gofRosenblattChisq contains the RosenblattChisq gof test for
copulae, 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 and the possible copulae are
"normal", "t", "clayton", "gumbel",
"frank", "joe", "amh", "galambos",
"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.
gofRosenblattChisq(
copula = c("normal", "t", "clayton", "gumbel", "frank", "joe", "amh", "galambos",
"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", "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.
This Anderson-Darling test statistic (supposedly) computes U[0,1]-distributed (under \(H_0\)) random variates via the distribution function of chi-square distribution with d degrees of freedom, see Hofert et al. (2014). The \(H_0\) hypothesis is $$C \in \mathcal{C}_0$$ with \(\mathcal{C}_0\) as the true class of copulae under \(H_0\).
This test is based on the Rosenblatt probability integral transform which uses the mapping \(\mathcal{R}: (0,1)^d \rightarrow (0,1)^d\). Following Genest et al. (2009) ensures this transformation the decomposition of a random vector \(\mathbf{u} \in [0,1]^d\) with a distribution into mutually independent elements with a uniform distribution on the unit interval. The mapping provides pseudo observations \(E_i\), given by $$E_1 = \mathcal{R}(U_1), \dots, E_n = \mathcal{R}(U_n).$$ The mapping is performed by assigning to every vector \(\mathbf{u}\) for \(e_1 = u_1\) and for \(i \in \{2, \dots, d\}\), $$e_i = \frac{\partial^{i-1} C(u_1, \dots, u_i, 1, \dots, 1)}{\partial u_1 \cdots \partial u_{i-1}} / \frac{\partial^{i-1} C(u_1, \dots, u_{i-1}, 1, \dots, 1)}{\partial u_1 \cdots \partial u_{i-1}}.$$
The Anderson-Darling test statistic of the variates
$$G(x_j) = \chi_d^2 \left( x_j \right)$$
is computed (via ADGofTest::ad.test), where \(x_j = \sum_{i=1}^d
(\Phi^{-1}(e_{ij}))^2\), \(\Phi^{-1}\)
denotes the quantile function of the standard normal distribution function,
\(\chi_d^2\) denotes the distribution function of the
chi-square distribution with d degrees of freedom, and \(u_{ij}\)
is the \(j\)th component in the \(i\)th row of \(\mathbf{u}\).
The test statistic is then given by $$T = -n - \sum_{j=1}^n \frac{2j - 1}{n} [\ln(G(x_j)) + \ln(1 - G(x_{n+1-j}))].$$
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.
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")
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)
gofRosenblattChisq("normal", IndexReturns2D, M = 10)
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