gofCopula {copula} | R Documentation |
Goodness-of-fit Tests for Copulas
Description
The goodness-of-fit tests are based, by default, on the empirical
process comparing the empirical copula with a parametric estimate of
the copula derived under the null hypothesis, the default test
statistic, "Sn", being the Cramer-von Mises functional S_n
defined in Equation (2) of Genest, Remillard and Beaudoin (2009). In
that case, approximate p-values for the test statistic can be obtained
either using a parametric bootstrap (see references two and
three) or by means of a faster multiplier approach (see
references four and five).
Alternative test statistics can be used, in particular if a parametric bootstrap is employed.
The prinicipal function is gofCopula()
which, depending on
simulation
either calls gofPB()
or gofMB()
.
Usage
## Generic [and "rotCopula" method] ------ Main function ------
gofCopula(copula, x, ...)
## S4 method for signature 'copula'
gofCopula(copula, x, N = 1000,
method = c("Sn", "SnB", "SnC", "Rn"),
estim.method = c("mpl", "ml", "itau", "irho", "itau.mpl"),
simulation = c("pb", "mult"), test.method = c("family", "single"),
verbose = interactive(), ties = NA,
ties.method = c("max", "average", "first", "last", "random", "min"),
fit.ties.meth = eval(formals(rank)$ties.method), ...)
## (Deprecated) internal 'helper' functions : ---
gofPB(copula, x, N, method = c("Sn", "SnB", "SnC"),
estim.method = c("mpl", "ml", "itau", "irho", "itau.mpl"),
trafo.method = if(method == "Sn") "none" else c("cCopula", "htrafo"),
trafoArgs = list(), test.method = c("family", "single"),
verbose = interactive(), useR = FALSE, ties = NA,
ties.method = c("max", "average", "first", "last", "random", "min"),
fit.ties.meth = eval(formals(rank)$ties.method), ...)
gofMB(copula, x, N, method = c("Sn", "Rn"),
estim.method = c("mpl", "ml", "itau", "irho"),
test.method = c("family", "single"), verbose = interactive(),
useR = FALSE, m = 1/2, zeta.m = 0, b = 1/sqrt(nrow(x)),
ties.method = c("max", "average", "first", "last", "random", "min"),
fit.ties.meth = eval(formals(rank)$ties.method), ...)
Arguments
copula |
object of class |
x |
a data matrix that will be transformed to pseudo-observations
using |
N |
number of bootstrap or multiplier replications to be used to obtain approximate realizations of the test statistic under the null hypothesis. |
method |
a |
estim.method |
a |
simulation |
a string specifying the resampling method for
generating approximate realizations of the test statistic under the null
hypothesis; can be either |
test.method |
a |
verbose |
a logical specifying if progress of the parametric bootstrap
should be displayed via |
... |
for |
trafo.method |
only for the parametric bootstrap ( |
trafoArgs |
only for the parametric bootstrap. A
|
useR |
logical indicating whether an R or C implementation is used. |
ties.method |
string specifying how ranks should be computed,
except for fitting, if there are ties in any of the coordinate
samples of |
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 |
ties |
only for the parametric bootstrap. Logical indicating
whether a version of the parametric boostrap adapted to the
presence of ties in any of the coordinate samples of |
m , zeta.m |
only for the multiplier with |
b |
only for the multiplier. |
Details
If the parametric bootstrap is used, the dependence parameters of the hypothesized copula family can be estimated by any estimation method available for the family, up to a few exceptions. If the multiplier is used, any of the rank-based methods can be used in the bivariate case, but only maximum pseudo-likelihood estimation can be used in the multivariate (multiparameter) case.
The price to pay for the higher computational efficiency of the
multiplier is more programming work as certain
partial derivatives need to be computed for each hypothesized
parametric copula family. When estimation is based on maximization of
the pseudo-likelihood, these have been implemented for six copula
families thus far: the Clayton, Gumbel-Hougaard, Frank, Plackett,
normal and t
copula families. For other families, numerical
differentiation based on grad()
from package
numDeriv is used (and a warning message is displayed).
Although the empirical processes involved in the multiplier and the parametric bootstrap-based test are asymptotically equivalent under the null, the finite-sample behavior of the two tests might differ significantly.
Both for the parametric bootstrap and the multiplier, the approximate p-value is computed as
(0.5 +\sum_{b=1}^N\mathbf{1}_{\{T_b\ge T\}})/(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.
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.
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
These tests were theoretically studied and implemented under the
assumption of continuous margins, which implies that ties in the
component samples occur with probability zero. The presence of ties in
the data might substantially affect the approximate p-values. Through
argument ties
, the user can however select a version of the
parametric bootstrap adapted to the presence of ties. No such adaption
exists for the multiplier for the moment.
References
Genest, C., Huang, W., and Dufour, J.-M. (2013). A regularized goodness-of-fit test for copulas. Journal de la Société française de statistique 154, 64–77.
Genest, C. and Rémillard, 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., Rémillard, 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, https://www.jstatsoft.org/v34/i09/.
Kojadinovic, I. (2017). Some copula inference procedures adapted to the presence of ties. Computational Statistics and Data Analysis 112, 24–41, https://arxiv.org/abs/1609.05519.
Pesarin, F. (2001). Multivariate Permutation Tests: With Applications in Biostatistics. Wiley.
See Also
fitCopula()
for the underlying estimation procedure and
gofTstat()
for details on *some* of the available test
statistics.
Examples
## The following example is available in batch through
## demo(gofCopula)
n <- 200; N <- 1000 # realistic (but too large for interactive use)
n <- 60; N <- 200 # (time (and tree !) saving ...)
## A two-dimensional data example ----------------------------------
set.seed(271)
x <- rCopula(n, claytonCopula(3))
## Does the Gumbel family seem to be a good choice (statistic "Sn")?
gofCopula(gumbelCopula(), x, N=N)
## With "SnC", really s..l..o..w.. --- with "SnB", *EVEN* slower
gofCopula(gumbelCopula(), x, N=N, method = "SnC", trafo.method = "cCopula")
## What about the Clayton family?
gofCopula(claytonCopula(), x, N=N)
## Similar with a different estimation method
gofCopula(gumbelCopula (), x, N=N, estim.method="itau")
gofCopula(claytonCopula(), x, N=N, estim.method="itau")
## A three-dimensional example ------------------------------------
x <- rCopula(n, tCopula(c(0.5, 0.6, 0.7), dim = 3, dispstr = "un"))
## Does the Gumbel family seem to be a good choice?
g.copula <- gumbelCopula(dim = 3)
gofCopula(g.copula, x, N=N)
## What about the t copula?
t.copula <- tCopula(dim = 3, dispstr = "un", df.fixed = TRUE)
if(FALSE) ## this is *VERY* slow currently
gofCopula(t.copula, x, N=N)
## The same with a different estimation method
gofCopula(g.copula, x, N=N, estim.method="itau")
if(FALSE) # still really slow
gofCopula(t.copula, x, N=N, estim.method="itau")
## The same using the multiplier approach
gofCopula(g.copula, x, N=N, simulation="mult")
gofCopula(t.copula, x, N=N, simulation="mult")
if(FALSE) # no yet possible
gofCopula(t.copula, x, N=N, simulation="mult", estim.method="itau")