bivariate Poisson

[drfg2008]

We are interested in the SPSS implementation of bivariate Poisson distributions, which allow the modelling of (negative and positive) correlated marginal Poisson distributions. Two different constructions of the bivariate distribution are of special interest:

    "trivariate reduction"-type:
        the bivariate distribution is constructed via three independet random variables (RV) Z1~Pois(lambda1), Z2~Pois(lambda2) and Z3~Pois(lambda3). Then the RV X=Z1+Z3 and Y=Z2+Z3 follow jointly a bivariate Poisson distribution BivPoiss(lambda1,lambda2,lambda3).
        the theoretical foundations are described in very detail in
            Johnson, N., Kotz, S. and Balakrishnan, N. (1997) Discrete Multivariate Distributions. New York: Wiley. (pages126ff)
        the implemtation in the open source statistical software R is illustrated in
            Karlis, D et al, JOURNAL OF STATISTICAL SOFTWARE 14, 10 (2005) (http://www.jstatsoft.org/v14/a10/)
    "multiplicative"-type:
        the bivariate dristribution is constructed from marginals with a multiplicative factor, leading to BivPois(lambda1,lambda2, alpha) where alpha describes the multiplicative coupling.
        the theoretical foundations are give in
            J. Lakshminarayana, S.N.N. Pandit & K. Srinivasa Rao (1999): On a bivariate poisson distribution, Communications in Statistics - Theory and Methods, 28:2, 267-276 (http://www.tandfonline.com/doi/abs/10.1080/03610929908832297#tabModule)
        a very recent publication extends this model to take into account possible over- and underdispersion in the marginal distributions, which will allow a much broader application of the SPSS implementation.
            Famoye, F.: A new bivariate generalized Poisson distribution; Statistica Neerlandica 64, 1; (2010)(http://dx.doi.org/10.1111/j.1467-9574.2009.00446.x)

[David Marso]

http://spssx-discussion.1045642.n5.nabble.com/bivariate-Poisson-td5145597.html
;-((

drfg2008 wrote

We are interested in the SPSS implementation of bivariate Poisson distributions, which allow the modelling of (negative and positive) correlated marginal Poisson distributions. Two different constructions of the bivariate distribution are of special interest:

    "trivariate reduction"-type:
        the bivariate distribution is constructed via three independet random variables (RV) Z1~Pois(lambda1), Z2~Pois(lambda2) and Z3~Pois(lambda3). Then the RV X=Z1+Z3 and Y=Z2+Z3 follow jointly a bivariate Poisson distribution BivPoiss(lambda1,lambda2,lambda3).
        the theoretical foundations are described in very detail in
            Johnson, N., Kotz, S. and Balakrishnan, N. (1997) Discrete Multivariate Distributions. New York: Wiley. (pages126ff)
        the implemtation in the open source statistical software R is illustrated in
            Karlis, D et al, JOURNAL OF STATISTICAL SOFTWARE 14, 10 (2005) (http://www.jstatsoft.org/v14/a10/)
    "multiplicative"-type:
        the bivariate dristribution is constructed from marginals with a multiplicative factor, leading to BivPois(lambda1,lambda2, alpha) where alpha describes the multiplicative coupling.
        the theoretical foundations are give in
            J. Lakshminarayana, S.N.N. Pandit & K. Srinivasa Rao (1999): On a bivariate poisson distribution, Communications in Statistics - Theory and Methods, 28:2, 267-276 (http://www.tandfonline.com/doi/abs/10.1080/03610929908832297#tabModule)
        a very recent publication extends this model to take into account possible over- and underdispersion in the marginal distributions, which will allow a much broader application of the SPSS implementation.
            Famoye, F.: A new bivariate generalized Poisson distribution; Statistica Neerlandica 64, 1; (2010)(http://dx.doi.org/10.1111/j.1467-9574.2009.00446.x)

[drfg2008]

@D.M.:
That was also meant as a chargeable request this time.

(By the way: This refers to the bivariate Poisson distribution with a negative intercorrelation (!). That does not exist yet as far as I know)

[David Marso]

Hi Frank,
If you are seeking a programmer/contractor to provide you with a
solution to your Bivariate Poisson questions I might be interested.  I
am not terribly familiar with the details of this area but if the
written explication is sufficiently clear I can likely program just
about anything.
If it would be possible to send me a copy of the papers you cited and
any others which might be relevant  I would be happy to look them over
and provide you with an estimate of the effort which would be required
and a rough quote of the compensation I would require to implement
such a solution.
Regards, David

On Tue, Jul 24, 2012 at 1:05 AM, drfg2008 [via SPSSX Discussion]
<[hidden email]> wrote:

> @D.M.:
> That was also meant as a chargeable request this time.
>
> (By the way: This refers to the bivariate Poisson distribution with a
> negative intercorrelation (!). That does not exist yet as far as I know)
> Dr. Frank Gaeth
> FU-Berlin
>
>
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> NAML

[drfg2008]

Hope this may help:

explanation of the algorithm and R-Code:
http://www.jstatsoft.org/v14/a10/
(including a version of R)


Books / articles (see first message):

Johnson, N., Kotz, S. and Balakrishnan, N. (1997)

Karlis, D et al, JOURNAL OF STATISTICAL SOFTWARE 14, 10 (2005)

J. Lakshminarayana, S.N.N. Pandit & K. Srinivasa Rao (1999)

Famoye, F.: A new bivariate generalized Poisson distribution; Statistica Neerlandica 64
http://onlinelibrary.wiley.com/doi/10.1111/j.1467-9574.2009.00446.x/abstract;jsessionid=2C79509593D87FE763099CD7B9B520C1.d03t04
(Registration required: http://onlinelibrary.wiley.com/user-registration)