# [SciPy-User] [OT] Transform (i.e., Fourier, Laplace, etc.) methods in Prob. & Stats.

josef.pktd@gmai... josef.pktd@gmai...
Wed Nov 25 14:19:54 CST 2009

```On Wed, Nov 25, 2009 at 2:53 PM, David Goldsmith
<d.l.goldsmith@gmail.com> wrote:
> Are there enough applications of transform methods (by which I mean,
> Fourier, Laplace, Z, etc.) in probability & statistics for this to be
> considered its own specialty therein?  Any text recommendations on it (even
> if it's only a chapter dedicated to it)?  Thanks,
>

Some information is in the thread on my recent question
"characteristic functions of probability distributions"

There is a large literature in econometrics and statistics about using
the characteristic function for estimation and testing.
The reference of Nicky for queuing theory uses mostly the Laplace
transform (for discrete distributions), while for continuous
distributions and mixtures the continuous fourier transform is used
(definition of characteristic function).

I started to work my way through part of the literature with
application in finance. Main use I looked at was using the inverse
Fourier transform when the characteristic function has an analytical
expression and the pdf does not, e.g used for estimating difffusion
processes by MLE.

I haven't looked much at the Laplace transform, because I'm more
interested in the continuous random variable case.

Related methods work directly with the empirical characteristic
function to do estimation and testing, but I haven't looked much at
that yet.

I looked at references from all over the place, essentially with
google searches and searches of the main stats journal collections.
(I have a unsorted collection of pdfs on my computer but no overview

Of course the biggest and oldest use of the Fourier transform is the
frequency domain analysis in time series analysis.

It's not off topic because I try to get some of these methods
programmed in python.

Josef

> DG
>
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```