The Fisher information matrix

Question

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.)

1. Zacks, The Theory of Statistical Inference (1971), p. 194: $$ \left[E_\theta\left(\frac{\partial}{\partial\theta_i}\ln f\left(X;\theta_1,\dots,\theta_r\right)\cdot\frac{\partial}{\partial\theta_j}\ln f\left(X;\theta_1,\dots,\theta_r\right)\right)\right]_{i,j} $$
2. Schervish, Theory of Statistics (1997), Definition 2.79, p. 111: $$ \mathrm{Cov}_\theta\left(\frac{\partial}{\partial\theta_i}\ln f\left(X;\theta_1,\dots,\theta_r\right),\frac{\partial}{\partial\theta_j}\ln f\left(X;\theta_1,\dots,\theta_r\right)\right) $$

Answer 1

Fortunately they are the same. All we need to prove is $E_{\theta} \left[ \frac{\partial}{\partial \theta} \log f(x|\theta) \right] = 0$, so here we go.

First note that

$$\int_{\mathbb{R}} \frac{\partial}{\partial \theta}\log(f(x|\theta)) \cdot f(x|\theta) dx = \int_{\mathbb{R}} \frac{\partial f(x|\theta)}{\partial \theta}dx. $$

Under the conditions of Leibniz's integral rule you can rewrite the above

$$ \frac{\partial}{\partial \theta} \int_{\mathbb{R}} f(x|\theta)dx = \frac{\partial}{\partial \theta} 1 = 0.$$

So, except pathological cases where the conditions mentioned above do not hold, the expected value of the score is 0, and the covariance is the same as the expected value of the product.