When trying to obtain point estimators, we can use the maximum likelihood method, which, given a random sample whose distribution depends on a parameter , consists of finding the value of that maximises the sample’s likelihood function.
This value of is called the maximum likelihood estimator and we can use it to build an estimator for .
Let be a random sample whose probability distribution depends on a parameter . The maximum likelihood method consists of maximising:
In other words, the method consists of finding the value of that maximises . According to the theory behind local extrema, this is just finding the that satisfies:
Sometimes, it might be a good idea to instead maximise: