I have a density which is defined on $[0,1)$. On 1 it is not because there is a positive probability of attaining the value 1. Can I still use Maximum Likelihood to estimate the parameters?
For example I have a density with parameter $p$ such that the probability of $\mathbb{P}[X=1]=p$ and on $[0,1)$ it is equally distributed.
My idea was that I can just sort of combine the MLE approaches of discrete / continuous variables: $$ \log \prod_{i=1}^n (p 1_{x_i=1}+ \dfrac{1}{1-p}1_{x_i \neq 1}) $$ I am not quite sure whether this is correct because I am mixing up the lebesgue and the counting measure as far as I understand. Thank you very much for your input!