[R-sig-ME] negative binomial distribution mixture model

[Fabian Amman]

Dear mixed-models group

I'd like to ask for your advice in the following matter:

I have two data vectors of observed count data: 'A' and 'B', whereas
count A[n] and B[n] refer to the same observation point.
'A' is assumed to follow a negative binomial distribution.
'B' is assumed to be the result from two underlying processes. For one
again an independent negative binomial distribution and additionally a
kind of shadowing effect from 'A', where a certain fraction 'f' of
counts from 'A' are also observed in 'B'.

Accordingly, one can simulate such data by:

        set.seed(13579)
        f <- runif(1)
        A <- rnbinom(100, mu = 100, size = 1)
        B <- floor(f*A) + rnbinom(100, mu = 20, size = 1)
        
Now to my question: Since 'B' is a mixed model of two negative binomial
distributed variables, how can I estimate the value of factor 'f'
explaining the underlying data best?

The final result should be a vector 'B2', correcting 'B' for its shadow
portion: 'B2 = B - (A/f)'; Since I know 'A' and 'B', to gain this I need
to estimate 'f' from the data and the underlying assumption of a
negative binomial distribution.

As I guess you could figure from my question, I m not really a statistic
nor an R expert, therefore any help is highly appreciated.

Thank you very much in advance.
Regards
Fabian