[1]
\(E[\bar{X}^2]\)\(=V[\bar{X}]+E[\bar{X}]^2\)
\(=\displaystyle \frac{\sigma^2}{n}+\mu^2\)
[2]
\(E[S^2]\)\(=\displaystyle E\left[\frac{1}{n-1}\sum^n_{i=1}(X_i-\bar{X})^2\right]\)
\(=\displaystyle \frac{1}{n-1}E\left[\sum^n_{i=1}(X_i-\bar{X})^2\right]\)
\(=\displaystyle \frac{1}{n-1}E\left[\sum^n_{i=1}((X_i-\mu)-(\bar{X}-\mu))^2\right]\)
\(=\displaystyle \frac{1}{n-1}E\left[\sum^n_{i=1}(X_i-\mu)^2\right . \)\(\displaystyle -2\sum^n_{i=1}(X_i-\mu)(\bar{X}-\mu) \)\(\displaystyle \left . +\sum^n_{i=1}(\bar{X}-\mu)^2\right]\)
\(=\displaystyle \frac{1}{n-1}E\left[\sum^n_{i=1}(X_i-\mu)^2\right . \)\(\displaystyle -2n(\bar{X}-\mu)^2 \)\(\displaystyle \left . +n(\bar{X}-\mu)^2\right]\)
\(=\displaystyle \frac{1}{n-1}E\left[\sum^n_{i=1}(X_i-\mu)^2 -n(\bar{X}-\mu)^2 \right]\)
\(=\displaystyle \frac{1}{n-1}(n\sigma^2 – \sigma^2)\)
\(=\sigma^2\)
[3]
\(E[(\bar{X}-\mu)^k]\)\(\displaystyle =\int^\infty_{-\infty} (\bar{X}-\mu)^k \frac{1}{\sqrt{2\pi}\sigma/\sqrt{n}}e^{-\frac{(\bar{X}-\mu)^2}{2\sigma^2/n}}d\bar{X}\)
\(\bar{X}-\mu=Y\)とすると、
\(\displaystyle =\int^\infty_{-\infty}y^k \frac{1}{\sqrt{2\pi}\sigma/\sqrt{n}}e^{-\frac{y^2}{2\sigma^2/n}}dy\)
\(k\)が奇数の時、積分内は奇関数になるので、\(0\)。
\(k\)が偶数の時、積分内は偶関数になるので、
\(\displaystyle =2\int^\infty_0y^k \frac{1}{\sqrt{2\pi}\sigma/\sqrt{n}}e^{-\frac{y^2}{2\sigma^2/n}}dy\)
\(y^2=t\)と変数変換を行うと、
\(\displaystyle =2\int^\infty_0 t^{\frac{k}{2}} \frac{1}{\sqrt{2\pi}\sigma/\sqrt{n}}e^{-\frac{t}{2\sigma^2/n}}\frac{1}{2\sqrt{t}}dt\)
\(\displaystyle =\frac{\sqrt{n}}{\sqrt{2\pi}\sigma}\int^\infty_0 t^{\frac{k-1}{2}}e^{-\frac{t}{2\sigma^2/n}}dt\)
\(\displaystyle =\frac{\sqrt{n}}{\sqrt{2\pi}\sigma}\Gamma \left(\frac{k+1}{2}\right)\left(\frac{2\sigma^2}{n}\right)^{\frac{k+1}{2}}\)
\(\displaystyle =\frac{1}{\sqrt{\pi}}\Gamma \left(\frac{k+1}{2}\right)\left(\frac{2\sigma^2}{n}\right)^{\frac{k}{2}}\)
[4]
チェビシェフの不等式により、
\(P(|S^2-\sigma^2| \geq \varepsilon) \leq \displaystyle \frac{V[S^2]}{\varepsilon^2}\)
\(\displaystyle \frac{(n-1)S^2}{\sigma^2}\sim \chi^2(n-1)\)なので、\(\displaystyle V\left[\frac{(n-1)S^2}{\sigma^2}\right]\)\(=2(n-1)\)となり、\(V[S^2]\)\(\displaystyle =\frac{2\sigma^4}{n-1}\)
\(P(|S^2-\sigma^2| \geq \varepsilon) \leq \displaystyle \frac{2\sigma^4}{\varepsilon^2(n-1)}\)\(\overset{n \to \infty}{\longrightarrow}\)\(0\)
よって一致推定量。
詳細
\(P(|S^2-\sigma^2| \geq \varepsilon) \)\(\overset{n \to \infty}{\longrightarrow}\)\(0\)
より、
\(\displaystyle \lim_{n \to \infty} P(|S^2-\sigma^2| \geq \varepsilon)=0\)
\(\iff \displaystyle \lim_{n \to \infty} P(|S^2-\sigma^2| < \varepsilon)\)\(=1\)
\(\iff \)「任意の\(\varepsilon\)に対して、\(n\)を限りなく大きくとれば、\(|S^2-\sigma^2| < \varepsilon\)が成り立つ」
\(\iff \)「\( \forall\varepsilon>0\)\(, \exists N \in \mathbb{N}\)\(;|S_N^2-\sigma^2| < \varepsilon\)」(\(\varepsilon\)-論法)
\(\iff\) \(\displaystyle \lim_{n \to \infty} S^2 =\sigma^2\)
[5]
\(MSE[cS^2]=E[(cS^2-\sigma^2)^2]\)
\(\displaystyle = V[cS^2-\sigma^2]+E[cS^2-\sigma^2]^2\)
\(\displaystyle = c^2V[S^2]+(cE[S^2]-\sigma^2)^2\)
\(\displaystyle = c^2 \frac{2\sigma^4}{n-1}+(c\sigma^2-\sigma^2)^2\)
\(\displaystyle =\left \{\frac{2c^2}{n-1}+(c-1)^2\right\}\sigma^4\)
\(\{\}\)内を\(f(c)\)とすると、
\(f'(c) = \displaystyle \frac{4}{n-1}c +2(c-1)\)\(=0\)
として、\(\displaystyle c = \frac{n-1}{n+1}\)となり、この時、\(MSE = \displaystyle \frac{2\sigma^4}{n+1}\)
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