MLE (maximum likelihood estimation) can be defined mathematically for discrete or continuous variables. But there is a technical specificity about variables being neither discrete nor continuous. Instead they are a mixture of discrete and continuous. More precisely:
Assume you have a model with a parameter $\theta$ and an observable variable $X$ whose distribution depends on $\theta$. You have $x_1,x_2...x_n$ an independent sample of real-life observations of $X$.
If $X$ is a discrete variable, the maximum likelihood estimator is:
$$\hat\theta=\text{argmax}_\theta \left(\displaystyle\prod_{i=1}^n P_\theta(X=x_i)\right)$$
If $X$ has a density $p_\theta$, then you just use the density instead:
$$\hat\theta=\text{argmax}_\theta \left(\displaystyle\prod_{i=1}^np_\theta(x_i)\right)$$
Sometimes, $X$ is essentially continuous but has one or several atoms: the mixture of a continuous distribution and a discrete distribution. A good example : ML estimate of exponential distribution (with censored data).
I want to find a way to define MLE mathematically when you have atoms for students or people with some mathematical background but not so much in statistics. Ideally :
- not too much theoretical or abstract
- rather general
- not uselessly confusing
I struggle. Any idea?
