enacopula {copula}

Estimation Procedures for (Nested) Archimedean Copulas

Description

A set of ten different estimators, currently for one-parameter Archimedean copulas, of possibly quite high dimensions.

Usage

enacopula(u, cop,
          method = c("mle", "smle", "dmle",
                     "mde.chisq.CvM", "mde.chisq.KS",
                     "mde.gamma.CvM", "mde.gamma.KS",
                     "tau.tau.mean", "tau.theta.mean", "beta"),
          n.MC = if (method == "smle") 10000 else 0,
          interval = initOpt(cop@copula@name),
          xargs = list(), ...)

Arguments

u

n\times d-matrix of (pseudo-)observations (each value in [0,1]) from the copula to be estimated, where n denotes the sample size and d the dimension. Consider applying the function pobs first in order to obtain u.

cop

outer_nacopula to be estimated (currently only Archimedean copulas are provided).

method

a character string specifying the estimation method to be used, which has to be one (or a unique abbreviation) of

"mle"

maximum likelihood estimator (MLE) computed via .emle.

"smle"

simulated maximum likelihood estimator (SMLE) computed with the function .emle, where n.MC gives the Monte Carlo sample size.

"dmle"

MLE based on the diagonal (DMLE); see edmle.

"mde.chisq.CvM"

minimum distance estimator based on the chisq distribution and Cramér-von Mises distance; see emde.

"mde.chisq.KS"

minimum distance estimation based on the chisq distribution and Kolmogorov-Smirnov distance; see emde.

"mde.gamma.CvM"

minimum distance estimation based on the Erlang distribution and Cramér-von Mises distance; see emde.

"mde.gamma.KS"

minimum distance estimation based on the Erlang distribution and Kolmogorov-Smirnov distance; see emde.

"tau.tau.mean"

averaged pairwise Kendall's tau estimator

"tau.theta.mean"

average of pairwise Kendall's tau estimators

"beta"

multivariate Blomqvist's beta estimator

n.MC

only for method = "smle": integer, sample size for simulated maximum likelihood estimation.

interval

bivariate vector denoting the interval where optimization takes place. The default is computed as described in Hofert et al. (2012). Used for all methods except "tau.tau.mean" and "tau.theta.mean".

xargs

list of additional arguments for the chosen estimation method.

...

additional arguments passed to optimize.

Details

enacopula serves as a wrapper for the different implemented estimators and provides a uniform framework to utilize them. For more information, see the single estimators as given in the section ‘See Also’.

Note that Hofert, Mächler, and McNeil (2013) compared these estimators. Their findings include a rather poor performance and numerically challenging problems of some of these estimators. In particular, the estimators obtained by method="mde.gamma.CvM", method="mde.gamma.KS", method="tau.theta.mean", and method="beta" should be used with care (or not at all). Overall, MLE performed best (by far).

Value

the estimated parameter, \hat{\theta}, that is, currently a number as only one-parameter Archimedean copulas are considered.

References

Hofert, M., Mächler, M., and McNeil, A. J. (2012). Likelihood inference for Archimedean copulas in high dimensions under known margins. Journal of Multivariate Analysis 110, 133–150.

Hofert, M., Mächler, M., and McNeil, A. J. (2013). Archimedean Copulas in High Dimensions: Estimators and Numerical Challenges Motivated by Financial Applications. Journal de la Société Française de Statistique 154(1), 25–63.

See Also

emle which returns an object of "mle" providing useful methods not available for other estimators. demo(opC-demo) and vignette("GIG", package="copula") for examples of two-parameter families. edmle for the diagonal maximum likelihood estimator. emde for the minimum distance estimators. etau for the estimators based on Kendall's tau. ebeta for the estimator based on Blomqvist's beta.

Examples

tau <- 0.25
(theta <- copGumbel@iTau(tau)) # 4/3
d <- 12
(cop <- onacopulaL("Gumbel", list(theta,1:d)))

set.seed(1)
n <- 100
U <- rnacopula(n, cop)

meths <- eval(formals(enacopula)$method)

fun <- function(meth, u, cop, theta) {
	run.time <- system.time(val <- enacopula(u, cop=cop, method=meth))
	list(value=val, error=val-theta, utime.ms=1000*run.time[[1]])
}
t(res <- sapply(meths, fun, u=U, cop=cop, theta=theta))