gofRosenblattSnB: The SnB test based on the Rosenblatt transformation

In gofCopula: Goodness-of-Fit Tests for Copulae

Description

gofRosenblattSnB contains the SnB gof test for copulae from Genest (2009) 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.

Usage

gofRosenblattSnB(
  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
)

Arguments

copula	The copula to test for. Possible are "normal", "t", "clayton", "gumbel", "frank", "joe", "amh", "galambos", "fgm" and "plackett".
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 TRUE or FALSE. TRUE means that param will be estimated.
df	Degrees of freedom, if not meant to be estimated. Only necessary if tested for "t"-copula.
df.est	Indicates if df shall be estimated. Has to be either FALSE or TRUE, where TRUE means that it will be estimated.
margins	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.
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.
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 copula.
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 NULL.
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 NULL.
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 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.
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 probability integral transform which uses the mapping R : (0,1)^d -> (0,1)^d to test the H0 hypothesis

C in Ccal0

with Ccal0 as the true class of copulae under H0. Following Genest et al. (2009) ensures this transformation the decomposition of a random vector 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 = R(U_1), ..., E_n = R(U_n).

The mapping is performed by assigning to every vector u for e[1] = u[1] and for i in {2, ..., d},

e[i] = (d^(i-1) C(u[1], ..., u[i], 1, ..., 1))/(d u[1] ... d u[i-1]) / (d^(i-1) C(u[1], ..., u[i-1], 1, ..., 1))/(d u[1] ... d u[i-1]).

The resulting independence copula is given by Cind(u) = u[1] x ... x u[d].

The test statistic T is then defined as

n int_{[0,1]^d} ( {Dn(u) - Cind(u)}^2 )du

with Dn(u) = 1/n sum(E[i] <= u, i = 1, ..., n).

The approximate p-value is computed by the formula, see copula,

sum(|T[b]| >= |T|, b=1, .., M) / 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 "ranks" or NULL.
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

Examples

data(IndexReturns2D)

gofRosenblattSnB("normal", IndexReturns2D, M = 10)

gofCopula documentation built on April 22, 2021, 5:10 p.m.