標本平均まわりの3次モーメントの和

“標本平均$\bar{X}$まわりの3次モーメントの和”を“母平均$\mu$まわりの3次モーメント${\mu }_3$”で表す

$$\begin{array}{rcl}{\displaystyle \sum _{k=1}^{n}E[(X_k-\bar{X}{)}^3]}& =& {\displaystyle E[\sum _{k=1}^n(X_k-\bar{X}{)}^3]}\\ & & {\displaystyle \dots E\left[X\right]+E\left[Y\right]=E[X+Y]}\\ & =& {\displaystyle E\left[\sum _{k=1}^n{\left\{{\displaystyle (X_k-\mu ){\displaystyle -(\bar{X}-\mu ){\displaystyle }}}\right\}}^3\right]}\\ & & {\displaystyle \dots (A-B)=(A-C)-(B-C)}\\ & =& {\displaystyle E\left[\sum _{k=1}^n\left\{{\displaystyle {(X_k-\mu )}^3{\displaystyle -3{(X_k-\mu )}^2(\bar{X}-\mu ){\displaystyle +3(X_k-\mu ){(\bar{X}-\mu )}^2{\displaystyle -{(\bar{X}-\mu )}^3{\displaystyle }}}}}\right\}\right]}\\ & & {\displaystyle \dots (A-B{)}^3=A^3-3A^{2}B+3A{B}^2-B^3}\\ & =& {\displaystyle E\left[{\displaystyle \sum _{k=1}^n{(X_k-\mu )}^3{\displaystyle -3\left\{(\bar{X}-\mu )\sum _{k=1}^n{(X_k-\mu )}^2\right\}{\displaystyle +3\left\{{(\bar{X}-\mu )}^2\sum _{k=1}^n(X_k-\mu )\right\}{\displaystyle -{(\bar{X}-\mu )}^3\sum _{k=1}^{n}1{\displaystyle }}}}}\right]}\\ & & {\displaystyle \dots \sum _{k=1}^n(X+Y)=\sum _{k=1}^{n}X+\sum _{k=1}^{n}Y}\\ & =& {\displaystyle E\left[\sum _{k=1}^n{(X_k-\mu )}^3\right]{\displaystyle -3E\left[(\bar{X}-\mu )\sum _{k=1}^n{(X_k-\mu )}^2\right]{\displaystyle +3E\left[{(\bar{X}-\mu )}^2\sum _{k=1}^n(X_k-\mu )\right]{\displaystyle -E\left[{(\bar{X}-\mu )}^3\sum _{k=1}^{n}1\right]{\displaystyle }}}}}\\ & & {\displaystyle \dots E[X+Y]=E\left[X\right]+E\left[Y\right]}\\ & =& {\displaystyle \sum _{k=1}^{n}E\left[{(X_k-\mu )}^3\right]{\displaystyle -3E\left[(\bar{X}-\mu )\sum _{k=1}^n{(X_k-\mu )}^2\right]{\displaystyle +3E\left[{(\bar{X}-\mu )}^2(\sum _{k=1}^n{X}_k-\sum _{k=1}^n\mu )\right]{\displaystyle -E\left[n{(\bar{X}-\mu )}^3\right]{\displaystyle }}}}}\\ & & {\displaystyle \dots {\displaystyle E[X+Y]=E\left[X\right]+E\left[Y\right],\sum _{k=1}^n(X+Y)=\sum _{k=1}^{n}X+\sum _{k=1}^{n}Y}}\\ & =& {\displaystyle \sum _{k=1}^{n}E\left[{(X_k-\mu )}^3\right]{\displaystyle -3E\left[(\bar{X}-\mu )\sum _{k=1}^n{(X_k-\mu )}^2\right]{\displaystyle +3E\left[{(\bar{X}-\mu )}^2(n\bar{X}-n\mu )\right]{\displaystyle -E\left[n{(\bar{X}-\mu )}^3\right]{\displaystyle }}}}}\\ & & {\displaystyle \dots \sum _{k=1}^n{X}_k=n\bar{X},\sum _{k=1}^n\mu =n\mu }\\ & =& {\displaystyle n{\mu }_3{\displaystyle -3E\left[(\bar{X}-\mu )\sum _{k=1}^n{(X_k-\mu )}^2\right]{\displaystyle +3E\left[n{(\bar{X}-\mu )}^3\right]{\displaystyle -nE\left[{(\bar{X}-\mu )}^3\right]{\displaystyle }}}}}\\ & & {\displaystyle \dots E\left[{(X_k-\mu )}^3\right]={\mu }_3}\\ & =& {\displaystyle n{\mu }_3{\displaystyle -3E\left[(\bar{X}-\mu )\sum _{k=1}^n{(X_k-\mu )}^2\right]{\displaystyle +3nE\left[{(\bar{X}-\mu )}^3\right]{\displaystyle -nE\left[{(\bar{X}-\mu )}^3\right]{\displaystyle }}}}}\\ & =& {\displaystyle n{\mu }_3{\displaystyle -3E\left[(\bar{X}-\mu )\sum _{k=1}^n{(X_k-\mu )}^2\right]{\displaystyle +2nE\left[{(\bar{X}-\mu )}^3\right]{\displaystyle }}}}\\ & =& {\displaystyle n{\mu }_3{\displaystyle -3E\left[(\bar{X}-\mu )\sum _{k=1}^n{(X_k-\mu )}^2\right]{\displaystyle +2n\frac{{\mu }_3}{n^2}{\displaystyle }}}}\\ & & {\displaystyle \dots E\left[{(\bar{X}-\mu )}^3\right]=\frac{{\mu }_3}{n^2}}\\ & =& {\displaystyle n{\mu }_3{\displaystyle -3E\left[(\bar{X}-\mu )\sum _{k=1}^n{(X_k-\mu )}^2\right]{\displaystyle +2\frac{{\mu }_3}{n}{\displaystyle }}}}\\ & =& {\displaystyle {\mu }_3\left(\frac{n^2+2}{n}\right){\displaystyle -3E\left[(\bar{X}-\mu )\sum _{k=1}^n{(X_k-\mu )}^2\right]{\displaystyle }}}\\ & =& {\displaystyle {\mu }_3\left(\frac{n^2+2}{n}\right){\displaystyle -3E\left[(\bar{X}-\mu )\sum _{k=1}^n(X_k^2-2\mu X_k+{\mu }^2)\right]{\displaystyle }}}\\ & =& {\displaystyle {\mu }_3\left(\frac{n^2+2}{n}\right){\displaystyle -3E\left[(\bar{X}-\mu )(\sum _{k=1}^n{X}_k^2-2\mu \sum _{k=1}^n{X}_k+{\mu }^2\sum _{k=1}^{n}1)\right]{\displaystyle }}}\\ & =& {\displaystyle {\mu }_3\left(\frac{n^2+2}{n}\right){\displaystyle -3E\left[(\bar{X}-\mu )(\sum _{k=1}^n{X}_k^2-2\mu n\bar{X}+n{\mu }^2)\right]{\displaystyle }}}\\ & & {\displaystyle \dots \sum _{k=1}^n{X}_k=n\bar{X},\sum _{k=1}^{n}1=n}\\ & =& {\displaystyle {\mu }_3\left(\frac{n^2+2}{n}\right){\displaystyle -3E\left[{\displaystyle \bar{X}(\sum _{k=1}^n{X}_k^2-2\mu n\bar{X}+n{\mu }^2){\displaystyle -\mu (\sum _{k=1}^n{X}_k^2-2\mu n\bar{X}+n{\mu }^2){\displaystyle }}}\right]{\displaystyle }}}\\ & =& {\displaystyle {\mu }_3\left(\frac{n^2+2}{n}\right){\displaystyle -3E\left[{\displaystyle (\bar{X}\sum _{k=1}^n{X}_k^2-2n\mu {\bar{X}}^2+n{\mu }^2\bar{X}){\displaystyle -(\mu \sum _{k=1}^n{X}_k^2-2{\mu }^{2}n\bar{X}+n{\mu }^3){\displaystyle }}}\right]{\displaystyle }}}\\ & =& {\displaystyle {\mu }_3\left(\frac{n^2+2}{n}\right){\displaystyle -3E\left[{\displaystyle \bar{X}\sum _{k=1}^n{X}_k^2-2n\mu {\bar{X}}^2+n{\mu }^2\bar{X}{\displaystyle -\mu \sum _{k=1}^n{X}_k^2+2{\mu }^{2}n\bar{X}-n{\mu }^3{\displaystyle }}}\right]{\displaystyle }}}\\ & =& {\displaystyle {\mu }_3\left(\frac{n^2+2}{n}\right){\displaystyle -3E\left[{\displaystyle \bar{X}\sum _{k=1}^n{X}_k^2{\displaystyle -2n\mu {\bar{X}}^2-\mu \sum _{k=1}^n{X}_k^2{\displaystyle +n{\mu }^2\bar{X}+2{\mu }^{2}n\bar{X}{\displaystyle -n{\mu }^3{\displaystyle }}}}}\right]{\displaystyle }}}\\ & =& {\displaystyle {\mu }_3\left(\frac{n^2+2}{n}\right){\displaystyle -3E\left[{\displaystyle \bar{X}\sum _{k=1}^n{X}_k^2{\displaystyle -\mu (2n{\bar{X}}^2+\sum _{k=1}^n{X}_k^2){\displaystyle +n{\mu }^2(\bar{X}+2\bar{X}){\displaystyle -n{\mu }^3{\displaystyle }}}}}\right]{\displaystyle }}}\\ & =& {\displaystyle {\mu }_3\left(\frac{n^2+2}{n}\right){\displaystyle -3E\left[{\displaystyle \bar{X}\sum _{k=1}^n{X}_k^2{\displaystyle -\mu (2n{\bar{X}}^2+\sum _{k=1}^n{X}_k^2){\displaystyle +3n{\mu }^2\bar{X}{\displaystyle -n{\mu }^3{\displaystyle }}}}}\right]{\displaystyle }}}\\ & & {\displaystyle \dots {\displaystyle \sum _{k=1}^n{X}_k^2=\sum _{k=1}^n(X_k-\bar{X}{)}^2+n{\bar{X}}^2}}\\ & =& {\displaystyle {\mu }_3\left(\frac{n^2+2}{n}\right){\displaystyle -3E\left[{\displaystyle \bar{X}(\sum _{k=1}^n{(X_k-\bar{X})}^2+n{\bar{X}}^2){\displaystyle -\mu (2n{\bar{X}}^2+(\sum _{k=1}^n{(X_k-\bar{X})}^2+n{\bar{X}}^2)){\displaystyle +3n{\mu }^2\bar{X}{\displaystyle -n{\mu }^3{\displaystyle }}}}}\right]{\displaystyle }}}\\ & =& {\displaystyle {\mu }_3\left(\frac{n^2+2}{n}\right){\displaystyle -3E\left[{\displaystyle \bar{X}\sum _{k=1}^n{(X_k-\bar{X})}^2{\displaystyle +n{\bar{X}}^3{\displaystyle -2n\mu {\bar{X}}^2{\displaystyle -\mu \sum _{k=1}^n{(X_k-\bar{X})}^2{\displaystyle -n\mu {\bar{X}}^2{\displaystyle +3n{\mu }^2\bar{X}{\displaystyle -n{\mu }^3{\displaystyle }}}}}}}}\right]{\displaystyle }}}\\ & =& {\displaystyle {\mu }_3\left(\frac{n^2+2}{n}\right){\displaystyle -3E\left[{\displaystyle (\bar{X}-\mu )\sum _{k=1}^n{(X_k-\bar{X})}^2{\displaystyle +n{\bar{X}}^3{\displaystyle -3n\mu {\bar{X}}^2{\displaystyle +3n{\mu }^2\bar{X}{\displaystyle -n{\mu }^3{\displaystyle }}}}}}\right]{\displaystyle }}}\\ & =& {\displaystyle {\mu }_3\left(\frac{n^2+2}{n}\right){\displaystyle -3E\left[{\displaystyle (\bar{X}-\mu )\sum _{k=1}^n{(X_k-\bar{X})}^2{\displaystyle }}\right]{\displaystyle -3E\left[{\displaystyle n{\bar{X}}^3{\displaystyle -3n\mu {\bar{X}}^2{\displaystyle +3n{\mu }^2\bar{X}{\displaystyle -n{\mu }^3{\displaystyle }}}}}\right]{\displaystyle }}}}\\ & =& {\displaystyle {\mu }_3\left(\frac{n^2+2}{n}\right){\displaystyle -3E[\bar{X}-\mu ]{\displaystyle \sum _{k=1}^n{(X_k-\bar{X})}^2{\displaystyle -3E\left[{\displaystyle n{\bar{X}}^3{\displaystyle -3n\mu {\bar{X}}^2{\displaystyle +3n{\mu }^2\bar{X}{\displaystyle -n{\mu }^3{\displaystyle }}}}}\right]{\displaystyle }}}}}\\ & =& {\displaystyle {\mu }_3\left(\frac{n^2+2}{n}\right){\displaystyle -3(E\left[\bar{X}\right]-\mu ){\displaystyle \sum _{k=1}^n{(X_k-\bar{X})}^2{\displaystyle -3E\left[{\displaystyle n{\bar{X}}^3{\displaystyle -3n\mu {\bar{X}}^2{\displaystyle +3n{\mu }^2\bar{X}{\displaystyle -n{\mu }^3{\displaystyle }}}}}\right]{\displaystyle }}}}}\\ & =& {\displaystyle {\mu }_3\left(\frac{n^2+2}{n}\right){\displaystyle -3(\mu -\mu ){\displaystyle \sum _{k=1}^n{(X_k-\bar{X})}^2{\displaystyle -3E\left[{\displaystyle n{\bar{X}}^3{\displaystyle -3n\mu {\bar{X}}^2{\displaystyle +3n{\mu }^2\bar{X}{\displaystyle -n{\mu }^3{\displaystyle }}}}}\right]{\displaystyle }}}}}\\ & =& {\displaystyle {\mu }_3\left(\frac{n^2+2}{n}\right){\displaystyle -3E\left[{\displaystyle n{\bar{X}}^3{\displaystyle -3n\mu {\bar{X}}^2{\displaystyle +3n{\mu }^2\bar{X}{\displaystyle -n{\mu }^3{\displaystyle }}}}}\right]{\displaystyle }}}\\ & =& {\displaystyle {\mu }_3\left(\frac{n^2+2}{n}\right){\displaystyle -3E\left[n{\displaystyle \left({\bar{X}}^3{\displaystyle -3\mu {\bar{X}}^2{\displaystyle +3{\mu }^2\bar{X}{\displaystyle -{\mu }^3}}}\right){\displaystyle }}\right]{\displaystyle }}}\\ & =& {\displaystyle {\mu }_3\left(\frac{n^2+2}{n}\right){\displaystyle -3E\left[{\displaystyle n{(\bar{X}-\mu )}^3{\displaystyle }}\right]{\displaystyle }}}\\ & & {\displaystyle \dots A^3-3A^{2}B+3A{B}^2-B^3=(A-B{)}^3}\\ & =& {\displaystyle {\mu }_3\left(\frac{n^2+2}{n}\right){\displaystyle -3nE\left[{\displaystyle {(\bar{X}-\mu )}^3{\displaystyle }}\right]{\displaystyle }}}\\ & & {\displaystyle \dots E[cX]=cE[X]}\\ & =& {\displaystyle {\mu }_3\left(\frac{n^2+2}{n}\right){\displaystyle -3n\frac{{\mu }_3}{n^2}{\displaystyle }}}\\ & & {\displaystyle \dots E\left[{(\bar{X}-\mu )}^3\right]=\frac{{\mu }_3}{n^2}}\\ & =& {\displaystyle {\mu }_3\left(\frac{n^2+2}{n}\right){\displaystyle -3\frac{{\mu }_3}{n}{\displaystyle }}}\\ & =& {\displaystyle {\mu }_3(\frac{n^2+2}{n}-3\frac{n}{n}){\displaystyle }}\\ & =& {\displaystyle {\mu }_3\frac{n^2-3n+2}{n}}\\ & =& {\displaystyle {\mu }_3\frac{(n-1)(n-2)}{n}}\end{array}$$

“標本平均$\bar{X}$まわりの3次モーメントの和”から“母平均$\mu$まわりの3次モーメント${\mu }_3$”を推定する${\hat{\mu }}_3$

$$\begin{array}{rcl}{\displaystyle {\hat{\mu }}_3}& =& {\displaystyle \frac{n}{(n-1)(n-2)}E[\sum _{k=1}^n(X_k-\bar{X}{)}^3]}\end{array}$$