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I have a problem where I have $m$ observations of the following IID quantities (the two types of quantities are independent):

$$N_i \sim \text{Pois}(d \lambda) \quad \quad \quad X_i \sim \text{Bernoulli}(p) \quad \quad \quad p = 1-e^{-d \lambda}.$$

I need to find the MLE for the parameter $\lambda>0$, where $0 \leqslant d < 1$ is a nuisance parameter. My attempt at this problem is written out here and is also shown in the image below; I have not been able to find a closed form solution. Can you please help me find a solution for the MLE.

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    $\begingroup$ If this is homework for a course, please add the self-study tag. If not, please explain context. Please add a clear statement of your starting pdf, and the relevance of the condition $x=1$ in this. $\endgroup$
    – Ben
    Commented Jun 22, 2018 at 22:38
  • $\begingroup$ This is not for a homework course. For now it's for fun... $\endgroup$
    – Corinne Wiesner
    Commented Jun 22, 2018 at 23:21
  • $\begingroup$ the context of the $x=1$ is basically that I have a bernoulli distribution representing a presence or absence of something that I am also separately counting, but from the same sample. So... I have X~Bernoulli(p) where $p=(1-e^{-d \lambda})$ This is assuming that you would have absence when the count is equal to 0. The likelihood function I am defining is then the joint distribution of the count n~poisson($d \lambda$) and x~bernoulli(p). Is that helpful? $\endgroup$
    – Corinne Wiesner
    Commented Jun 22, 2018 at 23:37
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    $\begingroup$ That is extremely helpful - I have edited the question to set out the starting problem. Can you please check this and confirm that it accurately summarises your problem. $\endgroup$
    – Ben
    Commented Jun 22, 2018 at 23:46

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The problem you have here is that your model depends on $\lambda$ only through the product term $d \lambda$, so the latter is a "minimal sufficient parameter" and the former is non-identifiable. That means that you are not able to use the data to separate inference for $\lambda$ out from the product parameter. You can certainly obtain an MLE for the latter, but that is the best you can hope for. If you try to get an MLE for $\lambda$ under the current model, you will get a set of non-unique solution pairs $(\hat{d}, \hat{\lambda})$ where the latter estimator varies over the whole allowable range of the parameter.

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