# Calibrating Generalized Hyperbolic distribution in R - which parameters are valid and allow for a numerical calculation of absolute moments

I am using the R-package ghyp in order to calibrate and model. In fact my coding is based on this paper.

I know that I could do quite a robust fit using fit.ghypuv but I try to maximize a rather complicated likelihood function bases on a Markov-chain with ghyp marginals given the state. Therefore I use a non-linear optimizer (nloptr with parameter NLOPT_LN_NELDERMEAD and optim with parameter Nelder-Mead).

In the alpha-bar parametrization I assure that $\bar{\alpha}$ and $\sigma$ are great zero. Do I have to restrict the parameters further? If yes, then how can I do this? I use $\log$ and $\exp$ transformations to transform the non-negative parameters to the real line and some $(0,1)\rightarrow \mathbb{R}$ mapping for the probabilities of the chain.

Part of the calibration is based on absolute centered moments of the ghyp distribution. These are calculated numerically using ghyp.moment. The problem is that the optimizer searches the whole space (it searches in $\mathbb{R}$ and in the objective function I transform the parameters to their domain, i.e. probabilities to $(0,1)$ and non-negative parameters to $\mathbb{R}^+$) and finds values such that I get the error message

Error in integrate(internal.moment, -Inf, Inf, tmp.order = order[i],
lambda = object@lambda,  :  the integral is probably divergent


E.g. I get the error in this parameter setting:

my.ghyp =ghyp(lambda = -1.570151, mu =  0.006935, sigma = 0.001719412,
gamma = -0.006574129, alpha.bar = 923391)
ghyp.moment(my.ghyp,order=1,absolute=TRUE,central=TRUE)


while it works, if I increase $\sigma$ slightly

my.ghyp =ghyp(lambda = -1.570151, mu =  0.006935, sigma = 0.003,
gamma = -0.006574129, alpha.bar = 923391)
ghyp.moment(my.ghyp,order=1,absolute=TRUE,central=TRUE)


EDIT: In the paper mentioned the absolute moments of $X$, ie. $|X|$, is used for the calibration for $X\sim \text{GHYP}$. This has to be done numerically. If I reformulate the calibration using $X^2$ can I then avoid a numerical procedure and get something explicite or some recursion?

EDIT: I have found the paper Moments of the generalized hyperbolic distribution. There one can find expressions using fractions of Bessel functions that can be calculated in a nummerically stable way for moments up to order four. I will use this.

No additional constraints are needed for the moments to exist. Most likely you are running into numerical inaccuracies in the implementation of ghyp.moment.
• Thanks for your answer. Maybe you can help me solve the problem. I edited the question and added:In the paper mentioned the absolute moment of $X$, ie. $|X|$, is used for the calibration for $X\sim \text{GHYP}$. This has to be done numerically. If I reformulate the calibration using $X^2$ can I then avoid a numerical procedure and get something explicite or some recursion? – Ric Oct 21 '14 at 6:47