My original question was:
>Can anybody tell me, how to generate random numbers for overdispersed Poisson
>with E(Y)=lambda and Var(Y)=(sigma**2)lambda ?
Most people pointed me to generate negative binomial random numbers (using the gamma distribution) as described in McCullagh and Nelder (1989, pp 198-199) but the resulting variance was not exactly what I needed.
Two mails pointed me to other solutions, where I have applied the first one with great success.
> There is, unfortunately, no distribution with the precise properties you
> are asking for! I think that one or more of Wedderburn, McCullagh or
> Nelder showed that no family of distributions on the non-negative integers
> have the property that variance = constant x mean, with abliity to
> pre-specify any value of the constant.
>
> If you are prepared to accept a distribution that has non-integral values,
> then it is easy. Just generate a Poisson distribution with mean =
> lambda/sigma**2 and multiply the values by sigma**2. The result has mean
> lambda and variance lambda*sigma**2 as you require.
>
> Peter Lane
> Research Statistics Unit, GlaxoSmithKline
> Malcolm Faddy (was at Canterbury, NZ) has done some good work on this. If
> you take a sequence
>
> p(x(t+dt)=i+1|x(t)=i) = l_i.dt + o(dt)
>
> with l_i an increasing sequence, then you can get _any_ overdispersed
> discrete distribution
>
> There aren't many sensible alternatives (beta-binomial?) that I know of.
>
> I can't remember any more details, but it's probably been published
> somewhere.
>
> Hope that helps
> Robert
Many thanks to all,
Martina
Dr. Martina Mittlböck
Department of Medical Computer Sciences
Section of Clinical Biometrics
Spitalgasse 23, A-1090 Vienna, Austria
e-mail: [log in to unmask]
Phone: ++43 (0) 1 40400 2276
Fax: ++43 (0) 1 40400 2278
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