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

nicky van foreest vanforeest@gmail....
Wed Nov 25 16:41:43 CST 2009

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.

Nicky