R: ArchmChisq Gof test using the Anderson-Darling test statistic...

gofArchmChisq {gofCopula}

R Documentation

ArchmChisq Gof test using the Anderson-Darling test statistic and the chi-square distribution

Description

gofArchmChisq contains the Chisq gof test with a Rosenblatt transformation for archimedean copulae, described in Hering and Hofert (2015). The test follows the RosenblattChisq test as 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 (except for the "amh" copula) and the possible copulae are "clayton", "gumbel", "frank", "joe" and "amh". The parameter estimation is performed with pseudo maximum likelihood method. In case the estimation fails, inversion of Kendall's tau is used.

Usage

gofArchmChisq(
  copula = c("clayton", "gumbel", "frank", "joe", "amh"),
  x,
  param = 0.5,
  param.est = TRUE,
  margins = "ranks",
  flip = 0,
  M = 1000,
  lower = NULL,
  upper = NULL,
  seed.active = NULL,
  processes = 1
)

Arguments

`copula`	The copula to test for. Possible are `"clayton"`, `"gumbel"`, `"frank"`, `"joe"` and `"amh"`.
`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.
`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.
`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 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 transformation for archimedean copula which uses the mapping \mathcal{R}: (0,1)^d \rightarrow (0,1)^d. Following Hering and Hofert (2015) the mapping provides pseudo observations E_i, given by

E_1 = \mathcal{R}(U_1), \dots, E_n = \mathcal{R}(U_n).

Let C be an Archimedean copula with d monotone generator \psi and continuous Kendall distribution function K_{C}. Then,

e_{j}=\left(\frac{\sum_{k=1}^{j} \psi^{-1}\left(u_{k} \right)}{\sum_{k=1}^{j+1} \psi^{-1}\left(u_{k}\right)}\right)^{j}, j \in\{1, \ldots, d-1\}

and

e_{d}= \frac{n}{n+1} K_{C}(C(u))

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 jth component in the ith 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.

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

Christian Hering, Marius Hofert (2015) Goodness-of-fit tests for Archimedean copulas in high dimensions. In: Glau K., Scherer M., Zagst R. (eds) Innovations in Quantitative Risk Management, Springer Proceedings in Mathematics & Statistics, Volume 99, Springer, Cham, 357-373. doi: 10.1007/978-3-319-09114-3_21

Examples


data(IndexReturns2D)

gofArchmChisq("clayton", IndexReturns2D, M = 10)

[Package gofCopula version 0.4-1 Index]