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We have likelihood function

$$ L\left(\mathbf{\Theta} \middle| x_1, \ldots, x_n \right) = \prod^n_{i=1} f\left( x_i \middle| \mathbf{\Theta} \right) $$

and a score function

$$ V\left(\mathbf{\Theta} \middle| x_1, \ldots, x_n \right) = \nabla_\Theta \sum^n_{i=1} \ln{f\left(x_i, \mathbf{\Theta} \right)}, $$

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

$$ V\left(\mathbf{\Theta^{\ast}}\right) = 0, $$

but we need $L\left(\mathbf{\Theta^{\ast}}\middle| x_1, \ldots, x_n \right) \neq 0$.

So, is this inequality always true or is there a situation, where the likelihood with the MLE could be equal to zero?

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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 $\theta\in\Theta$, that is, you essentially observe something that is impossible under that model, then yes, your maximum likelihood would be zero.

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