[R] How to standardize the generalized hyperbolic distribution?

[Hans Karl]

Hi,

I want to fit a standardized generalized hyperbolic distribution to my data. I am aware, that I can do this with the dsgh command of the fBasics package along with the optim command. My problem is, that I also want to have a derivation of it. So I need the theory behind it, i.e. I need the formula of the probability density function which they use and the derivation of it.

I thought about standardizing the generalized hyperbolic distribution. So I use the formula of the mean and the variance (e.g. can be found on wikipedia) and set them to zero and one. Then I try to solve for single paramters and insert them in the original pdf and use this along with the optim command. BUT the problem is, that there is no unique solution, so the mean and the variance, as you can see on the wikipedia page depends on several parameters and terms. So I could have different set of parameter combinations which all fulfill the requirement of the mean to be zero and the variance to be equal to one. What are people doing in this case? How do they get a "unique" solution for the standardized version?

e.g. I do a simplified example, so you see, what I mean


Suppose, the pdf is given by

mu + alpha + 2* beta + 3* delta



the mean is given by

mu + delta*beta

and the variance given by

beta*delta + delta/(alpha-beta)



I set them to zero and one:

0 = mu + delta*beta

1 = beta*delta + delta/(alpha-beta)


now I can solve in different ways and insert in different ways in the original pdf.

So the result is, that I can get a pdf formula, which depends on alpha, beta, and delta. The mu is

fixed.

Or I can get a pdf, which depends on mu and delta. The alpha and delta is then fixed.

Both would fulfill the requirement of mean zero and variance one.


What should one do in such a case? Thank your very much.