標本平均の分布の尖度

標本平均$\bar{X}$の尖度${\beta }_2\left(\bar{X}\right)$

$$\begin{array}{rcl}{\beta }_2\left(\bar{X}\right)& =& \frac{\mathrm{E}[(\bar{X}-\mu {)}^4]}{\mathrm{V}{\left[\bar{X}\right]}^{\frac{4}{2}}}-3\cdots {\beta }_2={\beta }_2\left(X\right)=\frac{\mathrm{E}[(X-\mu {)}^4]}{\mathrm{V}{\left[X\right]}^{\frac{4}{2}}}-3=\frac{{\mu }_4}{{\sigma }^4}-3:\mathrm{尖}\mathrm{度}\\ & =& \frac{\frac{1}{n^3}({\mu }_4+3(n-1){\sigma }^4)}{{\left(\frac{{\sigma }^2}{n}\right)}^2}-3\\ & & \cdots \mathrm{E}[(\bar{X}-\mu {)}^4]=\frac{1}{n^3}({\mu }_4+3(n-1){\sigma }^4):\mathrm{標}\mathrm{本}\mathrm{平}\mathrm{均}\mathrm{の}\mathrm{母}\mathrm{平}\mathrm{均}\mathrm{ま}\mathrm{わ}\mathrm{り}\mathrm{の}4\mathrm{次}\mathrm{モ}\mathrm{ー}\mathrm{メ}\mathrm{ン}\mathrm{ト}(\mathrm{標}\mathrm{本}\mathrm{平}\mathrm{均}\mathrm{の}4\mathrm{次}\mathrm{の}\mathrm{中}\mathrm{心}(\mathrm{化})\mathrm{モ}\mathrm{ー}\mathrm{メ}\mathrm{ン}\mathrm{ト})\\ & & \cdots \mathrm{V}\left[\bar{X}\right]=\frac{{\sigma }^2}{n}:\mathrm{標}\mathrm{本}\mathrm{平}\mathrm{均}\mathrm{の}\mathrm{分}\mathrm{散}\\ & =& \frac{n^2}{n^3}\frac{{\mu }_4+3(n-1){\sigma }^4}{{\sigma }^4}-3\\ & =& \frac{1}{n}(\frac{{\mu }_4}{{\sigma }^4}+3(n-1))-3\\ & =& \frac{1}{n}(\frac{{\mu }_4}{{\sigma }^4}+3n-3)-3\\ & =& \frac{\frac{{\mu }_4}{{\sigma }^4}-3}{n}+3-3\\ & =& \frac{\frac{{\mu }_4}{{\sigma }^4}-3}{n}\\ & =& \frac{{\beta }_2}{n}\cdots {\beta }_2=\frac{{\mu }_4}{{\sigma }^4}-3\end{array}$$ よって標本数$n$を増やすことで${\beta }_2\left(\bar{X}\right)$は$0$に近づいていく．