tests_vine2: 2 dimensional gof tests based on White's information matrix...

In gofCopula: Goodness-of-Fit Tests for Copulae

Description

gofWhite tests a given 2 dimensional dataset for a copula with the gof test based on White's information matrix equality. The possible copulae are "normal", "t", "gumbel", "clayton" and "frank". See for reference Schepsmeier et al. (2015). The parameter estimation is performed with pseudo maximum likelihood method. In case the estimation fails, inversion of Kendall's tau is used. 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 computation of the test statistic and p-values is performed by corresponding functions from the VineCopula package.

Usage

gofWhite(copula, x, M = 1000, param = 0.5, param.est = T, df = 4, df.est = T, 
          margins = "ranks", execute.times.comp = T, processes = 1)

Arguments

copula	The copula to test for. Possible are the copulae "normal", "clayton", "gumbel" and "frank".
x	A 2 dimensional matrix containing the residuals of the data.
M	Number of bootstrap samples.
param	The parameter to be used.
param.est	Shall be either TRUE or FALSE. TRUE means that param will be estimated with a maximum likelihood estimation.
df	The degrees of freedom for the "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 shall be used in case that the input data are not in the range [0,1]. The default is "ranks", which is the standard approach to convert data in such a case. Alternatively can the following distributions be specified: "beta", "cauchy", Chi-squared ("chisq"), "f", "gamma", Log normal ("lnorm"), Normal ("norm"), "t", "weibull", Exponential ("exp").
execute.times.comp	Logical. Defines if the time which the estimation most likely takes shall be computed. It'll be just given if M is at least 100.
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

The details are obtained from Schepsmeier et al. (2015) who states that this test uses the information matrix equality of White (1982). Under correct model specification is the Fisher Information equivalently calculated as minus the expected Hessian matrix or as the expected outer product of the score function. The null hypothesis is

H0 : H(theta) + S(theta) = 0

where H(theta) is the expected Hessian matrix and S(theta) is the expected outer product of the score function.

The test statistic is derived by

T_n = n(dbar(theta_n))^T V_(theta_n)^(-1) dbar(theta_n)

with

dbar(theta_n) = 1/n sum(vech(H(theta_n|u) + S(theta_n|u)), i=1, ..., n),

d(theta_n) = vech(H(theta_n|u) + S(theta_n|u)),

Vtheta_n] = 1/n sum(d(theta_n) - D(theta_n) H_n(theta_n)^(-1) d l(theta_n)d(theta_n) - D(theta_n) H_n(theta_n)^(-1) d l(theta_n)^T, i=1, ..., n)

and

D(theta_n = 1/n sum(d_(theta_k) d[l](theta_n)_(l=1, ...,(p(p+1))/2, k=1, ..., p), i=1, ..., n)

where l(theta_n) represents the log likelihood function and p is the length of the parameter vector theta.

The test statistic will be rejected if

T > (1 - alpha) (chi^2_(p(p+1)/2))^(-1).

For small values of M, initializing the parallization via processes does not make sense. The registration of the parallel processes increases the computation time. Please consider to enable parallelization just for high values of M.

Value

A object of the class gofCOP with the components

method	a character which informs about the performed analysis
erg.tests	a matrix with the p-value and test statistic of test

References

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

Schepsmeier, U. and J. Stoeber (2014). Derivatives and Fisher information of bivariate copulas. Statistical Papers, 55(2), 525-542. http://link.springer.com/article/10.1007/s00362-013-0498-x

Stoeber, J. and U. Schepsmeier (2013). Estimating standard errors in regular vine copula models Computational Statistics, 28 (6), 2679-2707

Schepsmeier, U. (2015). Efficient information based goodness-of-fit tests for vine copula models with fixed margins. Journal of Multivariate Analysis 138, 34-52. Schepsmeier, U. (2014). A goodness-of-fit test for regular vine copula models.

Examples

data(IndexReturns)

gofWhite("normal", IndexReturns[c(1:100),c(1:2)], M = 10)

gofCopula documentation built on May 29, 2017, 6:33 p.m.