copula (version 0.999-7)

fitCopula: Estimation of the Parameters in Copula Models

Description

Fits a copula model to multivariate data belonging to the unit hypercube. The data can be pseudo-observations constructed from empirical or parametric marginal distribution functions, or true observations from the copula.

Usage

loglikCopula(param, x, copula, hideWarnings)
fitCopula(copula, data, method = c("mpl", "ml", "itau", "irho"),
          start = NULL, lower = NULL, upper = NULL,
          optim.method = "BFGS", optim.control = list(maxit=1000),
          estimate.variance = TRUE, hideWarnings = TRUE)

Arguments

param

a vector of parameter values.

x, data

$n\times d$-matrix of (pseudo-)observations (for "mpl" and "ml" with values necessarily in $[0,1]$) from the copula to be estimated, where $n$ denotes the sample size and $d$ the dimension. Consider applying the func

copula

a "copula" object.

hideWarnings

deprecated and unused for loglikCopula(); logical; if TRUE, warning messages from likelihood maximization (mostly evaluating at invalid parameter values) are suppressed.

method

a character string specifying the method; can be either "ml" (maximum likelihood), "mpl" (maximum pseudo-likelihood), "itau" (inversion of Kendall's

start

a vector of starting values for param.

lower, upper

bounds on the variables for the "Brent" or "L-BFGS-B" method.

optim.control

a list of control parameters to be passed to optim(*, control=optim.control).

optim.method

the method for optim().

estimate.variance

logical; if TRUE, the asymptotic variance is estimated.

Value

• loglikCopula() returns the log likelihood evaluated at the given value of "param".

  The return value of fitCopula() is an object of class "fitCopula" (see there), containing slots (among others!)

• estimate
  the estimate of the parameters.
• var.est
  large-sample (i.e., asymptotic) variance estimate of the parameter estimator (filled with NA if estimate.variance = FALSE).
• copula
  the fitted copula.
• The summary() method for "fitCopula" objects returns a S3 class "summary.fitCopula", simply a list with components method, loglik, and convergence, all three from corresponding slots of the "fitCopula" objects, and
• coefficients
  a matrix of estimated coefficients, standard errors, t values and p-values.

Details

There are methods for vcov(), coef(), logLik(), and nobs().

References

Genest, C. (1987). Frank's family of bivariate distributions. Biometrika 74, 549--555.

Genest, C. and Rivest, L.-P. (1993). Statistical inference procedures for bivariate Archimedean copulas. Journal of the American Statistical Association 88, 1034--1043.

Rousseeuw, P. and Molenberghs, G. (1993). Transformation of nonpositive semidefinite correlation matrices. Communications in Statistics: Theory and Methods 22, 965--984.

Genest, C., Ghoudi, K., and Rivest, L.-P. (1995). A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika 82, 543--552.

Joe, H. (2005). Asymptotic efficiency of the two-stage estimation method for copula-based models. Journal of Multivariate Analysis 94, 401--419.

Demarta, S. and McNeil, A. J. (2005). The t copula and related copulas. International Statistical Review 73, 111--129.

Genest, C. and Favre, A.-C. (2007). Everything you always wanted to know about copula modeling but were afraid to ask. Journal of Hydrologic Engineering 12, 347--368.

Kojadinovic, I. and Yan, J. (2010). Comparison of three semiparametric methods for estimating dependence parameters in copula models. Insurance: Mathematics and Economics 47, 52--63.

See Also

Copula, mvdc for fitting multivariate distributions including the margins aka meta copulas; gofCopula.

Examples

gumbel.cop <- gumbelCopula(3, dim=2)

(Xtras <- copula:::doExtras())
n <- if(Xtras) 200 else 64

x <- rCopula(n, gumbel.cop)## "true" observations
u <- pobs(x)               ## pseudo-observations
## inverting Kendall's tau
fit.tau <- fitCopula(gumbel.cop, u, method="itau")
fit.tau
coef(fit.tau)# named vector
## inverting Spearman's rho
fit.rho <- fitCopula(gumbel.cop, u, method="irho")
fit.rho
## maximum pseudo-likelihood
fit.mpl <- fitCopula(gumbel.cop, u, method="mpl")
fit.mpl
## maximum likelihood
fit.ml <- fitCopula(gumbel.cop, x, method="ml")
fit.ml # print()ing works via summary() ...
## and of that, what's the log likelihood (in two different ways):
(ll. <- logLik(fit.ml))
stopifnot(all.equal(as.numeric(ll.),
    loglikCopula(coef(fit.ml), x=x, copula=gumbel.cop)))

## a multiparameter example
normal.cop <- normalCopula(c(0.6,0.36, 0.6),dim=3,dispstr="un")
x <- rCopula(n, normal.cop) ## "true" observations
u <- pobs(x)                ## pseudo-observations
## inverting Kendall's tau
fit.tau <- fitCopula(normal.cop, u, method="itau")
fit.tau
## inverting Spearman's rho
fit.rho <- fitCopula(normal.cop, u, method="irho")
fit.rho
## maximum pseudo-likelihood
fit.mpl <- fitCopula(normal.cop, u, method="mpl")
fit.mpl
coef(fit.mpl) # named vector
str(sf.mpl <- summary(fit.mpl))
coef(sf.mpl)# the matrix, with SE, t-value, ...

## maximum likelihood
fit.ml <- fitCopula(normal.cop, x, method="ml")
fit.ml
## with dispstr="toep"
normal.cop.toep <- normalCopula(c(0, 0), dim=3, dispstr="toep")
## inverting Kendall's tau
fit.tau <- fitCopula(normal.cop.toep, u, method="itau")
fit.tau
## inverting Spearman's rho
fit.rho <- fitCopula(normal.cop.toep, u, method="irho")
fit.rho
## maximum pseudo-likelihood
fit.mpl <- fitCopula(normal.cop.toep, u, method="mpl")
fit.mpl
## maximum likelihood
fit.ml <- fitCopula(normal.cop.toep, x, method="ml")
fit.ml
## with dispstr="ar1"
normal.cop.ar1 <- normalCopula(c(0), dim=3, dispstr="ar1")
## inverting Kendall's tau
fit.tau <- fitCopula(normal.cop.ar1, u, method="itau")
fit.tau
## inverting Spearman's rho
fit.rho <- fitCopula(normal.cop.ar1, u, method="irho")
fit.rho
## maximum pseudo-likelihood
fit.mpl <- fitCopula(normal.cop.ar1, u, method="mpl")
fit.mpl
## maximum likelihood
fit.ml <- fitCopula(normal.cop.ar1, x, method="ml")
fit.ml

## a t copula with variable df  (df.fixed=FALSE):
(tCop <- tCopula(c(0.2,0.4,0.6), dim=3, dispstr="un", df=5))
set.seed(101)
x <- rCopula(n, tCop) ## "true" observations
u <- pobs(x)          ## pseudo-observations
## maximum likelihood;   start := (rho[1:3], df)
(tc.ml  <- fitCopula(tCop, x, method="ml", start=c(0,0,0, 10)))
(tc.ml. <- fitCopula(tCop, x, method="ml")) # without 'start'
## maximum pseudo-likelihood; the asymptotic variance cannot be estimated
(tc.mpl <- fitCopula(tCop, u, method="mpl", estimate.variance=FALSE,
                     start= c(0,0,0,10)))

if(Xtras) { ##---- typically not run with CRAN checking: ---

## without start:
(tc.mp. <- fitCopula(tCop, u, method="mpl", estimate.variance=FALSE))
all.eqCop <- function(x,y, ...) {
   [e-mail]$counts <- [e-mail]$counts <- NULL
   all.equal(x,y, ...) }
stopifnot(all.eqCop(tc.ml , tc.ml., tol= .005),
          all.eqCop(tc.mpl, tc.mp., tol= .005))

## same t copula but with df.fixed=TRUE	 (--> use same data!)
(tC.f <- tCopula(c(0.2,0.4,0.6), dim=3, dispstr="un", df=5, df.fixed=TRUE))
## maximum likelihood;  start := rho[1:3]		    -------------
(tcF.ml  <- fitCopula(tC.f, x, method="ml", start=c(0,0,0)))
(tcF.ml. <- fitCopula(tC.f, x, method="ml"))# without 'start'
stopifnot(all.eqCop(tcF.ml,tcF.ml., tol= 4e-4))
## the (estimated, asymptotic) var-cov matrix:
vcov(tcF.ml)

## maximum pseudo-likelihood; the asymptotic variance cannot be estimated
(tcF.mpl <- fitCopula(tC.f, u, method="mpl", estimate.variance=FALSE,
                      start=c(0,0,0)))
(tcF.mp. <- fitCopula(tC.f, u, method="mpl", estimate.variance=FALSE))
stopifnot(all.eqCop(tcF.mpl,tcF.mp., tol= 1e-5))

}## end{typically not run ...}