# absdpsiMC: Absolute Value of Generator Derivatives via Monte Carlo In copula: Multivariate Dependence with Copulas

 absdPsiMC R Documentation

## Absolute Value of Generator Derivatives via Monte Carlo

### Description

Computes the absolute values of the dth 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

 `t` `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.

### Details

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

(-1)^d psi^{(d)}(t) = int_0^Inf x^d exp(-tx) dF(x) ~= (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`.

### Value

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

### 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.

`acopula-families`.

### Examples

```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...
```

copula documentation built on June 15, 2022, 5:07 p.m.