Estimating the mean of a Poisson distribution

Question

Imagine that an event has occurred at times $t_1, t_2,…,t_N$, where $t_i < t_{i+1}$. Assuming that the event occurs following a Poisson distribution, what would be the best estimate of $\lambda$, the mean of the distribution? For example, one possible estimate could be $\frac{N}{t_N}$, but this ignores a lot of useful information. I would appreciate your input.

Answer 1

Apparently, the meaning of this question is as follows: Let $t_1,\dots,t_n$ be the times of the jumps of the Poisson process with intensity $\lambda$, so that $t_j=X_1+\dots+X_j$ for all $j$, where the $X_i$'s are iid random variables with the exponential distribution with mean $1/\lambda$, so that $t_n$ has the Gamma distribution with parameters $n,1/\lambda$. We need to produce a "best" estimator of $\lambda$ based on $t_1,\dots,t_n$ or, equivalently, on $X_1,\dots,X_n$.

To answer this question, note first that \begin{equation} E\frac1{t_n}=\frac{\lambda^n}{\Gamma(n)}\int_0^\infty \frac1x\,x^{n-1}e^{-\lambda x}\,dx=\frac\lambda{n-1}, \end{equation} so that \begin{equation} T_n:=\frac{n-1}{t_n} \end{equation} is an unbiased estimator of $\lambda$, for $n\ge2$. Moreover, $t_n$ is a complete sufficient statistic for $\lambda$ (so, one does not have to worry about losing information on $\lambda$), and $T_n$ is a function of $t_n$. Therefore, $T_n$ is the best -- that is, uniformly minimum variance -- unbiased estimator (UMVUE) of $\lambda$. (This question is a standard exercise in elementary mathematical statistics).