |
Shall be either |
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. |
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
This test is based on the Rosenblatt transformation for archimedean copula
which uses the mapping \mathcal{R}: (0,1)^d
\rightarrow (0,1)^d
. Following
Hering and Hofert (2015) the mapping provides
pseudo observations E_i
, given by
E_1 = \mathcal{R}(U_1),
\dots, E_n = \mathcal{R}(U_n).
Let C
be an Archimedean copula with d
monotone generator
\psi
and continuous Kendall distribution function K_{C}
. Then,
e_{j}=\left(\frac{\sum_{k=1}^{j} \psi^{-1}\left(u_{k}
\right)}{\sum_{k=1}^{j+1} \psi^{-1}\left(u_{k}\right)}\right)^{j},
j \in\{1, \ldots, d-1\}
and
e_{d}= \frac{n}{n+1} K_{C}(C(u))
.
The resulting independence copula is given by
C_{\bot}(\mathbf{u}) = u_1 \cdot \dots \cdot u_d
.
The test statistic T
is then defined as
T = n \int_{[0,1]^d} \{ D_n(\mathbf{u}) - C_{\bot}(\mathbf{u}) \}^2 d
D_n(\mathbf{u})
with
D_n(\mathbf{u}) = \frac{1}{n} \sum_{i = 1}^n \mathbf{I}(E_i \leq
\mathbf{u})
.
The approximate p-value is computed by the formula, see copula,
\sum_{b=1}^M \mathbf{I}(|T_b| \geq |T|) / M,
where T
and T_b
denote the test statistic and the
bootstrapped test statistc, respectively.
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
Christian Genest, Bruno Remillard, David Beaudoin (2009).
Goodness-of-fit tests for copulas: A review and a power study.
Insurance: Mathematics and Economics, Volume 44, Issue 2, April 2009,
Pages 199-213, ISSN 0167-6687.
doi: 10.1016/j.insmatheco.2007.10.005
Marius
Hofert, Ivan Kojadinovic, Martin Maechler, Jun Yan (2014). copula:
Multivariate Dependence with Copulas. R package version 0.999-15..
https://cran.r-project.org/package=copula
Christian Hering, Marius Hofert (2015) Goodness-of-fit tests for
Archimedean copulas in high dimensions. In: Glau K., Scherer M.,
Zagst R. (eds) Innovations in Quantitative Risk Management,
Springer Proceedings in Mathematics & Statistics, Volume 99,
Springer, Cham, 357-373.
doi: 10.1007/978-3-319-09114-3_21
Examples
data(IndexReturns2D)
gofArchmSnC("clayton", IndexReturns2D, M = 10)