I am working on analyzing some data that come from the following experimental setup:
Imagine doing repeated trials of an experiment where a lever is pulled and a random number shows up drawn from a poisson distribution with parameter $\lambda$ (here representing the expected number/trial). Unfortunately you are really bad at reading numbers and so you are only able to ascertain if the number was 0 or $\ge 1$. What you are interested in though is deriving $\lambda$ from this censored data set.
For a single fixed value of $\lambda$ the current literature approach to solving this problem is to do the following: run the trials a large number of time and count how many times $\ge 1$ comes up and ratio that to the number of trials let's call this $\mu$. $\mu$ is then set equal to 1 - $p_0$ (the probability of getting 0) because it represents the sum of the probabilities of having gotten 1, 2, 3, etc and solved for $\lambda$ by the definition of the poisson distribution: $\lambda = -ln(1-\mu)$.
My question is this: is there a way to solve this problem if $\lambda$ is free to vary from trial to trial and I only care about getting the average value of $\lambda$?
As a note I realize the literature procedure doesn't work if $\lambda$ is large and so the experiments are usually configured to avoid that.