gofKendallKS tests a given dataset for a copula based on
Kendall's process with the Kolmogorov-Smirnov test statistic. The margins
can be estimated by a bunch of distributions and the time which is necessary
for the estimation can be given. The possible copulae are "normal",
"t", "clayton", "gumbel", "frank", "joe",
"amh", "galambos", "huslerReiss", "tawn",
"tev", "fgm" and "plackett". See for
reference Genest et al. (2009). 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.
gofKendallKS(
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
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 bootstrap samples.
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 \(C_n(\mathbf{u}) = \frac{1}{n} \sum_{i = 1}^n \mathbf{I}(U_{i1} \leq u_1, \dots, U_{id} \leq u_d).\) Let the rescaled pseudo observations be \(V_1 = C_n(U_1), \dots, V_n = C_n(U_n)\) and the distribution function of \(V\) shall be \(K\). The estimated version is given by $$K_n(v) = \frac{1}{n} \sum_{i=1}^n \mathbf{I}(V_i \leq v)$$ with \(v \in [0,1]^d.\) The testable \(H_0^{'}\) hypothesis is then $$K \in \mathcal{K}_0 = \{K_{\theta} : \theta \in \Theta \}$$ with \(\Theta\) being an open subset of \(R^p\) for an integer \(p \geq 1\), see Genest et al. (2009). The resulting Kolmogorov-Smirnof test statistic is then given by $$T = \sqrt{n} \sup_{v \in [0,1]} |K_n(v) - K_{\theta_n}| .$$
Because \(H_0^{'}\) consists of more distributions than the \(H_0\) is the test not necessarily consistent.
The approximate p-value is computed by the formula
$$\sum_{b=1}^M \mathbf{I}(|T_b| \geq |T|) / M,$$
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.
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.
tools:::Rd_expr_doi("10.1016/j.insmatheco.2007.10.005")
Christian
Genest, Jean-Francois Quessy, Bruno Remillard (2006). Goodness-of-fit
Procedures for Copula Models Based on the Probability Integral
Transformation. Scandinavian Journal of Statistics, Volume 33, Issue
2, 2006, Pages 337-366.
tools:::Rd_expr_doi("10.1111/j.1467-9469.2006.00470.x")
Ulf
Schepsmeier, Jakob Stoeber, Eike Christian Brechmann, Benedikt Graeler
(2015). VineCopula: Statistical Inference of Vine Copulas. R package
version 1.4.. https://cran.r-project.org/package=VineCopula
data(IndexReturns2D)
gofKendallKS("normal", IndexReturns2D, M = 10)
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