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Assume events occur as a Poission process with parameter $\lambda$ in a fixed size interval.

Assume $Y$ is a random variable, $Y = \frac{\text{count of intervals with 0 events}}{\text{total count of intervals}}$

Therefore we know that

$E(Y) = P(0 \text{ events occur})$

or $E(Y) = e^{-\lambda}$.

Now I observe a value of $Y$ from a single data point as $Y=y$. Can I estimate $\lambda$ as a function of $y$?

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  • $\begingroup$ I don't see any estimator here: by definition, an estimator is a function of data, but you haven't exhibited any data at all. What then could you possibly mean by "unbiased" and "estimator"?? $\endgroup$ – whuber Feb 25 '17 at 2:31
  • $\begingroup$ I know J, and I would like to know if this is valid way to estimate lambda. If yes, could we comment on the quality of this estimation in anyway? Sorry if this is currently ambiguous, it has been a long time since I studied probability concepts. $\endgroup$ – CHIRAG Feb 25 '17 at 2:47
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    $\begingroup$ If you know P(0 events) (rather than observe a proportion of cases with 0 events) this is not an estimation problem at all -- you have a known strictly monotonic function of the parameter so when you solve that to get $\lambda$, you know $\lambda$ exactly. It makes no sense to talk about this as estimation unless you have information from a sample rather than knowing the probability. $\endgroup$ – Glen_b -Reinstate Monica Feb 25 '17 at 5:19
  • $\begingroup$ @Glen_b I have improved the formulation of my question. You are right, I have an observation of proportion of cases with 0 events. How should I go about it? $\endgroup$ – CHIRAG Feb 28 '17 at 4:04
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Note that -- conditional on the total number of fixed-size intervals ($n$) -- the number of them with counts of 0 will be $\textit{binomial}(n,e^{-\lambda})$.

In that case, this looks like a straightforward inference problem for which a simple maximum likelihood (ML) estimator for $e^{-\lambda}$ exists, and hence one for $\lambda$ is available (though you need the number of counts of 0 - let's call that $N_0$ - to be non-zero for it to make any sense, and even conditioning on $N_0>0$, it will be biased).

You might consider if you can find ways to modify it (/regularize it) to make a more suitable estimator. For example, it might help to get started along those lines if you consider ways (other than ML) to estimate the binomial parameter when the sample proportion is 0 (the ML estimate is 0, but given that the parameter is in (0,1) that's not much use).

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