= Solution
<Maximum likelihood estimation> treats the observations as fixed and chooses admissible parameter values maximizing their joint <likelihood function>, or equivalently its logarithm. Interior score equations identify candidates, but parameter boundaries and global maximality still need checking; an unattained supremum is not an estimate.
The <statistical method of moments> equates enough empirical moments to the corresponding theoretical moments to solve for the unknown parameters. For example, for a one-parameter exponential rate, the relation $\mathbb EX=1/\lambda$ gives the moment estimate $1/\bar X$; maximizing $\lambda^ne^{-\lambda\sum X_i}$ gives the same estimate. Agreement of these two methods in this example does not imply that they agree in every model, nor that either is unbiased.
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