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gofEVCopula

Goodness-of-fit Tests for Bivariate Extreme-Value Copulas

Description

Goodness-of-fit tests for extreme-value copulas based on the empirical process comparing one of the two nonparameteric rank-based estimator of the Pickands dependence function studied in Genest and Segers (2009) with a parametric estimate of the Pickands dependence function derived under the null hypothesis. The test statistic is the Cramer-von Mises functional Sn defined in Equation (5) of Genest, Kojadinovic, G. Nešlehová, and Yan (2010). Approximate p-values for the test statistic are obtained using a parametric bootstrap.

Usage

gofEVCopula(copula, x, N = 1000,
            method = c("mpl", "ml", "itau", "irho"),
            estimator = c("CFG", "Pickands"), m = 1000,
            verbose = interactive(),
            ties.method = c("max", "average", "first", "last", "random", "min"),
            fit.ties.meth = eval(formals(rank)$ties.method), ...)

Arguments

`copula`	object of class `"evCopula"` representing the hypothesized extreme-value copula family.
`x`	a data matrix that will be transformed to pseudo-observations.
`N`	number of bootstrap samples to be used to simulate realizations of the test statistic under the null hypothesis.
`method`	estimation method to be used to estimate the dependence parameter(s); can be either `"mpl"` (maximum pseudo-likelihood), `"itau"` (inversion of Kendall's tau) or `"irho"` (inversion of Spearman's rho).
`estimator`	specifies which nonparametric rank-based estimator of the unknown Pickands dependence function to use; can be either `"CFG"` (Caperaa-Fougeres-Genest) or `"Pickands"`.
`m`	number of points of the uniform grid on [0,1] used to compute the test statistic numerically.
`verbose`	a logical specifying if progress of the bootstrap should be displayed via `txtProgressBar`.
`ties.method`	string specifying how ranks should be computed, except for fitting, if there are ties in any of the coordinate samples of `x`; passed to `pobs`.
`fit.ties.meth`	string specifying how ranks should be computed when fitting by maximum pseudo-likelihood if there are ties in any of the coordinate samples of `x`; passed to `pobs`.
`...`	further optional arguments, passed to `fitCopula()`, notably `optim.method`, the `method` for `optim()`. In copula versions `0.999-14` and earlier, the default for that was `"Nelder-Mead"`, but now is the same as for `fitCopula()`.

Details

More details can be found in the second reference.

The former argument print.every is deprecated and not supported anymore; use verbose instead.

Value

An object of class htest which is a list, some of the components of which are

`statistic`	value of the test statistic.
`p.value`	corresponding approximate p-value.
`parameter`	estimates of the parameters for the hypothesized copula family.

Note

For a given degree of dependence, the most popular bivariate extreme-value copulas are strikingly similar.

References

Genest, C. and Segers, J. (2009). Rank-based inference for bivariate extreme-value copulas. Annals of Statistics 37, 2990–3022.

Genest, C. Kojadinovic, I., G. Nešlehová, J., and Yan, J. (2011). A goodness-of-fit test for bivariate extreme-value copulas. Bernoulli 17(1), 253–275.

Examples

n <- 100; N <- 1000 # realistic (but too large currently for CRAN checks)
n <-  60; N <-  200 # (time (and tree !) saving ...)
x <- rCopula(n, claytonCopula(3))


## Does the Gumbel family seem to be a good choice?
gofEVCopula(gumbelCopula(), x, N=N)


## The same with different (and cheaper) estimation methods:
gofEVCopula(gumbelCopula(), x, N=N, method="itau")
gofEVCopula(gumbelCopula(), x, N=N, method="irho")


## The same with different extreme-value copulas
gofEVCopula(galambosCopula(), x, N=N)
gofEVCopula(galambosCopula(), x, N=N, method="itau")
gofEVCopula(galambosCopula(), x, N=N, method="irho")

gofEVCopula(huslerReissCopula(), x, N=N)
gofEVCopula(huslerReissCopula(), x, N=N, method="itau")
gofEVCopula(huslerReissCopula(), x, N=N, method="irho")

gofEVCopula(tevCopula(df.fixed=TRUE), x, N=N)
gofEVCopula(tevCopula(df.fixed=TRUE), x, N=N, method="itau")
gofEVCopula(tevCopula(df.fixed=TRUE), x, N=N, method="irho")

copula

Multivariate Dependence with Copulas

v1.0-1

GPL (>= 3) | file LICENCE

Authors

Marius Hofert [aut] (<https://orcid.org/0000-0001-8009-4665>), Ivan Kojadinovic [aut] (<https://orcid.org/0000-0002-2903-1543>), Martin Maechler [aut, cre] (<https://orcid.org/0000-0002-8685-9910>), Jun Yan [aut] (<https://orcid.org/0000-0003-4401-7296>), Johanna G. Nešlehová [ctb] (evTestK(), <https://orcid.org/0000-0001-9634-4796>), Rebecca Morger [ctb] (fitCopula.ml(): code for free mixCopula weight parameters)

Initial release

2020-12-07