Let \(X\) be a random variable following a distribution \(F(x \mid \theta)\) where \(\theta \in \Omega\). Assume that \(F(x \mid \theta)\) has continuous density \(f(x \mid \theta)\). The maximum likelihood estimator of \(\theta\) is given by the value of \(\theta\) that maximizes \(F(x \mid \theta)\) with respect to \(\theta\). Assuming that \(F(x \mid \theta)\) is maximized at a unique interior point of \(\Omega\), argue that the functional corresponding to the maximum likelihood estimator can be written implicitly as

\[\int_{-\infty}^{\infty} \frac{f^{\prime}(x \mid \theta)}{f(x \mid \theta)} d F(x \mid \theta)=0\]

where the derivative in the integral is taken with respect to \(\theta\). Discuss any additional assumptions that you need to make.