 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 $\mathbb{E}X=1/\lambda$ gives the moment estimate $1/\bar{X}$; maximizing ${\lambda }^n{e}^{-\lambda \sum X_i}$ 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.