標本平均の分布の尖度

標本平均\(\overline{X}\)の尖度\(\beta_2\left(\overline{X}\right)\)

$$ \begin{eqnarray} \beta_2\left(\overline{X}\right) &=&\frac{\mathrm{E}\left[(\overline{X}-\mu)^4\right]}{\mathrm{V}\left[\overline{X}\right]^{\frac{4}{2}}}-3 \;\cdots\; \href{https://shikitenkai.blogspot.com/2020/08/blog-post_27.html}{\beta_2=\beta_2\left(X\right)=\frac{\mathrm{E}\left[(X-\mu)^4\right]}{\mathrm{V}\left[X\right]^{\frac{4}{2}}}-3=\frac{\mu_4}{\sigma^4}-3\;:尖度} \\&=&\frac{\frac{1}{n^3}\left(\mu_4+3(n-1)\sigma^4\right)}{\left(\frac{\sigma^2}{n}\right)^2}-3 \\&&\;\cdots\;\href{https://shikitenkai.blogspot.com/2020/08/4-4.html}{\mathrm{E}\left[(\overline{X}-\mu)^4\right]=\frac{1}{n^3}\left(\mu_4+3(n-1)\sigma^4\right) \;:標本平均の母平均まわりの4次モーメント (標本平均の4次の中心(化)モーメント) } \\&&\;\cdots\;\href{https://shikitenkai.blogspot.com/2019/06/specimen-random-variable_3.html}{\mathrm{V}\left[\overline{X}\right]=\frac{\sigma^2}{n}\;:標本平均の分散} \\&=&\frac{n^2}{n^3}\frac{\mu_4+3(n-1)\sigma^4}{\sigma^4}-3 \\&=&\frac{1}{n}\left(\frac{\mu_4}{\sigma^4}+3(n-1)\right)-3 \\&=&\frac{1}{n}\left(\frac{\mu_4}{\sigma^4}+3n-3\right)-3 \\&=&\frac{\frac{\mu_4}{\sigma^4}-3}{n}+3-3 \\&=&\frac{\frac{\mu_4}{\sigma^4}-3}{n} \\&=&\frac{\beta_2}{n} \;\cdots\;\beta_2=\frac{\mu_4}{\sigma^4}-3 \end{eqnarray} $$ よって標本数\(n\)を増やすことで\(\beta_2\left(\overline{X}\right)\)は\(0\)に近づいていく．