       Re: Characteristic Function of Pareto distribution

• To: mathgroup at smc.vnet.net
• Subject: [mg104608] Re: Characteristic Function of Pareto distribution
• From: sashap <pavlyk at gmail.com>
• Date: Wed, 4 Nov 2009 01:41:53 -0500 (EST)
• References: <hconpj\$1kn\$1@smc.vnet.net>

```On Nov 3, 1:57 am, fd <fdi... at gmail.com> wrote:
> All
>
> I'm computing the Characteristic function of a Pareto distribution
> with Mathematica, but I'm trying to make sense of the answers.
>
> If I define
>
> P=ParetoDistribution[1,3]
>
> In:= PDF[P][x]
>
> Out= 3/x^4
>
> Then, I try
>
> In:= CharacteristicFunction[P, t]
>
> During evaluation of In:= \[Infinity]::indet: Indeterminate
> expression 0 (t^2)^(3/2) ComplexInfinity encountered. >>
>
> Out= Indeterminate
>
> If I try calculating the Fourier transform I get
>
> In:= FourierTransform[PDF[P][x], x, \[Omega],  FourierParameters=
-
>
> > {1, 1}]
>
> Out= 1/2 \[Pi] \[Omega]^3 Sign[\[Omega]]
>
> I have some limited understanding of what's going on. In the first
> case the the integration does not include the origin, while with the
> Fourier transform it makes the integration around the origin and
> Mathematica uses the Cauchy integral, thus the result it shows, does
> that make sense?
>
> What seems a problem to me appears if I do not define values for the
> parameters in the Pareto distribution, I get a rather strange answer
>
> P=ParetoDistribution[xm,alpha]
>
> In:= CharacteristicFunction[P, t]
>
> Out=
> alpha (t^2)^(alpha/2) xm^alpha Cos[(alpha \[Pi])/2] Gamma[-alpha] +
>  HypergeometricPFQ[{-(alpha/2)}, {1/2,
>    1 - alpha/2}, -(1/4) t^2 xm^2] - (
>  I alpha Sqrt[t^2]
>    xm HypergeometricPFQ[{1/2 - alpha/2}, {3/2,
>     3/2 - alpha/2}, -(1/4) t^2 xm^2] Sign[t])/(1 - alpha) +
>  I (t^2)^(alpha/2) xm^
>   alpha Gamma[1 - alpha] Sign[t] Sin[(alpha \[Pi])/2]
>
> The expression I find in reference textbooks don't involve
> Hypergoemetric function of any sort, and is much simpler
>
> alpha (-I xm \[Omega])^alpha Gamma[alpha, I xm \[Omega]]
>

An expression used by Mathematica for characteristic
function of Pareto distribution has removable singularities
for integer values of alpha.

The unexpanded form involves MeijerG function:

alpha /2 1/Sqrt[Pi]
MeijerG[{{}, {1 + alpha/2}}, {{1/2, 0, alpha/2}, {}}, -I (
k  t)/2, 1/2]

For alpha == 3 and k ==1:

In:= With[{alpha = 3, k = 1}, alpha/2 1/Sqrt[Pi]
MeijerG[{{}, {1 + alpha/2}}, {{1/2, 0, alpha/2}, {}}, -I (k t)/2,
1/2]] // FunctionExpand // Simplify

Out= 1/4 (4 Cos[t] + 2 I t Cos[t] - 2 t^2 Cos[t] +
2 I t^3 CosIntegral[t] + I t^3 Log + 2 I t^3 Log[-((I t)/2)] -
2 I t^3 Log[t] + 4 I Sin[t] - 2 t Sin[t] - 2 I t^2 Sin[t] -
2 t^3 SinIntegral[t])

Compare with direct integration:

In:= Integrate[
Exp[I t x] PDF[ParetoDistribution[1, 3], x], {x, 1, Infinity},
Assumptions -> Im[t] == 0]

Out= 1/4 (4 E^(I t) + 2 I E^(I t) t - 2 E^(I t) t^2 + \[Pi] t^3 +
2 I t^3 CosIntegral[t] - 2 t^3 SinIntegral[t])

In:= % - %% // FullSimplify[#, t \[Element] Reals] &

Out= 0

Oleksandr Pavlyk
Wolfram Research

>