gofCvM performs the "CvM" gof test, described in Genest
et al. (2009), for copulae 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 and 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.
gofCvM(
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,
dispstr = "ex",
lower = NULL,
upper = NULL,
seed.active = NULL,
processes = 1
)An object of the class gofCOP with the components
a character which informs about the performed analysis
the copula tested for
the method used to estimate the margin distribution.
the parameters of
the estimated margin distributions. Only applicable if the margins were not
specified as "ranks" or NULL.
dependence parameters of the copulae
the degrees of freedem of the copula. Only applicable for t-copula.
a matrix with the p-values and test statistics of the hybrid and the individual tests
The copula to test for. Possible are "normal",
"t", "clayton", "gumbel", "frank", "joe",
"amh", "galambos", "huslerReiss", "tawn",
"tev", "fgm" and "plackett".
A matrix containing the data with rows being observations and columns being variables.
The copula parameter to use, if it shall not be estimated.
Shall be either TRUE or FALSE. TRUE
means that param will be estimated.
Degrees of freedom, if not meant to be estimated. Only necessary
if tested for "t"-copula.
Indicates if df shall be estimated. Has to be either
FALSE or TRUE, where TRUE means that it will be
estimated.
Specifies which estimation method for the margins shall be
used. The default is "ranks", which is the standard approach to
convert data in such a case. Alternatively the following distributions can
be specified: "beta", "cauchy", Chi-squared ("chisq"),
"f", "gamma", Log normal ("lnorm"), Normal
("norm"), "t", "weibull", Exponential ("exp").
Input can be either one method, e.g. "ranks", which will be used for
estimation of all data sequences. Also an individual method for each margin
can be specified, e.g. c("ranks", "norm", "t") for 3 data sequences.
If one does not want to estimate the margins, set it to NULL.
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.
Number of bootstrapping loops.
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
copula.
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 NULL.
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 NULL.
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 NULL, then R generates
the seeds from the computer runtime. Controlling the seeds is useful for
reproducibility of a simulation study to compare the power of the tests or
for reproducibility of an empirical study.
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.
With the pseudo observations \(U_{ij}\) for \(i = 1, \dots,n\), \(j = 1, \dots,d\) and \(\mathbf{u} \in [0,1]^d\) is the empirical copula given by \(C_n(\mathbf{u}) = \frac{1}{n} \sum_{i = 1}^n \mathbf{I}(U_{i1} \leq u_1, \dots, U_{id} \leq u_d).\) It shall be tested the \(H_0\) hypothesis: $$C \in \mathcal{C}_0$$ with \(\mathcal{C}_0\) as the true class of copulae under \(H_0\). The test statistic \(T\) is then defined as
$$T = n \int_{[0,1]^d} \{ C_n(\mathbf{u}) - C_{\theta_n}(\mathbf{u}) \}^2 d C_n(\mathbf{u})$$ with \(C_{\theta_n}(\mathbf{u})\) the estimation of \(C\) under the \(H_0\).
The approximate p-value is computed by the formula,
$$\sum_{b=1}^M \mathbf{I}(|T_b| \geq |T|) / M,$$
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.
Rosenblatt, M. (1952). Remarks on a Multivariate Transformation.
The Annals of Mathematical Statistics 23, 3, 470-472.
Hering,
C. and Hofert, M. (2014). Goodness-of-fit tests for Archimedean copulas in
high dimensions. Innovations in Quantitative Risk Management.
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
data(IndexReturns2D)
gofCvM("normal", IndexReturns2D, M = 10)
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