Here is the motivation for my question. I have a sensor that reports data to me. The occurrence of the reports from the sensor follows a Poisson process (so, obviously, the inter-event times are exponential). I assume a constant event rate $\lambda$.
The device, however, can fail. Let $T_F$ be the failure time. After failure, the event occurrences are not reported. So what I observe are event times $t_1,t_2,\ldots,t_n$ that have occurred on some interval $(0,T_F)$. I do not have prior information.
So this is just a standard Poisson "set-up" except that I don't know the length of the interval over which the events can be observed. I want to estimate both the rate $\lambda$ and the interval length $T_F$.
I have tried writing down the equations for the maximum likelihood estimates for $\lambda$ and $T_F$, but I am finding that they have no solution. (Maybe I have made a mistake.)
It seems that this should be a simple enough standard problem. I have not been able to find an answer (in part because searches that involve the term "interval" return large numbers of pages/answers about confidence intervals). Any help or pointers to references would be greatly appreciated.