Re: weighted sum of independent beta random variables

From: Batu Chalise (chalise_at_sent5.uni-duisburg.de)
Date: 10/27/04 

Date: Wed, 27 Oct 2004 15:39:04 +0000 (UTC)

On Wed, 27 Oct 2004 13:13:55 +0100, David Jones wrote:
>Batu Chalise wrote:
>> Dear all,
>>
>> Is there any closed form expression or approximation for the pdf or
>> cdf of a weighted sum of independent beta random variables?
>> For example, if W=a_1X_1+a_2X_2+----+ a_nX_n, where a_i's are
>> constants and X_i's are beta random variables with parameters
(a=1)
>> and (b>0), what would be exact or approx. pdf of W? Looking for
your
>> help.
>>
>> Best regards,
>> Batu
>
>Are these beta bistributions on (0,1) or one of the other forms of
>beta distributions? If bounded, then you know the bounds on W, which
>you should take into account in any approximation. You can work out
>the moments of W and base an approximation on these. You may be able
>to do something analytical which may help to determine the shape
>(power behaviour) of the density at the lower and upper bounds, which
>you could use to help construct an approximation.
>
>You can get and exact formula for the characteristic function of the
>sum, and an exact formula for inverting this to get the pdf, but I
>expect this is not what you are looking for.
>
>David Jones

Dear David,

yes, in my question X_i's are beta random variables with parameters
(a,b) where a=1 and b>1 (i made a mistake last time). Since a=1, the
pdf of X_i's becomes simplified.

I do not know the bounds on W. But bounds on X_i's are known because
they are beta random. You are also right that the characteristic
function of W can be calculated but I do not know how to compute
analytically the pdf from this characteristic function. Can you
provide me some references that deal with inversion formula?

 Thank you very much for your help in advance.

Best regards,
Batu

Relevant Pages

  • Re: weighted sum of independent beta random variables
    ... Batu Chalise wrote: ... > cdf of a weighted sum of independent beta random variables? ... pdf of W? ... you should take into account in any approximation. ...
    (sci.stat.math)
  • Re: The First Variation of a PDF
    ... the pdf is symmetric about x=0 and the pdf takes a very simple form. ... to assume the approximation is not negative. ... Since the support for fas a function of x is compact, ... pdf of fhas the same compact support? ...
    (sci.math)
  • Re: The First Variation of a PDF
    ... that is a function of a real parameter c. ... When c is non-zero, the form of the pdf is non-symmetric about x=0 ... One way I considered was to construct an approximation to the pdf: ...
    (sci.math)
  • weighted sum of independent beta random variables
    ... Is there any closed form expression or approximation for the pdf or ... cdf of a weighted sum of independent beta random variables? ... pdf of W? ...
    (sci.stat.math)