问题 设总体$p(x;\theta)$的费希尔信息量存在，若二阶导数$\frac{\partial^2}{\partial\theta^2}p(x;\theta)$对一切的$\theta\in\Theta$存在，证明费希尔信息量

$$ I(\theta)=-E\left(\frac{\partial^2}{\partial\theta^2}\ln p(x;\theta)\right) $$

## 答案 证明： 设 $S_\theta = \dfrac{\partial \ln p(x; \theta)}{\partial \theta}$，则有：

$$ E[S_\theta] = \int_{-\infty}^{+\infty} S_\theta \, p(x; \theta) \, \mathrm{d}x. $$

计算 $E[S_\theta]$： $$ \begin{aligned} E[S_\theta] &= \int_{-\infty}^{+\infty} \frac{\partial \ln p(x; \theta)}{\partial \theta} , p(x; \theta) , \mathrm{d}x \ &= \int_{-\infty}^{+\infty} \frac{1}{p(x; \theta)} \frac{\partial p(x; \theta)}{\partial \theta} , p(x; \theta) , \mathrm{d}x \ &= \int_{-\infty}^{+\infty} \frac{\partial p(x; \theta)}{\partial \theta} , \mathrm{d}x \ &= \frac{\partial}{\partial \theta} \int_{-\infty}^{+\infty} p(x; \theta)

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