We have likelihood function
$$
Lleft(mathbf{Theta} middle| x_1, ldots, x_n right) = prod^n_{i=1} fleft( x_i middle| mathbf{Theta} right)
$$
and a score function
$$
Vleft(mathbf{Theta} middle| x_1, ldots, x_n right) = nabla_Theta sum^n_{i=1} ln{fleft(x_i, mathbf{Theta} right)},
$$
where $x_i$ are observations and $mathbf{Theta}$ is the vector of parameters.
If we consider the maximum likelihood estimator $mathbf{Theta^{ast}}$, it's the extreme of $L$, so
$$
Vleft(mathbf{Theta^{ast}}right) = 0,
$$
but we need $Lleft(mathbf{Theta^{ast}}middle| x_1, ldots, x_n right) neq 0$.
So, is this inequality always true or is there a situation, where the likelihood with the MLE could be equal to zero?
Best Answer
If you have a very inadequate model such that at least one discretely (or continuously) distributed observation has zero probability (or probability density) for any parameter value $thetainTheta$, that is, you essentially observe something that is impossible under that model, then yes, your maximum likelihood would be zero.
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