Volatility-adjusted log returns of the two Banks Citigroup and Bank of America.

Description

A dataset containing the volatility-adjusted log returns of two Banks in the years 2004-2012.

Format

A list with 9 entries containing the Banks log returns divided by years.

Source

Yahoo-Finance.

Applicable dimensions and copula for each test

Description

CopulaTestTable returns a table which shows the applicable dimensions and copula for each test.

Usage

CopulaTestTable()

Details

Before performing a gof test with the package on a dataset, it pays out to know the implemented copula and dimensions for each test. This function is dedicated to help finding the applicable tests depending on the copula and dimensions available.

Value

A character matrix which consists of dimensions for the combination of tests and copula.

Examples

CopulaTestTable()

Volatility-adjusted log returns of the two Cryptocurrencies Bitcoin and Litecoin.

Description

A dataset containing the volatility-adjusted log returns of two Cryptocurrencies in the years 2015-2018.

Format

A list with 4 entries containing the Cryptocurrencies log returns divided by years.

Source

CoinMetrics. https://coinmetrics.io/

Log returns of european stock indices

Description

A dataset containing the log returns of two european stock indices in the year 1998.

Format

A matrix with 100 rows and 2 columns

Source

The raw data are part of the R datasets and were kindly provided by Erste Bank AG, Vienna, Austria.

Log returns of european stock indices

Description

A dataset containing the log returns of three european stock indices in the year 1998.

Format

A matrix with 200 rows and 3 columns

Source

The raw data are part of the R datasets and were kindly provided by Erste Bank AG, Vienna, Austria.

Combining function for tests

Description

gof computes for a given dataset and based on the choices of the user different tests for different copulae. If copulae are given, all the implemented tests for those copulae are calculated. If tests are given, all the implemented copulae for every test are used. If both copulae and tests are given, all possible combinations are calculated.

Usage

gof(
  x,
  priority = "copula",
  copula = NULL,
  tests = NULL,
  customTests = NULL,
  param = 0.5,
  param.est = TRUE,
  df = 4,
  df.est = TRUE,
  margins = "ranks",
  flip = 0,
  M = 1000,
  MJ = 100,
  dispstr = "ex",
  m = 1,
  delta.J = 0.5,
  nodes.Integration = 12,
  lower = NULL,
  upper = NULL,
  seed.active = NULL,
  processes = 1
)

Arguments

Details

If a character vector is given for the argument copula and nothing for tests, then all tests are performed for which the given copulae are implemented. If tests contains a character vector of tests and copula = NULL, then this tests will be performed for all implemented copulae. If character vectors are given for copula and tests, then the tests are performed with the given copulae. If tests = NULL and copula = NULL, then the argument priority catches in and defines the procedure.

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.

Note that this function does not display warning() messages. Due to the large amount of tests run at once, the messages are not tracable to the situation when they appeared. Hence they are omitted for this function.

Value

A list containing several objects of class gofCOP with the following components for each copulae

Examples

data(IndexReturns2D)

gof(IndexReturns2D, priority = "tests", copula = "normal", 
tests = c("gofRosenblattSnB", "gofRosenblattSnC"), M = 5)

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

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

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)

The ArchmGamma Gof test using the Anderson-Darling test statistic and the gamma distribution

Description

gofArchmGamma contains the Gamma 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

gofArchmGamma(
  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

Details

This Anderson-Darling test statistic (supposedly) computes U[0,1]-distributed (under H_0) random variates via the distribution function of the gamma distribution, see Hofert et al. (2014). As written in Hofert et al. (2014) computes this Anderson-Darling test statistic for (supposedly) U[0,1]-distributed (under H_0) random variates via the distribution function of the gamma distribution. 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) = \Gamma_d \left( x_j \right)

is computed (via ADGofTest::ad.test), where x_j = \sum_{i=1}^d (- \ln e_{ij}), \Gamma_d() denotes the distribution function of the gamma distribution with shape parameter d and shape parameter one (being equal to an Erlang(d) distribution function).

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

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)

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

The ArchmSnB test based on the Rosenblatt transformation for archimedean copula

Description

gofArchmSnB contains the SnB 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

gofArchmSnB(
  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

Details

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 resulting independence copula is given by C_{\bot}(\mathbf{u}) = u_1 \cdot \dots \cdot u_d.

The test statistic T is then defined as

T = n \int_{[0,1]^d} \{ D_n(\mathbf{u}) - C_{\bot}(\mathbf{u}) \}^2 d (\mathbf{u})

with D_n(\mathbf{u}) = \frac{1}{n} \sum_{i = 1}^n \mathbf{I}(E_i \leq \mathbf{u}).

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

\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

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)

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

The ArchmSnC test based on the Rosenblatt transformation for archimedean copula

Description

gofArchmSnC contains the SnC 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

gofArchmSnC(
  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

Details

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 resulting independence copula is given by C_{\bot}(\mathbf{u}) = u_1 \cdot \dots \cdot u_d.

The test statistic T is then defined as

T = n \int_{[0,1]^d} \{ D_n(\mathbf{u}) - C_{\bot}(\mathbf{u}) \}^2 d D_n(\mathbf{u})

with D_n(\mathbf{u}) = \frac{1}{n} \sum_{i = 1}^n \mathbf{I}(E_i \leq \mathbf{u}).

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

\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

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)

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

Combining function for tests

Description

The computation of a gof test can take very long, especially when the number of bootstrap rounds is high. The function gofCheckTime computes the time which the estimation most likely takes.

Usage

gofCheckTime(
  copula,
  x,
  tests = NULL,
  customTests = NULL,
  param = 0.5,
  param.est = TRUE,
  df = 4,
  df.est = TRUE,
  margins = "ranks",
  flip = 0,
  M = 1000,
  MJ = 100,
  dispstr = "ex",
  print.res = TRUE,
  m = 1,
  delta.J = 0.5,
  nodes.Integration = 12,
  lower = NULL,
  upper = NULL,
  seed.active = NULL,
  processes = 1
)

Arguments

Details

The function estimates the time which the entire gof test will take.

Examples

## Not run: 
data(IndexReturns2D)

gofCheckTime("normal", IndexReturns2D, "gofKendallKS", M = 10000)

## End(Not run)

Implemented copula for a certain test

Description

gofCopula4Test returns for a given test the applicable implemented copula.

Usage

gofCopula4Test(test)

Arguments

Details

In case that the decision for a certain gof test was already done, it is interesting to know which copula can be used with this test.

Value

A character vector which consists of the names of the copula.

Examples

gofCopula4Test("gofRosenblattSnB")

gofCopula4Test("gofPIOSTn")

Function to derive own tests

Description

gofCustomTest allows to include own Goodness-of-Fit tests and perform the test with the package. 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. It is possible to insert datasets of all dimensions above 1 and the possible copulae are "normal", "t", "clayton", "gumbel", "frank", "joe", "amh", "galambos", "huslerReiss", "tawn", "tev", "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

gofCustomTest(
  copula = c("normal", "t", "clayton", "gumbel", "frank", "joe", "amh", "galambos",
    "huslerReiss", "tawn", "tev", "fgm", "plackett"),
  x,
  customTest = NULL,
  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

Details

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

\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

Examples

# For illustration we load here the test statistic of the gofSn test
Testfunc = function(x, copula) {
            C.theo = pCopula(x, copula = copula)
            C.n = F.n(x, X = x)
            CnK = sum((C.n - C.theo)^2)
            return(CnK)
}

data(IndexReturns2D)

gofCustomTest(copula = "normal", x = IndexReturns2D, 
              customTest = "Testfunc", M=10)

The CvM gof test using the empirical copula

Description

gofCvM performs the "CvM" gof test, described in Genest et al. (2009), for copulae 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", "huslerReiss", "tawn", "tev", "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

gofCvM(
  copula = c("normal", "t", "clayton", "gumbel", "frank", "joe", "amh", "galambos",
    "huslerReiss", "tawn", "tev", "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

Details

With the pseudo observations U_{ij} for i = 1, \dots,n, j = 1, \dots,d and \mathbf{u} \in [0,1]^d is the empirical copula given by C_n(\mathbf{u}) = \frac{1}{n} \sum_{i = 1}^n \mathbf{I}(U_{i1} \leq u_1, \dots, U_{id} \leq u_d). It shall be tested the H_0 hypothesis:

C \in \mathcal{C}_0

with \mathcal{C}_0 as the true class of copulae under H_0. The test statistic T is then defined as

T = n \int_{[0,1]^d} \{ C_n(\mathbf{u}) - C_{\theta_n}(\mathbf{u}) \}^2 d C_n(\mathbf{u})

with C_{\theta_n}(\mathbf{u}) the estimation of C under the H_0.

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

References

Rosenblatt, M. (1952). Remarks on a Multivariate Transformation. The Annals of Mathematical Statistics 23, 3, 470-472.

Hering, C. and Hofert, M. (2014). Goodness-of-fit tests for Archimedean copulas in high dimensions. Innovations in Quantitative Risk Management.

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)

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

GetHybrid gof test

Description

gofGetHybrid computes based on previous test results from this package and on p-values from your own goodness-of-fit tests the hybrid test p-values for the specified testing size.

Usage

gofGetHybrid(result, p_values = NULL, nsets = NULL)

Arguments

Details

In most of scenarios for goodness-of-fit tests, including the one for copula models (e.g. Genest et al. (2009)) there exists no single dominant optimal test. Zhang et al. (2015) proposed a hybrid test which performed in their simulation study more desirably compared to the applied single tests.

The p-value is a combination of the single tests in the following way:

p_n^{hybrid} = \min(q \cdot \min{(p_n^{(1)}, \dots, p_n^{(q)})}, 1)

where q is the number of tests and p_n^{(i)} the p-value of the test i. It is ensured that the hybrid test is consistent as long as at least one of the tests is consistent.

The computation of the individual p-values is performed as described in the details of this tests. Note that the derivation differs.

Value

An object of the class gofCOP with the components

References

Zhang, S., Okhrin, O., Zhou, Q., and Song, P.. Goodness-of-fit Test For Specification of Semiparametric Copula Dependence Models. Journal of Econometrics, 193, 2016, pp. 215-233 doi:10.1016/j.jeconom.2016.02.017

Examples

data(IndexReturns2D)

res_2 = gof(x = IndexReturns2D, copula = "normal", 
            tests = c("gofKernel", "gofKendallCvM", "gofWhite"), M = 10)
gofGetHybrid(result = res_2, 
             p_values = c("MyTest" = 0.3, "AnotherTest" = 0.7), nsets = 3)

The KS gof test using the empirical copula

Description

gofKS performs the "KS" gof test for copulae 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", "huslerReiss", "tawn", "tev", "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

gofKS(
  copula = c("normal", "t", "clayton", "gumbel", "frank", "joe", "amh", "galambos",
    "huslerReiss", "tawn", "tev", "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

Details

With the pseudo observations U_{ij} for i = 1, \dots,n, j = 1, \dots,d and \mathbf{u} \in [0,1]^d is the empirical copula given by C_n(\mathbf{u}) = \frac{1}{n} \sum_{i = 1}^n \mathbf{I}(U_{i1} \leq u_1, \dots, U_{id} \leq u_d). It shall be tested the H_0 hypothesis:

C \in \mathcal{C}_0

with \mathcal{C}_0 as the true class of copulae under H_0. The resulting Kolmogorov-Smirnof test statistic is then given by

T = \sqrt{n} \sup_{v \in [0,1]} |C_n(v) - C_{\theta_n}|

with C_{\theta_n}(\mathbf{u}) the estimation of C under the H_0.

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

References

Rosenblatt, M. (1952). Remarks on a Multivariate Transformation. The Annals of Mathematical Statistics 23, 3, 470-472.

Hering, C. and Hofert, M. (2014). Goodness-of-fit tests for Archimedean copulas in high dimensions. Innovations in Quantitative Risk Management.

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)

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

gof test (Cramer-von Mises) based on Kendall's process

Description

gofKendallCvM tests a given dataset for a copula based on Kendall's process with the Cramer-von Mises test statistic. The margins can be estimated by a bunch of distributions and the time which is necessary for the estimation can be given. The possible copulae are "normal", "t", "clayton", "gumbel", "frank", "joe", "amh", "galambos", "huslerReiss", "tawn", "tev", "fgm" and "plackett". See for reference Genest et al. (2009). The parameter estimation is performed with pseudo maximum likelihood method. In case the estimation fails, inversion of Kendall's tau is used. The approximate p-values are computed with a parametric bootstrap, which computation can be accelerated by enabling in-build parallel computation.

Usage

gofKendallCvM(
  copula = c("normal", "t", "clayton", "gumbel", "frank", "joe", "amh", "galambos",
    "huslerReiss", "tawn", "tev", "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

Details

With the pseudo observations U_{ij} for i = 1, \dots,n, j = 1, \dots,d and \mathbf{u} \in [0,1]^d is the empirical copula given by C_n(\mathbf{u}) = \frac{1}{n} \sum_{i = 1}^n \mathbf{I}(U_{i1} \leq u_1, \dots, U_{id} \leq u_d). Let the rescaled pseudo observations be V_1 = C_n(U_1), \dots, V_n = C_n(U_n) and the distribution function of V shall be K. The estimated version is given by

K_n(v) = \frac{1}{n} \sum_{i=1}^n \mathbf{I}(V_i \leq v)

with v \in [0,1]^d. The testable H_0^{'} hypothesis is then

K \in \mathcal{K}_0 = \{K_{\theta} : \theta \in \Theta \}

with \Theta being an open subset of R^p for an integer p \geq 1, see Genest et al. (2009). The resulting Cramer-von Mises test statistic is then given by

T = n \int_0^1 (K_n(v) - K_{\theta_n})^2 d K_{\theta_n}(v).

Because H_0^{'} consists of more distributions than H_0 the test is not necessarily consistent.

The approximate p-value is computed by the formula

\sum_{b=1}^M \mathbf{I}(|T_b| \geq |T|) / M,

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

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

Christian Genest, Jean-Francois Quessy, Bruno Remillard (2006). Goodness-of-fit Procedures for Copula Models Based on the Probability Integral Transformation. Scandinavian Journal of Statistics, Volume 33, Issue 2, 2006, Pages 337-366. doi:10.1111/j.1467-9469.2006.00470.x

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

Examples

data(IndexReturns2D)

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

gof test (Kolmogorov-Smirnov) based on Kendall's process

Description

gofKendallKS tests a given dataset for a copula based on Kendall's process with the Kolmogorov-Smirnov test statistic. The margins can be estimated by a bunch of distributions and the time which is necessary for the estimation can be given. The possible copulae are "normal", "t", "clayton", "gumbel", "frank", "joe", "amh", "galambos", "huslerReiss", "tawn", "tev", "fgm" and "plackett". See for reference Genest et al. (2009). The parameter estimation is performed with pseudo maximum likelihood method. In case the estimation fails, inversion of Kendall's tau is used. The approximate p-values are computed with a parametric bootstrap, which computation can be accelerated by enabling in-build parallel computation.

Usage

gofKendallKS(
  copula = c("normal", "t", "clayton", "gumbel", "frank", "joe", "amh", "galambos",
    "huslerReiss", "tawn", "tev", "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

Details

With the pseudo observations U_{ij} for i = 1, \dots,n, j = 1, \dots,d and \mathbf{u} \in [0,1]^d is the empirical copula given by C_n(\mathbf{u}) = \frac{1}{n} \sum_{i = 1}^n \mathbf{I}(U_{i1} \leq u_1, \dots, U_{id} \leq u_d). Let the rescaled pseudo observations be V_1 = C_n(U_1), \dots, V_n = C_n(U_n) and the distribution function of V shall be K. The estimated version is given by

K_n(v) = \frac{1}{n} \sum_{i=1}^n \mathbf{I}(V_i \leq v)

with v \in [0,1]^d. The testable H_0^{'} hypothesis is then

K \in \mathcal{K}_0 = \{K_{\theta} : \theta \in \Theta \}

with \Theta being an open subset of R^p for an integer p \geq 1, see Genest et al. (2009). The resulting Kolmogorov-Smirnof test statistic is then given by

T = \sqrt{n} \sup_{v \in [0,1]} |K_n(v) - K_{\theta_n}| .

Because H_0^{'} consists of more distributions than the H_0 is the test not necessarily consistent.

The approximate p-value is computed by the formula

\sum_{b=1}^M \mathbf{I}(|T_b| \geq |T|) / M,

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

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

Christian Genest, Jean-Francois Quessy, Bruno Remillard (2006). Goodness-of-fit Procedures for Copula Models Based on the Probability Integral Transformation. Scandinavian Journal of Statistics, Volume 33, Issue 2, 2006, Pages 337-366. doi:10.1111/j.1467-9469.2006.00470.x

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

Examples

data(IndexReturns2D)

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

2 dimensional gof test of Scaillet (2007)

Description

gofKernel tests a 2 dimensional dataset with the Scaillet test for a copula. The possible copulae are "normal", "t", "clayton", "gumbel", "frank", "joe", "amh", "galambos", "huslerReiss", "tawn", "tev", "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. The approximate p-values are computed with a parametric bootstrap, which computation can be accelerated by enabling in-build parallel computation.

Usage

gofKernel(
  copula = c("normal", "t", "clayton", "gumbel", "frank", "joe", "amh", "galambos",
    "huslerReiss", "tawn", "tev", "fgm", "plackett"),
  x,
  param = 0.5,
  param.est = TRUE,
  df = 4,
  df.est = TRUE,
  margins = "ranks",
  flip = 0,
  M = 1000,
  MJ = 100,
  dispstr = "ex",
  delta.J = 0.5,
  nodes.Integration = 12,
  lower = NULL,
  upper = NULL,
  seed.active = NULL,
  processes = 1
)

Arguments

Details

The Scaillet test is a kernel-based goodness-of-fit test with a fixed smoothing parameter. For the copula density c(\mathbf{u}, \theta), the corresponding kernel estimator is given by

c_n(\mathbf{u}) = \frac{1}{n} \sum_{i=1}^n K_H[\mathbf{u} - (U_{i1}, \dots, U_{id})^{\top}],

where U_{ij} for i = 1, \dots,n; j = 1, \dots,d are the pseudo observations, \mathbf{u} \in [0,1]^d and K_H(y) = K(H^{-1}y)/\det(H) for which a bivariate quadratic kernel is used, as in Scaillet (2007). The matrix of smoothing parameters is H = 2.6073n^{-1/6} \hat{\Sigma}^{1/2} with \hat{\Sigma} the sample covariance matrix. The test statistic is then given by

T = \int_{[0,1]^d} \{c_n(\mathbf{u}) - K_H * c(\mathbf{u}, \theta_n)\} \omega(\mathbf{u}) d \mathbf{u},

where * denotes the convolution operator and \omega is a weight function, see Zhang et al. (2015). The bivariate Gauss-Legendre quadrature method is used to compute the integral in the test statistic numerically, see Scaillet (2007).

The approximate p-value is computed by the formula

\sum_{b=1}^M \mathbf{I}(|T_b| \geq |T|) / M,

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

References

Zhang, S., Okhrin, O., Zhou, Q., and Song, P.. Goodness-of-fit Test For Specification of Semiparametric Copula Dependence Models. Journal of Econometrics, 193, 2016, pp. 215-233 doi:10.1016/j.jeconom.2016.02.017

Scaillet, O. (2007). Kernel based goodness-of-fit tests for copulas with fixed smoothing parameters. Journal of Multivariate Analysis, 98:533-543

Examples

data(IndexReturns2D)

gofKernel("normal", IndexReturns2D, M = 5, MJ = 5)

Output Hybrid gof test

Description

gofOutputHybrid outputs the desired Hybrid tests from previous test results from this package for the specified testing size.

Usage

gofOutputHybrid(result, tests = NULL, nsets = NULL)

Arguments

Details

In most of scenarios for goodness-of-fit tests, including the one for copula models (e.g. Genest et al. (2009)) there exists no single dominant optimal test. Zhang et al. (2015) proposed a hybrid test which performed in their simulation study more desirably compared to the applied single tests.

The p-value is a combination of the single tests in the following way:

p_n^{hybrid} = \min(q \cdot \min{(p_n^{(1)}, \dots, p_n^{(q)})}, 1)

where q is the number of tests and p_n^{(i)} the p-value of the test i. It is ensured that the hybrid test is consistent as long as at least one of the tests is consistent.

The computation of the individual p-values is performed as described in the details of this tests. Note that the derivation differs.

Value

An object of the class gofCOP with the components

References

Zhang, S., Okhrin, O., Zhou, Q., and Song, P.. Goodness-of-fit Test For Specification of Semiparametric Copula Dependence Models. Journal of Econometrics, 193, 2016, pp. 215-233 doi:10.1016/j.jeconom.2016.02.017

Examples

data(IndexReturns2D)

res1 = gof(IndexReturns2D, priority = "tests", copula = "normal", 
           tests = c("gofKendallCvM", "gofRosenblattSnC", "gofKendallKS"), 
           M = 5)
gofOutputHybrid(res1, tests = 1, nsets = 2)
# mind the difference to the regular output
res1

2 and 3 dimensional gof test based on the in-and-out-of-sample approach

Description

gofPIOSRn tests a 2 or 3 dimensional dataset with the approximate PIOS test for a copula. 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. The approximate p-values are computed with a semiparametric bootstrap, which computation can be accelerated by enabling in-build parallel computation.

Usage

gofPIOSRn(
  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

Details

The "Rn" test is introduced in Zhang et al. (2015). It is a information ratio statistic which is approximately equivalent to the "Tn" test, which is the PIOS test. Both test the H_0 hypothesis

H_0 : C_0 \in \mathcal{C}.

"Rn" is introduced because the "Tn" test has to estimate n/m parameters which can be computationally demanding. The test statistic of the "Tn" test is defined as

T = \sum_{b=1}^M \sum_{k=1}^m \{l(U_k^b;\theta_n ) - l(U_k^b;\theta_n^{-b} )\}

with l the log likelihood function, the pseudo observations U_{ij} for i = 1, \dots,n; j = 1, \dots,d and

\theta_n = \arg \min_{\theta} \sum_{i=1}^n l(U_i; \theta)

and

\theta_n^{-b} = \arg \min_{\theta} \sum_{b^{'} \neq b}^M \sum_{k=1}^m l(U_k^{b^{'}}; \theta), b=1, \dots, M.

By defining two information matrices

S(\theta) = - E_0 [\frac{\partial^2}{\partial \theta \partial \theta^{\top}}l \{U_1; \theta \} ],

V(\theta) = E_0 [\frac{\partial}{\partial \theta} l (U_1; \theta) \{\frac{\partial}{\partial \theta} l (U_1; \theta) \}^{\top} ]

where S(\cdot) represents the negative sensitivity matrix, V(\cdot) the variability matrix and E_0 is the expectation under the true copula C_0. Under suitable regularity conditions, given in Zhang et al. (2015), holds then in probability, that

T = tr\{S(\theta^{*})^{-1} V(\theta^{*}) \}

as n \rightarrow \infty.

The approximate p-value is computed by the formula

\sum_{b=1}^M \mathbf{I}(|T_b| \geq |T|) / M,

For more details, see Zhang et al. (2015). The applied estimation method is the two-step pseudo maximum likelihood approach, see Genest and Rivest (1995).

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

References

Zhang, S., Okhrin, O., Zhou, Q., and Song, P.. Goodness-of-fit Test For Specification of Semiparametric Copula Dependence Models. Journal of Econometrics, 193, 2016, pp. 215-233 doi:10.1016/j.jeconom.2016.02.017

Genest, C., K. G. and Rivest, L.-P. (1995). A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika, 82:534-552

Examples

data(IndexReturns2D)

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

2 and 3 dimensional gof test based on the in-and-out-of-sample approach

Description

gofPIOSTn tests a 2 or 3 dimensional dataset with the PIOS test for a copula. 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. The approximate p-values are computed with a semiparametric bootstrap, which computation can be accelerated by enabling in-build parallel computation.

Usage

gofPIOSTn(
  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",
  m = 1,
  lower = NULL,
  upper = NULL,
  seed.active = NULL,
  processes = 1
)

Arguments

Details

The "Tn" test is introduced in Zhang et al. (2015). It tests the H_0 hypothesis

H_0 : C_0 \in \mathcal{C}.

For the test blocks of length m are constructed out of the data. The test compares then the pseudo likelihood of the data in each block with the overall parameter and with the parameter by leaving out the data in the block. By this procedure can be determined if the data in the block influence the parameter estimation significantly. The test statistic is defined as

T = \sum_{b=1}^M \sum_{k=1}^m [l\{U_k^b;\theta_n \} - l\{U_k^b;\theta_n^{-b} \}]

with the pseudo observations U_{ij} for i = 1, \dots,n; j = 1, \dots,d and

\theta_n = \arg \min_{\theta} \sum_{i=1}^n l(U_i; \theta)

and

\theta_n^{-b} = \arg \min_{\theta} \sum_{b^{'} \neq b}^M \sum_{i=1}^m l(U_i^{b^{'}}; \theta), b=1, \dots, M.

The approximate p-value is computed by the formula

\sum_{b=1}^M \mathbf{I}(|T_b| \geq |T|) / M,

The applied estimation method is the two-step pseudo maximum likelihood approach, see Genest and Rivest (1995).

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

References

Zhang, S., Okhrin, O., Zhou, Q., and Song, P.. Goodness-of-fit Test For Specification of Semiparametric Copula Dependence Models. Journal of Econometrics, 193, 2016, pp. 215-233 doi:10.1016/j.jeconom.2016.02.017

Genest, C., K. G. and Rivest, L.-P. (1995). A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika, 82:534-552

Examples

data(IndexReturns2D)

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

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

Description

gofRosenblattChisq contains the RosenblattChisq gof test for copulae, 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 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

gofRosenblattChisq(
  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

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 probability integral transform which uses the mapping \mathcal{R}: (0,1)^d \rightarrow (0,1)^d. Following Genest et al. (2009) ensures this transformation the decomposition of a random vector \mathbf{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 = \mathcal{R}(U_1), \dots, E_n = \mathcal{R}(U_n).

The mapping is performed by assigning to every vector \mathbf{u} for e_1 = u_1 and for i \in \{2, \dots, d\},

e_i = \frac{\partial^{i-1} C(u_1, \dots, u_i, 1, \dots, 1)}{\partial u_1 \cdots \partial u_{i-1}} / \frac{\partial^{i-1} C(u_1, \dots, u_{i-1}, 1, \dots, 1)}{\partial u_1 \cdots \partial u_{i-1}}.

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

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)

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

Gof test using the Anderson-Darling test statistic and the gamma distribution

Description

gofRosenblattGamma contains the RosenblattGamma gof tests for copulae, 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 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

gofRosenblattGamma(
  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

Details

This Anderson-Darling test statistic (supposedly) computes U[0,1]-distributed (under H_0) random variates via the distribution function of the gamma distribution, see Hofert et al. (2014). As written in Hofert et al. (2014) computes this Anderson-Darling test statistic for (supposedly) U[0,1]-distributed (under H_0) random variates via the distribution function of the gamma distribution. 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 probability integral transform which uses the mapping \mathcal{R}: (0,1)^d \rightarrow (0,1)^d. Following Genest et al. (2009) ensures this transformation the decomposition of a random vector \mathbf{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 = \mathcal{R}(U_1), \dots, E_n = \mathcal{R}(U_n).

The mapping is performed by assigning to every vector \mathbf{u} for e_1 = u_1 and for i \in \{2, \dots, d\},

e_i = \frac{\partial^{i-1} C(u_1, \dots, u_i, 1, \dots, 1)}{\partial u_1 \cdots \partial u_{i-1}} / \frac{\partial^{i-1} C(u_1, \dots, u_{i-1}, 1, \dots, 1)}{\partial u_1 \cdots \partial u_{i-1}}.

The Anderson-Darling test statistic of the variates

G(x_j) = \Gamma_d \left( x_j \right)

is computed (via ADGofTest::ad.test), where x_j = \sum_{i=1}^d (- \ln e_{ij}), \Gamma_d() denotes the distribution function of the gamma distribution with shape parameter d and shape parameter one (being equal to an Erlang(d) distribution function).

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

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)

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

The SnB test based on the Rosenblatt transformation

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

Details

This test is based on the Rosenblatt probability integral transform which uses the mapping \mathcal{R}: (0,1)^d \rightarrow (0,1)^d to test the H_0 hypothesis

C \in \mathcal{C}_0

with \mathcal{C}_0 as the true class of copulae under H_0. Following Genest et al. (2009) ensures this transformation the decomposition of a random vector \mathbf{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 = \mathcal{R}(U_1), \dots, E_n = \mathcal{R}(U_n).

The mapping is performed by assigning to every vector \mathbf{u} for e_1 = u_1 and for i \in \{2, \dots, d\},

e_i = \frac{\partial^{i-1} C(u_1, \dots, u_i, 1, \dots, 1)}{\partial u_1 \cdots \partial u_{i-1}} / \frac{\partial^{i-1} C(u_1, \dots, u_{i-1}, 1, \dots, 1)}{\partial u_1 \cdots \partial u_{i-1}}.

The resulting independence copula is given by C_{\bot}(\mathbf{u}) = u_1 \cdot \dots \cdot u_d.

The test statistic T is then defined as

T = n \int_{[0,1]^d} \{ D_n(\mathbf{u}) - C_{\bot}(\mathbf{u}) \}^2 d (\mathbf{u})

with D_n(\mathbf{u}) = \frac{1}{n} \sum_{i = 1}^n \mathbf{I}(E_i \leq \mathbf{u}).

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

\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

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)

The SnC test based on the Rosenblatt transformation

Description

gofRosenblattSnC contains the SnC gof test from Genest (2009) for copulae 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

gofRosenblattSnC(
  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

Details

This test is based on the Rosenblatt probability integral transform which uses the mapping \mathcal{R}: (0,1)^d \rightarrow (0,1)^d to test the H_0 hypothesis

C \in \mathcal{C}_0

with \mathcal{C}_0 as the true class of copulae under H_0. Following Genest et al. (2009) ensures this transformation the decomposition of a random vector \mathbf{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 = \mathcal{R}(U_1), \dots, E_n = \mathcal{R}(U_n).

The mapping is performed by assigning to every vector \mathbf{u} for e_1 = u_1 and for i \in \{2, \dots, d\},

e_i = \frac{\partial^{i-1} C(u_1, \dots, u_i, 1, \dots, 1)}{\partial u_1 \cdots \partial u_{i-1}} / \frac{\partial^{i-1} C(u_1, \dots, u_{i-1}, 1, \dots, 1)}{\partial u_1 \cdots \partial u_{i-1}}.

The resulting independence copula is given by C_{\bot}(\mathbf{u}) = u_1 \cdot \dots \cdot u_d.

The test statistic T is then defined as

T = n \int_{[0,1]^d} \{ D_n(\mathbf{u}) - C_{\bot}(\mathbf{u}) \}^2 d D_n(\mathbf{u})

with D_n(\mathbf{u}) = \frac{1}{n} \sum_{i = 1}^n \mathbf{I}(E_i \leq \mathbf{u}).

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

\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

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)

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

Applicable gof tests for testing problem

Description

gofTest4Copula returns for a given copula and a dimension the applicable implemented tests.

Usage

gofTest4Copula(copula = NULL, d = 2)

Arguments

Details

Before performing a gof test on a dataset, it pays out to have a close look at the Scatterplot to receive an idea about the possible type of copula. Afterwards follows the decision about the test. The tests in this package can be used for different types of copulae functions and dimensions. This function is dedicated to help finding the applicable gof tests for the dataset.

Value

A character vector which consists of the names of the tests.

Examples

gofTest4Copula("clayton", d = 2)

gofTest4Copula("gumbel", d = 5)

The function gofWhich was renamed to gofTest4Copula. Please use gofTest4Copula.

Description

Please see and use the function gofTest4Copula.

Usage

gofWhich(copula = NULL, d = 2)

Arguments

Value

A character vector which consists of the names of the tests.

The function gofWhichCopula was renamed to gofCopula4Test. Please use gofCopula4Test.

Description

Please see and use the function gofCopula4Test.

Usage

gofWhichCopula(test)

Arguments

Value

A character vector which consists of the names of the copula.

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

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", "clayton", "gumbel", "frank" and "joe". 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 = c("normal", "t", "clayton", "gumbel", "frank", "joe"),
  x,
  param = 0.5,
  param.est = TRUE,
  df = 4,
  df.est = TRUE,
  margins = "ranks",
  flip = 0,
  M = 1000,
  lower = NULL,
  upper = NULL,
  seed.active = NULL,
  processes = 1
)

Arguments

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

H_0 : \mathbf{V}(\theta) + \mathbf{S}(\theta) = 0

where \mathbf{V}(\theta) is the expected Hessian matrix and \mathbf{S}(\theta) is the expected outer product of the score function.

The test statistic is derived by

T_n = n(\bar{d}(\theta_n))^\top A_{\theta_n}^{-1} \bar{d}(\theta_n)

with

\bar{d}(\theta_n) = \frac{1}{n} \sum_{i=1}^n vech(\mathbf{V}_n(\theta_n|\mathbf{u}) + \mathbf{S}_n(\theta_n|\mathbf{u})),

d(\theta_n) = vech(\mathbf{V}_n(\theta_n|\mathbf{u}) + \mathbf{S}_n(\theta_n|\mathbf{u})),

A_{\theta_n} = \frac{1}{n} \sum_{i=1}^n (d(\theta_n) - D_{\theta_n} \mathbf{V}_n(\theta_n)^{-1} \delta l(\theta_n))(d(\theta_n) - D_{\theta_n} \mathbf{V}_n(\theta_n)^{-1} \delta l(\theta_n))^\top

and

D_{\theta_n} = \frac{1}{n} \sum_{i=1}^n [\delta_{\theta_k} d_l(\theta_n)]_{l=1, \dots, \frac{p(p+1)}{2}, k=1, \dots, p}

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 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.

Please note, the test gofWhite may be unstable for t-copula. Please handle the results carefully.

Value

An object of the class gofCOP with the components

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. https://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(IndexReturns2D)

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

Interface with copula class

Description

gofco is an interface with the copula package. It reads out the information from a copula class object and hands them over to a specified gof test or set of tests.

Usage

gofco(
  copulaobject,
  x,
  tests = c("gofPIOSRn", "gofKernel"),
  customTests = NULL,
  margins = "ranks",
  flip = 0,
  M = 1000,
  MJ = 100,
  dispstr = "ex",
  m = 1,
  delta.J = 0.5,
  nodes.Integration = 12,
  lower = NULL,
  upper = NULL,
  seed.active = NULL,
  processes = 1
)

Arguments

Details

The function reads out the arguments in the copula class object. If the dependence parameter is not specified in the object, it is estimated. In case that the object describes a "t"-copula, then the same holds for the degrees of freedom. The dimension is not extracted from the object. It is obtained from the inserted dataset.

When more than one test shall be performed, the hybrid test is computed too.

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

References

Yan, Jun. Enjoy the joy of copulas: with a package copula. Journal of Statistical Software 21.4 (2007): 1-21.

Examples

data(IndexReturns2D)
copObject = normalCopula(param = 0.5)

gofco(copObject, x = IndexReturns2D, tests = c("gofPIOSRn", "gofKernel"), 
      M = 20)

Plotting function for objects of class gofCOP

Description

Plots an object of class gofCOP.

Usage

## S3 method for class 'gofCOP'
plot(
  x,
  copula = NULL,
  hybrid = NULL,
  point.bg = "white",
  point.col = "black",
  point.cex = 0.7,
  jitter.val = 0.1,
  pal = "xmen",
  bean.b.o = 0.2,
  inf.method = "hdi",
  theme = 2,
  ...
)

Arguments

Details

The plotting function is constructed around pirateplot from the yarrr package. Please see respective package for more details on the non-default specifications of the plotting function.

We recommend not to amend the arguments xlim, data, formula, sortx, xaxt, ylim and ylab from pirateplot. The arguments were defined such that the resulting plot displays the test results in a proper manner.

Value

None

References

Phillips, N. (2017). yarrr: A Companion to the e-Book "YaRrr!: The Pirate's Guide to R". R package version 0.1.5. https://CRAN.R-project.org/package=yarrr

Printing function for objects of class gofCOP

Description

print.gofCOP prints the values of an object of class gofCOP.

Usage

## S3 method for class 'gofCOP'
print(x, ...)

Arguments

Value

None

Printing function for an object of class goftime. goftime objects are created by the function gofCheckTime.

Description

Printing function for an object of class goftime. goftime objects are created by the function gofCheckTime.

Usage

## S3 method for class 'goftime'
print(x, ...)

Arguments