 Solution

Assume an identifiable smooth parametric model, an interior true parameter, a consistent interior maximum-likelihood estimator, interchange of differentiation and integration, a uniform local law of large numbers for the Hessian, and nonsingular finite Fisher information. For one observation define the score function $s_{\theta }(X)={\nabla }_{\theta }\log p_{\theta }(X)$ andThe score function has expectation zero by differentiating the normalization of the density. Its sample sum satisfies $n^{-1/2}{S}_n(\theta )\stackrel{d}{\to }N(0,J)$ by the vector central limit theorem. The score function equation and Taylor's formula giveThe bracket divided by $n$ converges in probability to $-J$ by consistency and the assumed uniform law. Solving the equation and applying Slutsky theorem yieldsThe integral Hessian avoids incorrectly assuming a single intermediate point for a vector-valued Taylor formula.