I came across this problem in survival analysis where an event can happen an a number of time points.
For discrete time $t_i$, the likelihood of an event happening at any of the time points is $x(t_j) \prod_{i:t_i < t_j} (1 - x(t_i))$ where $x(t_j) = P(T = t | T \ge t)$ and $0 \le x(t_i) \le 1$.
Imagine two observations. Observation 1 happened at either $t_1$ or $t_2$ and observation 2 happened at either $t_2$ or $t_3$.
The likelihood then looks like this (using $x_i \equiv x(t_i))$:
$$ L = \{ x_1 + (1 - x_1)x_2 \} \times \{(1 - x_1)x_2 + (1 - x_1)(1 - x_2)x_3\} $$
The maximum likelihood estimator is then $x_1 =0, x_2 = 1, x_3 = 0$. But I have no intuition as to why this is - why is there only weight in the overlap?
I think this is true for any number of observations with overlaps - the maximum likelihood estimate always seems to result in the non-overlapping sections having value $0$. Though I can't seem to prove it.
Any help?