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Problem. Eggs are thought to be infected with the bacterium salmonella enteritidis so that the number of organisms, $Y$, in each has a Poisson distribution with mean $\mu$. The value of $Y$ cannot be observed directly, but after a period it becomes certain whether the egg is infected ($Y > 0$) or not ($Y = 0$). Out of $n$ such eggs, $r$ are found to be infected. Find the maximum likelihood estimator of $µ$ and its asymptotic variance.

I'm practicing for an exam and I can't understand this question. I need to find the MLE of $\mu$, but what distribution should I use?

In the exercise, it says that we do not observe $Y$, but we know if the egg is infected or not. I'm confused... should I use the Poisson distribution, not with $n$, but with $n-r$ (un-infected eggs); is that a binomial? I don't really understand what to do...

I'll try and solve the exercise this way, using a new variable with binomial distribution, since we don't observe Y we need to use X. Is it correct? I

whuber
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Disappointed Mom
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    As this is preparing for an exam, please add the self-study tag to the question and read the site policy about how they get answered. Then review the probability mass function at zero events (in this case, no infection) for a Poisson distribution with mean $\mu$. – EdM Oct 16 '22 at 19:10
  • Thanks for the answer, I have added the self-study tag! Also I know that in order to compute the MLE I need to create the joint distribution (likelihood), find the log and maximixe its first derivative. My question was aimed at the distribution (PMF in this case) to use, since it is clearly stated that we do not observe each Y... – Disappointed Mom Oct 17 '22 at 09:43
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    The problem tells you to assume a Poisson distribution. A Poisson distribution with mean $\mu$ has a PMF of $\mu^k e^{-\mu}/k!$, where $k$ is the number of events (per egg, here). Although you don't know $Y$ for each egg, you do know for each egg if $Y=0$ and thus have an estimate of $P(k=0)$. For the likelihood of the data given a value of $\mu$, consider the PMF for $k=0$ (for the probability of $Y=0$ given $\mu$) and your data showing that $n-r$ eggs out of $n$ had $k=0$. If that gives you the solution, please post it as an answer to your question. – EdM Oct 17 '22 at 13:35
  • It might be safer to return to first principles. You have $(n-r)$ observations at $Y=0$ with $P(Y=0|\mu)= \exp(-\mu)$, as you show. You have $r$ observations with $Y>0$, and $P(Y>0|\mu)=1-P(Y=0|\mu)$. Set up the log-likelihood on that basis. Remember that you ae maximizing with respect to $\mu$, which might be important for the asymptotic variance estimate. – EdM Oct 19 '22 at 16:35

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