Solved – Can the likelihood function in MLE be equal to zero

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?

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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