absdPsiMC: Absolute Value of Generator Derivatives via Monte Carlo

Description

Computes the absolute values of the $d$th generator derivative $psi^{(d)}$ via Monte Carlo simulation.

Usage

absdPsiMC(t, family, theta, degree = 1, n.MC, method = c("log", "direct", "pois.direct", "pois"), log = FALSE, is.log.t = FALSE)

Arguments

numeric vector of evaluation points.

family

Archimedean family (name or object).

theta

parameter value.

degree

order $d$ of the derivative.

n.MC

Monte Carlo sample size.

method

different methods:

log

if TRUE the logarithm of absdPsi is returned.

is.log.t

if TRUE the argument t contains the logarithm of the “mathematical” $t$, i.e., conceptually, psi(t, *) == psi(log(t), *, is.log.t=TRUE), where the latter may potentially be numerically accurate, e.g., for $t = 10^{500}$, where as the former would just return $psi(Inf, *) = 0$.

Value

numeric vector of the same length as t containing the absolute values of the generator derivatives.

Details

The absolute value of the $d$th derivative of the Laplace-Stieltjes transform $psi=LS[F]$ can be approximated via

(- 1)^{d} ψ^{(d)} (t) = \int_{0}^{\infty} x^{d} \exp (- t x) d F (x) \approx \frac{1}{N} \sum_{k = 1}^{N} V_{k}^{d} \exp (- V_{k} t), t > 0,

where $V_k ~ F, k in {1,...,N}$. This approximation is used where $d=$degree and $N=$n.MC. Note that this is comparably fast even if t contains many evaluation points, since the random variates $V_k ~ F, k in {1,...,N}$ only have to be generated once, not depending on t.

References

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.

Examples

Run this code

t <- c(0:100,Inf)
set.seed(1)
(ps <- absdPsiMC(t, family="Gumbel", theta=2, degree=10, n.MC=10000, log=TRUE))
## Note: The absolute value of the derivative at 0 should be Inf for
## Gumbel, however, it is always finite for the Monte Carlo approximation
set.seed(1)
ps2 <- absdPsiMC(log(t), family="Gumbel", theta=2, degree=10,
                 n.MC=10000, log=TRUE, is.log.t = TRUE)
stopifnot(all.equal(ps[-1], ps2[-1], tolerance=1e-14))
## Now is there an advantage of using "is.log.t" ?
sapply(eval(formals(absdPsiMC)$method), function(MM)
       absdPsiMC(780, family="Gumbel", method = MM,
                 theta=2, degree=10, n.MC=10000, log=TRUE, is.log.t = TRUE))
## not really better, yet...

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