[R-SIG-Finance] R-SIG-Finance Digest, Vol 51, Issue 19

Dale W.R. Rosenthal daler at uic.edu
Wed Aug 27 16:18:15 CEST 2008

Matthias --

You can try a few approaches.

1) You could find a four-parameter distribution.  Some of those have 
been mentioned; and, I'd add the Normal Inverse Gaussian as a 
suggestion.  Barndorff-Nielsen and Sorensen have both looked at the 
properties of this distribution as applied to finance.

2) Another option would be to go with an Edgeworth expansion -- very 
similar to a Taylor expansion of the characteristic function.  The 
advantage is that Edgeworth expansions are designed precisely for 
matching cumulants (almost the same as moments, up to the fourth).  The 
disadvantage is that Edgeworth expansions can yield "pdf"s which go 
negative for a short time.  However, if you have a specific case in 
mind; you could check for that.

3) You could also try something like an Edgeworth expansion of the 
log-density; then, exponentiate the result.  You will get a pdf 
approximation that is non-negative; but, these expansions can explode in 
the tails -- so they need even more careful handling.  They can also be 
tough to integrate.

The reference for the Edgeworth expansions is McCullagh's _Tensor 
Methods in Statistics_ or Kolassa's _Series Approximation Methods in 
Statistics, 3d Ed._.  McCullagh uses tensor notation -- which some 
people find hard to get used to; his book can also be tough to find 
outside of libraries.  Kolassa doesn't cover the log-density expansion; 
but, his development is free from tensor notation.

4) You could also try using a saddlepoint method with an approximate 
cumulant generating function.  Easton and Ronchetti (JASA, 1986) would 
be the reference if you want to try this method.  I haven't tried it 
myself; but, if it works for you, it would probably be a very nice 

Hope that helps.


Dale W.R. Rosenthal
Assistant Professor, Department of Finance
University of Illinois at Chicago
SSRN: http://ssrn.com/author=906862 

> Message: 1
> Date: Tue, 26 Aug 2008 17:53:18 +0200
> From: Matthias.Koberstein at hsbctrinkaus.de
> Subject: [R-SIG-Finance] Generating Distributions with set skewness
> 	and	kurtosis
> To: r-sig-finance at stat.math.ethz.ch
> Hello,
> I am reaching out to you for help since I am struggeling to find a function
> to generate distributions with a set statistical properties as kurtosis and
> skewdness.
> Lets say I want to generate random variables following a "normal"
> distribution, but with skewness 2 and kurtosis 5.
> How would I do that, the most efficient way? Are there any packages for
> that? I had a quick look but were only able to find packages which
> calculate statistical
> distribution properties after having the data.
> Thank you very much
> Matthias

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