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Science Forum Index » Statistics - Math Forum » question on binomial random variables
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| manuhack |
 Posted: Fri Apr 11, 2008 6:35 am |
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Guest
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Hi,
I encounter the following problem:
X_i ~ Binomial(n_i, p_i), I would like to find the distribution of X_i
conditioned on \sum_i^N X_i = S, where S is given. I guess the
distribution cannot be found with closed form. But is it possible to
get sample (maybe via MCMC) or estimate the expectation?
If I use Poisson(\lambda_i), \lambda_i = n_i * p_i, to approximate,
then I get X_i | \sum_i^N X_i = S ~ multinomial.
I couldn't find anything in the literture on that. Would appreciate
any paper or book which could give me a clue.
Thanks a lot!
Manu |
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| Herman Rubin |
| Posted: Tue Apr 15, 2008 4:28 pm |
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In article <5bf9320b-4274-4334-bfc3-05713c971f5d@y21g2000hsf.googlegroups.com>,
manuhack <manuhack@gmail.com> wrote:
Quote: Hi,
I encounter the following problem:
X_i ~ Binomial(n_i, p_i), I would like to find the distribution of X_i
conditioned on \sum_i^N X_i = S, where S is given. I guess the
distribution cannot be found with closed form. But is it possible to
get sample (maybe via MCMC) or estimate the expectation?
If the p_i are all equal, the distribution is
hypergeometric for one X_i, and "multihypergeometric" for
all, and does not depend on the common value of the p_i.
If the p_i are known, then for one X_i the distribution
can easily be obtained numerically`with little difficulty.
If one wants all, it will be necessary to compute the
unconditioned probabilities separately and tabulate
them; if one is not interested in the large number of
vary small probabilities, this can be done relatively
efficiently.
If you are interested in computing the means, the number
of computations needed, particularly if S is not too
unlikely, is O(sum n_i)^3; I doubt that FFT methods will
give an improvement, as more accuracy will be needed.
I doubt that simulation methods with the Gibbs sampler
or something similar can match the speed of direct
calculation, if the calculation is intelligently done.
Quote: If I use Poisson(\lambda_i), \lambda_i = n_i * p_i, to approximate,
then I get X_i | \sum_i^N X_i = S ~ multinomial.
I couldn't find anything in the literture on that. Would appreciate
any paper or book which could give me a clue.
Thanks a lot!
Manu
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 |
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