Maximum likelihood of a no-closed form density

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Supposed we have density function $f(y_i)$ which have no closed form expression. Now can we find the estimating equation of the log likelihood of $f(y_i)$ by taking the derivative(I have tried this multiple times and failed) or we need to do optimization technique? I am asking just to make sure that either I am doing something wrong or it's usually not possible to find estimating equation by taking derivative of the log-likelihood of no closed-form density.