Goodness-of-fit Tests for Copulas

Description

Goodness-of-fit tests for copulas based on the empirical process comparing the empirical copula with a parametric estimate of the copula derived under the null hypothesis. Approximate p-values for the test statistic can be obtained either using the parametric bootstrap (see the two first references) or by means of a fast multiplier approach (see references three and four).

The default test statistic, "Sn", is the Cramer-von Mises functional S[n] defined in Equation (2) of Genest, Remillard and Beaudoin (2009).

The prinicipal function is gofCopula() which, depending on simulation either calls gofPB() or gofMB().

Usage

gofCopula(copula, x, N = 1000,
          method = "Sn",
          estim.method = eval(formals(fitCopula)$method),
          simulation = c("pb", "mult"),
	  verbose = TRUE, print.every = NULL, ...)

gofPB(copula, x, N, method = eval(formals(gofTstat)$method),
      estim.method = eval(formals(fitCopula)$method),
      trafo.method = c("none", "rtrafo", "htrafo"),
      trafoArgs = list(), verbose = TRUE, ...)

gofMB(copula, x, N, method = c("Sn", "Rn"),
      estim.method = eval(formals(fitCopula)$method), verbose = TRUE,
      useR = FALSE, m = 1/2, zeta.m = 0, b = 0.05, ...)

Arguments

Details

If the parametric bootstrap is used, the dependence parameters of the hypothesized copula family can be estimated either by maximizing the pseudo-likelihood, by inverting Kendall's tau, or by inverting Spearman's rho. If the multiplier is used, any estimation method can be used in the bivariate case, but only maximum pseudo-likelihood estimation can be used in the multivariate (multiparameter) case.

For the normal and t copulas, several dependence structures can be hypothesized: "ex" for exchangeable, "ar1" for AR(1), "toep" for Toeplitz, and "un" for unstructured (see ellipCopula()). For the t copula, "df.fixed" has to be set to TRUE, which implies that the degrees of freedom are not considered as a parameter to be estimated.

Thus far, the multiplier approach is implemented for six copula families: the Clayton, Gumbel, Frank, Plackett, normal and t.

Although the processes involved in the multiplier and the parametric bootstrap-based test are asymptotically equivalent under the null, note that the finite-sample behavior of the two tests might differ significantly.

Also note that in the case of the parametric and multiplier bootstraps, the approximate p-value is computed as

(0.5 + sum(T[b] >= T, b=1, .., N)) / (N+1),

where T and T[b] denote the test statistic and the bootstrapped test statistc, respectively. This ensures that the approximate p-value is a number strictly between 0 and 1, which is sometimes necessary for further treatments. See Pesarin (2001) for more details.

Value

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

Note

These tests were derived under the assumption of continuous margins, which implies that ties occur with probability zero. The presence of ties in the data might substantially affect the approximate p-values. One way of dealing with ties was suggested in the Journal of Statistical Software reference.

Since R is widely used by practitioners, a word of warning concerning goodness-of-fit tests in general is also advisable. Goodness-of-fit tests are often (ab)used in practice to “justify” an assumption under which one then continues to work (carelessly). From a mathematical point of view, this is not correct.

References

Genest, C., Huang, W., and Dufour, J.-M. (2013). A regularized goodness-of-fit test for copulas. Journal de la Soci<c3><83><c2><a9>t<c3><83><c2><a9> fran<c3><83><c2><a7>aise de statistique 154, 64–77.

Genest, C. and R<c3><83><c2><a9>millard, B. (2008). Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models. Annales de l'Institut Henri Poincare: Probabilites et Statistiques 44, 1096–1127.

Genest, C., R<c3><83><c2><a9>millard, B., and Beaudoin, D. (2009). Goodness-of-fit tests for copulas: A review and a power study. Insurance: Mathematics and Economics 44, 199–214.

Kojadinovic, I., Yan, J., and Holmes M. (2011). Fast large-sample goodness-of-fit tests for copulas. Statistica Sinica 21, 841–871.

Kojadinovic, I. and Yan, J. (2011). A goodness-of-fit test for multivariate multiparameter copulas based on multiplier central limit theorems. Statistics and Computing 21, 17–30.

Kojadinovic, I. and Yan, J. (2010). Modeling Multivariate Distributions with Continuous Margins Using the copula R Package. Journal of Statistical Software 34(9), 1–20. http://www.jstatsoft.org/v34/i09/.

Pesarin, F. (2001). Multivariate Permutation Tests: With Applications in Biostatistics. Wiley.

See Also

fitCopula() for the underlying estimation procedure and gofTstat() for the available test statistics.

Examples

## the following example is available in batch through
## demo(gofCopula)% == ../demo/gofCopula.R __keep >> EXACTLY << in sync!__

## Not run: 
## A two-dimensional data example ----------------------------------
x <- rCopula(200, claytonCopula(3))

(tau. <- cor(x, method="kendall")[1,2]) # around 0.5 -- 0.6
## Does the Gumbel family seem to be a good choice?
(thG <- iTau(gumbelCopula(), tau.)) # 3.02
gofCopula(gumbelCopula(thG), x)
# SnC: really s..l..o..w.. --- SnB is *EVEN* slower
gofCopula(gumbelCopula(thG), x, method = "SnC")
## What about the Clayton family?
(thC <- iTau(claytonCopula(), tau.)) # 4.05
gofCopula(claytonCopula(thC), x)
gofCopula(claytonCopula(thC), x, method = "AnChisq")

## The same with a different estimation method
gofCopula(gumbelCopula (thG), x, estim.method="itau")
gofCopula(claytonCopula(thC), x, estim.method="itau")


## A three-dimensional example  ------------------------------------
x <- rCopula(200, tCopula(c(0.5, 0.6, 0.7), dim = 3, dispstr = "un"))

## Does the Clayton family seem to be a good choice?
## here starting with the "same" as indepCopula(3) :
(gCi3 <- gumbelCopula(1, dim = 3, use.indepC="FALSE"))
gofCopula(gCi3, x)
## What about the t copula?
t.copula <- tCopula(rep(0, 3), dim = 3, dispstr = "un", df.fixed=TRUE)
## this is *VERY* slow currently 
gofCopula(t.copula, x)

## The same with a different estimation method
gofCopula(gCi3,     x, estim.method="itau")
gofCopula(t.copula, x, estim.method="itau")

## The same using the multiplier approach
gofCopula(gCi3,     x, simulation="mult")
gofCopula(t.copula, x, simulation="mult")

## End(Not run)

Results