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

josef.pktd@gmai... josef.pktd@gmai...
Wed Nov 25 17:23:26 CST 2009

On Wed, Nov 25, 2009 at 5:41 PM, nicky van foreest <vanforeest@gmail.com> wrote:
> Hi,
> 2009/11/25  <josef.pktd@gmail.com>:
>> 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),
> It has been some time ago (more than 5 years...), but I recall that
> Whitt, in his articles on the numerical inversion of Laplace
> transforms, discretized Laplace transforms to facilitate the
> inversion, The distributions themselves are not necessarily discrete.
> One example would be the waiting time distribution of customers in a
> queue, which is continuous for most service and arrival processes.
> There is certainly potential for dedicated numerical inversion algo's
> for the Laplace transforms of density and distribution functions. The
> latter form a somewhat specialized sort of function. Distribution
> functions are 0 at -\infty, and 1 at \infty, and are non decreasing.
> They may also have discontinuities, but not too many. These properties
> may affect the inversion.  Besides these properties, the transforms
> are used to obtain insight into the behavior of the sum of independent
> random variables. Such sums can be rewritten as the product of the
> transforms of distribution. This product in turn requires inversion
> to, as some people call it, take away the Laplacian curtain.

Is there an advantage to using Laplace instead of Fourier transform
in this context?

I had to stop working on this, because I have to finish up some other
The advantages that I saw for the Fourier transform are that it has
directly the interpretation as characteristic function with explicit
formulas for many distributions, e.g. stable distribution which has no
analytical expression for pdf or cdf, and the availability of fft to do fast
inversion instead of pointwise integration.

Except reading the definition of the Laplace transform, I don't know
much about it and have no idea what the numerical advantages might

Another application, besides the sum of rvs,  that I looked at, are mixture
distributions, e.g. Poisson mixture of continuous (lognormal) distributions,
which are also easy to calculate in terms of the characteristic function, and
 I guess the Laplace transform.

This is an older reference that is cited quite a bit:
Waller, Lance A., Bruce W. Turnbull, and J. Michael Hardin. “Obtaining
Distribution Functions by Numerical Inversion of Characteristic
Functions with Applications.” The American Statistician 49, no. 4
(November 1995): 346-350.


> Nicky
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