Stuck on a maximum likelihood estimation I am given a random variable with the following density function : $e^{\beta-y}$. I need to find the ML estimator of $\beta$. To find ML estimators, I usually apply the following steps (as explained here and in many other texts):


*

*Find the likelihood function $L(\beta)$, which is the joint probability function of n observations ; under the assumption of independence, the join probability can be split in a product of n times the initial probability function.

*Take the log of $L(\beta)$

*Find $\beta$ so that the derivative of $log L(\beta) = 0$


Which usually works fine. However, I am apparently doing something wrong in this case. Here is what I obtain:


*

*$L(\beta) = \Pi_{i=1}^{n}e^{\beta-y_i} = e^{n\beta}e^{-\sum_{i=1}^{n}y_i} $

*$Ln L(\beta) = n\beta-\sum_{i=1}^{n}y_i $

*$\frac{\partial lnL(\beta)}{\partial\beta} = n = 0$


Which is a dead end as it doesn't give me any information regarding $\beta$. I probably made a very obvious mathematical mistake on the way but I don't seem to figure out what. Or maybe this function has a property I'm unaware of but I haven't found traces of that in my researches. So far I never had trouble with ML.
 A: I'm guessing that what you really have is $Y \sim f(y) = e^{\beta - y}\times I(y \geq \beta)$. This gives us
$$
L(\beta) = \exp(n\beta - \sum_i y_i) \times \prod_i I(y_i \geq \beta) = \exp(n\beta - \sum_i y_i) \times  I(y_{(1)} \geq \beta).
$$
Any MLE is going to have that indicator function equal 1 since there are places with a non-zero likelihood. Note that $\exp(n\beta - \sum_i y_i)$ is an increasing function of $\beta$ so the MLE occurs for the largest value of$\beta$  such that the indicator is 1. This means that $\hat{\beta} = y_{(1)}$.
In general: if your likelihood $L$ is strictly monotonic over the entire domain of your parameter(s) (i.e. either always increasing or always decreasing) then the derivative will not have any roots and so the MLE will occur at a boundary (which might be $\pm \infty$). As you discovered, the derivative of this likelihood (or log-likelihood, but the $\log$ function preserves monotonicity so that doesn't change anything) never equals 0 so it is strictly monotonic. The MLE was then found to be at a boundary, as expected. 
Even if the likelihood is monotonic but not strictly so (i.e. always non-decreasing or always non-increasing) we'll get that the MLE is either at the boundary or could be (by "could be" I mean that if $L$ is flat where it attains its maximum then a whole set of points can be the MLE, but if it is monotonic then a boundary point will always be in the set of possible MLE points). One last remark: in the not strictly monotonic case we could have that $L$ has a shape like $f(x) = x^3$ where it has a horizontal tangent line but not a maximum. Because of this you should always verify that the roots of the derivative are indeed maxima.
