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Note that the maximum likelihood estimator of \(\sigma^2\) for the normal model is not the sample variance \(S^2\). Then, the joint probability mass (or density) function of \(X_1, X_2, \cdots, X_n\), which we'll (not so arbitrarily) call \(L(\theta)\) is: But how would we implement the method in practice? Well, suppose we have a random sample \(X_1, X_2, \cdots, X_n\) for which the probability density (or mass) function of each \(X_i\) is \(f(x_i \theta)\). (So, do you see from where the name "maximum likelihood" comes?) So, that is, in a nutshell, the idea behind the method of maximum likelihood estimation.
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