クラメール-ラオの下限
$$
\begin{eqnarray}
\mathrm{E}\left[\hat{\theta}\right]
&=&\theta\;\cdots\;\hat{\theta}を\thetaの不偏推定量とする\\
\frac{\partial}{\partial \theta}\mathrm{E}\left[\hat{\theta}\right]
&=&\frac{\partial}{\partial \theta}\theta\;\cdots\;両辺を\thetaで微分する\\
\frac{\partial}{\partial \theta}\int{\hat{\theta}f(x;\theta)\mathrm{d}x}
&=&1
\;\cdots\;右辺は\thetaを\thetaで微分したので1\\
&&\;\cdots\;左辺は期待値の定義通りE\left[\hat{\theta}\right]=\int_X \hat{\theta} f(x;\theta) \mathrm{d}xと展開した\\
&&\;\cdots\;f(x;\theta)は\thetaをパラメタとしたxの確率密度分布である\\
\end{eqnarray}
$$
$$
\begin{eqnarray}
1=\frac{\partial}{\partial \theta}\int{\hat{\theta}f(x;\theta)\mathrm{d}x}
&=&\int{\hat{\theta}\frac{\partial f(x;\theta)}{\partial \theta}\mathrm{d}x}
\;\cdots\;微分と積分の順番を入れ替えられる場合を考える\\
&=&\int{\hat{\theta} \frac{\partial f(x;\theta)}{\partial \theta}\left\{\frac{1}{f(x;\theta)}f(x;\theta)\right\}\mathrm{d}x}\\
&=&\int{\hat{\theta} \left\{\frac{\partial f(x;\theta)}{\partial \theta}\frac{1}{f(x;\theta)}\right\}f(x;\theta)\mathrm{d}x}\\
&=&\int{\hat{\theta} \frac{\partial \log{f(x;\theta)}}{\partial \theta} f(x;\theta)\mathrm{d}x}
\;\cdots\; \frac{\partial f(x;\theta)}{\partial \theta} \frac{1}{f(x;\theta)}=\frac{\partial \log{f(x;\theta)}}{\partial \theta}\\
&=&\mathrm{E}\left[\hat{\theta} \frac{\partial \log{f(x;\theta)}}{\partial \theta} \right]\\
&=&\mathrm{E}\left[\hat{\theta} \frac{\partial \log{f(x;\theta)}}{\partial \theta} \right]
-\theta \mathrm{E}\left[\frac{\partial \log{f(x;\theta)}}{\partial \theta} \right]
\;\cdots\;\href{https://shikitenkai.blogspot.com/2020/04/blog-post_19.html}{スコア凾数の期待値は0\left(\mathrm{E}\left[\frac{\partial \log{f(x;\theta)}}{\partial \theta} \right]=0\right)}なので加えても変わらない\\
&=&\mathrm{E}\left[\hat{\theta} \frac{\partial \log{f(x;\theta)}}{\partial \theta}
-\theta \frac{\partial \log{f(x;\theta)}}{\partial \theta} \right]
\;\cdots\;c\mathrm{E}\left[X\right]=\mathrm{E}\left[cX\right],\;\mathrm{E}\left[X\right]+\mathrm{E}\left[Y\right]=\mathrm{E}\left[X+Y\right]\\
&=&\mathrm{E}\left[(\hat{\theta}-\theta) \frac{\partial \log{f(x;\theta)}}{\partial \theta} \right]\\
&=&\mathrm{E}\left[
\left((\hat{\theta}-\theta)-\mathrm{E}\left[(\hat{\theta}-\theta)\right]\right)
\left(\frac{\partial \log{f(x;\theta)}}{\partial \theta}-\mathrm{E}\left[\frac{\partial \log{f(x;\theta)}}{\partial \theta}\right]\right)
\right]
\;\cdots\;\mathrm{E}\left[(\hat{\theta}-\theta)\right]=\mathrm{E}\left[\hat{\theta}\right]-\mathrm{E}\left[\theta\right]=\theta-\theta=0
,\;\mathrm{E}\left[\frac{\partial \log{f(x;\theta)}}{\partial \theta}\right]=0\\
&=&\mathrm{Cov}\left[\hat{\theta}, \frac{\partial \log{f(x;\theta)}}{\partial \theta} \right]
\;\cdots\;E\left[\left(X-E\left[X\right]\right) \left(Y-E\left[Y\right]\right) \right]\\
\end{eqnarray}
$$
$$
\begin{eqnarray}
-1\leq\rho&\leq&1\;\cdots\;\rhoを相関係数とする\\
-1\leq\frac{\mathrm{Cov}\left[X,\;Y\right]}{
\sqrt{ \mathrm{V}\left[ X \right] }
\sqrt{ \mathrm{V}\left[ Y \right] }
}&\leq&1\;\cdots\;\rho=\frac{\mathrm{Cov}\left[X,\;Y\right]}{
\sqrt{ \mathrm{V}\left[ X \right] }
\sqrt{ \mathrm{V}\left[ Y \right] }
}\\
-1\leq\frac{\mathrm{Cov}\left[\hat{\theta}, \frac{\partial \log{f(x;\theta)}}{\partial \theta} \right]}
{\sqrt{\mathrm{V}\left[\hat{\theta}\right]}\sqrt{\mathrm{V}\left[ \frac{\partial \log{f(x;\theta)}}{\partial \theta} \right]}}&\leq&1
\;\cdots\;\rho=\frac{\mathrm{Cov}\left[\hat{\theta}, \frac{\partial \log{f(x;\theta)}}{\partial \theta} \right]}
{\sqrt{\mathrm{V}\left[\hat{\theta}\right]}\sqrt{\mathrm{V}\left[ \frac{\partial \log{f(x;\theta)}}{\partial \theta} \right]}}\\
\end{eqnarray}
$$
$$
\begin{eqnarray}
\rho^2&\leq&1\\
\frac{\mathrm{Cov}\left[\hat{\theta}, \frac{\partial \log{f(x;\theta)}}{\partial \theta} \right]^2}
{\mathrm{V}\left[\hat{\theta}\right]\mathrm{V}\left[ \frac{\partial \log{f(x;\theta)}}{\partial \theta} \right]}&\leq&1\\
\frac{1^2}{\mathrm{V}\left[\hat{\theta}\right]\mathrm{V}\left[ \frac{\partial \log{f(x;\theta)}}{\partial \theta} \right]}&\leq&1
\;\cdots\;\mathrm{Cov}\left[\hat{\theta}, \frac{\partial \log{f(x;\theta)}}{\partial \theta} \right]=1\\
\frac{1}{\mathrm{V}\left[
\frac{ \partial \log{ f(x;\theta) }}{ \partial \theta }
\right]} &\leq& \mathrm{V} \left[ \hat{\theta} \right]\;\cdots\;\href{https://shikitenkai.blogspot.com/2020/04/blog-post_76.html}{\mathrm{V} \left[ \hat{\theta} \right]:不偏推定量の分散}\\
\mathrm{V}\left[
\frac{ \partial \log{ f(x;\theta) }}{ \partial \theta }
\right]^{-1} &\leq& \mathrm{V} \left[ \hat{\theta} \right]\\
\mathcal{I}^{-1}&\leq& \mathrm{V} \left[ \hat{\theta} \right]
\;\cdots\;\href{https://shikitenkai.blogspot.com/2020/04/blog-post_19.html}{\mathcal{I}=\mathrm{V}\left[\frac{ \partial \log{ f(x;\theta) }}{ \partial \theta }\right]:フィッシャー情報量}\\
\end{eqnarray}
$$
パラメタの推定(不偏推定量)の分散(パラメタの推定値と真の値(標本が従っている確率分布凾数のパラメタという意味で)との平均二乗誤差)は，フィッシャー情報量の逆数以下にはならないことを示す．
