I have two observations from a Poisson distribution. The first one ($N_r$) come with a Poisson distribution with mean $k_1$. For the second one ($N_e$) I know that $N_e - M$ also come from the same Poisson distribution with mean $k_1$, but the parameter $M$ is unknown. My observations are $N_r$ and $N_e$. I want to do the MLE for $M$ and $k_1$.

If I write the equations: $$ f(N_r, N_e|k_1, M) = \frac{e^{-k_1}k_1^{N_r}}{N_r!}\frac{e^{-k_1}k_1^{N_e-M}}{(N_e-M)!}=\frac{e^{-2k_1}k_1^{N_r+N_e-M}}{N_r!(N_e-M)!}$$

Taking logarithms: $$ ln(f) = -2k_1 + (N_r+N_e-M)ln(k_1)-ln(N_r!(N_e-M)!)$$

I can estimate the mean parameter $k_1$ as $$ k_1 = \frac{N_r+N_e-M}{2}$$

But I don't know how to estimate the $M$ parameter as I have to deal with the factorial in the derivative. Can anybody help me?

Thank you very much!!


Are you familiar with Stirling's Approximation? It is an approximation, but widely used. You can convert the factorial expression into a multiplicative logarithmic expression.

Edit: You can also convert the factorial expression into an expression that uses gamma functions. gamma(n+1) = n!


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