I have read two definitions of the Fisher information matrix. Are they equivalent? (The formulas given below are slightly modified versions of those given in the indicated sources, so as to bring out the differences between them.)
- Zacks, The Theory of Statistical Inference (1971), p. 194: $$ \left[E_\theta\left(\frac{\partial}{\partial\theta_i}\ln f\left(X;\theta_1,\dots,\theta_r\right)\cdot\frac{\partial}{\partial\theta_j}\ln f\left(X;\theta_1,\dots,\theta_r\right)\right)\right]_{i,j} $$
- Schervish, Theory of Statistics (1997), Definition 2.79, p. 111: $$ \mathrm{Cov}_\theta\left(\frac{\partial}{\partial\theta_i}\ln f\left(X;\theta_1,\dots,\theta_r\right),\frac{\partial}{\partial\theta_j}\ln f\left(X;\theta_1,\dots,\theta_r\right)\right) $$