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

 ```1 2``` ```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

 ```1 2 3``` ```data(IndexReturns) gofWhite("normal", IndexReturns[c(1:100),c(1:2)], M = 10) ```

Example output

```Loading required package: copula
R.methodsS3 v1.7.1 (2016-02-15) successfully loaded. See ?R.methodsS3 for help.
R.oo v1.21.0 (2016-10-30) successfully loaded. See ?R.oo for help.

Attaching package: 'R.oo'

The following objects are masked from 'package:methods':

getClasses, getMethods

The following objects are masked from 'package:base':

attach, detach, gc, load, save

R.utils v2.5.0 (2016-11-07) successfully loaded. See ?R.utils for help.

Attaching package: 'R.utils'

The following object is masked from 'package:utils':

timestamp

The following objects are masked from 'package:base':

cat, commandArgs, getOption, inherits, isOpen, parse, warnings

Parametric bootstrap goodness-of-fit test with White test and normal
copula

Tests results:
p.value test statistic     rho.1
White     0.1       2.198297 0.6265811
Warning message:
In .margins.param.est(copula = copula, margins = margins, x = x,  :
The observations aren't in [0,1]. The margins will be estimated by the ranks of the observations.
```

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