Suppose that people arrive at a location $l_1$ and move to a nearby location $l_2$ at various points in time. We are interested in modelling the transition from $l_1$ to $l_2$. We suspect that the transitions follow an exponential distribution with parameter $\lambda$, and that people transition independently with the same probability. By watching people make the transition, we make a list of transition times $T$ at which a person moves from $l_1$ to $l_2$. Also, we count the number of people at $l_1$(including the ones that made the transition) at these same times, and record them in a list $N$. We also assume that the transition time is zero.
Given $T$ and $N$, what is the MLE estimate of $\lambda$?
Assume that $|N|=|T|=m$. I suppose that one crude estimate of $\lambda$ would be by $1/\lambda = \sum_{i \in [1,m]}{t_i}/m$. But this is inaccurate , because the count of entities is different at different times.
So perhaps we could weight the times by the population counts , to get $1/\lambda = \sum_{i \in [1,m]} t_i/(n_i\cdot m)$. I am unsure however that this is an MLE, and if so, I don't know where to find a proof. I've looked up Grimmett and Stirzaker and Kai Lai Chung's Elementary Probability, but couldn't really come up with an apt theorem.
Added I had hoped that the given information would suffice to get a solution.But if it helps, the arrival times at $l_1$ are distributed exponentially with parameter $\lambda'$.
