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":: evaluates the logarithm of the sum involved in the Monte Carlo approximation in a numerically stable way;
"direct":: directly evaluates the sum;
"pois.direct":: interprets the sum in terms of the density of a Poisson distribution and evaluates this density directly;
"pois":: as for method="pois" but evaluates the logarithm of the Poisson density in a numerically stable way.

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 $\psi=\mathcal{LS}[F]$ can be approximated via $$(-1)^d\psi^{(d)}(t)=\int_0^\infty x^d\exp(-tx)\,dF(x)\approx\frac{1}{N}\sum_{k=1}^NV_k^d\exp(-V_kt),\ t> 0,$$ where $V_k\sim F,\ k\in\{1,\dots,N\}$. 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 $V_k\sim F,\ k\in\{1,\dots,N\}$ only have to be generated once, not depending on t.

References

Hofert, M., M<U+00E4>chler, M., and McNeil, A. J. (2013). Archimedean Copulas in High Dimensions: Estimators and Numerical Challenges Motivated by Financial Applications. Journal de la Soci<U+00E9>t<U+00E9> Fran<U+00E7>aise de Statistique 154(1), 25--63.

Examples

Run this code

# NOT RUN {
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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