MLE first-order optimality: d ln L/d theta = 0 at theta-hat; Fisher information

Maximum-likelihood-estimation framework (Fisher 1922 Phil Trans R Soc A 222, 309). Setup: given iid sample x_1, ..., x_n with density f(x; theta), the MLE theta-hat maximises log-likelihood ln L(theta) = sum ln f(x_i; theta). First-order…