Re: weighted sum of independent beta random variables
From: David Jones (dajxxx_at_ceh.ac.uk)
Date: 10/27/04
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Date: Wed, 27 Oct 2004 17:54:03 +0100
Batu Chalise wrote:
> 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.
If the X_i are all on (0,1) and the a_i are all non-negative, then W
has 0 as a lower bound and the sum of the a_i as an upper bound. More
generally the lower bound is
min(0, sum of negative a_i)
and the upper bound is
max(0, sum of postive a_i)
>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?
The characteristic function is essentially the Fourier transform of
the pdf, so the pdf is the inverse Fourier transform of the
characteristic function.
>
> Thank you very much for your help in advance.
>
> Best regards,
> Batu
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