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Suppose $x_1, x_2, ..., x_n$ are observed values of IID random variables $X_1, X2, ..., X_n$ from Poisson distribution, i.e., $P(x)=e^{-\lambda} \frac{\lambda^x}{x!}$.

The MLE for $\lambda$ is only defined (as the average of observed values) when there is at least one value of $x_i$ greater than zero; otherwise the log likelihood function is $l(\lambda)=-\lambda n$ and the MLE is undefined ($?$).

So, I am confused when computing the bias of this estimator, since the case of all-zero valued observations actually has non-zero probability, how to account this when finding the average of $\hat{\lambda}$.

Any answer is appreciated.

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    $\begingroup$ what do you mean mle is undefined? $\endgroup$ – Deep North Sep 26 '17 at 3:18
  • $\begingroup$ I mean when all observed values are zero, you simply can't estimate $\lambda$ as the average, which is zero, since we always have $\lambda>0$ $\endgroup$ – Tri Nguyen Sep 26 '17 at 3:21
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The likelihood function of the Poisson given observations $x_1, x_2, \ldots, x_n$ is

$$ l(\lambda; x) = \prod_i e^{-\lambda}\frac{\lambda^{x_i}}{x_i!} = \frac{e^{-n\lambda}}{x_1!x_2!\cdots x_n!}\lambda^{x_1 + x_2 + \cdots + x_n}$$

If $x_1 = x_2 = \cdots = x_n = 0$ then this becomes

$$ l(\lambda; x) = e^{- n \lambda} $$

Which is maximized when $\lambda = 0$.

So the MLE does exist in this case, it is $\lambda = 0$.

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  • $\begingroup$ Does it violate the assumption that $\lambda$ is always greater than zeros? $\endgroup$ – Tri Nguyen Sep 26 '17 at 3:28
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    $\begingroup$ I suppose that depends on who you are communicating with, and their level of fussiness about such things. It seems to me that a poisson distributed variable of rate $\lambda = 0$ is just a random variable that always gives zero. $\endgroup$ – Matthew Drury Sep 26 '17 at 3:30
  • $\begingroup$ Thank you for your answer, if it is supposed that $\lambda>=0$, then we simply have no mystery anymore :). $\endgroup$ – Tri Nguyen Sep 26 '17 at 3:36
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I still keep the opinion that MLE is $undefined$ when all the observations are zero. However, this does not affect on the expectation or variance of this MLE, since the average of all-zero observations is zero, which does not have any effect on these computation, whether MLE is defined or undefined at this point.

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