“標本平均\(\overline{X}\)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\left[(X_k-\overline{X})^3\right]
&=&\displaystyle E\left[\sum_{k=1}^{n}(X_k-\overline{X})^3\right]\\
&&\displaystyle\,\dotso\,E\left[X\right]+E\left[Y\right]=E\left[X+Y\right]\\
&=&\displaystyle E\left[\sum_{k=1}^{n}\left\{
\displaystyle \left( X_k-\mu \right)
\displaystyle -\left( \overline{X}-\mu \right)
\displaystyle \right\}^3\right]\\
&&\displaystyle\,\dotso\,(A-B)=(A-C)-(B-C)\\
&=&\displaystyle E\left[\sum_{k=1}^{n}\left\{
\displaystyle \left( X_k-\mu \right)^3
\displaystyle -3\left( X_k-\mu \right)^2\left( \overline{X}-\mu \right)
\displaystyle +3\left( X_k-\mu \right)\left( \overline{X}-\mu \right)^2
\displaystyle - \left( \overline{X}-\mu \right)^3
\displaystyle \right\}\right]\\
&&\displaystyle\,\dotso\,(A-B)^3=A^3-3A^2B+3AB^2-B^3\\
&=&\displaystyle E\left[
\displaystyle \sum_{k=1}^{n}\left( X_k-\mu \right)^3
\displaystyle -3\left\{ \left( \overline{X}-\mu \right) \sum_{k=1}^{n} \left( X_k-\mu \right)^2 \right\}
\displaystyle +3\left\{ \left( \overline{X}-\mu \right)^2 \sum_{k=1}^{n} \left( X_k-\mu \right) \right\}
\displaystyle - \left( \overline{X}-\mu \right)^3 \sum_{k=1}^{n}1
\displaystyle \right]\\
&&\displaystyle\,\dotso\,\sum_{k=1}^{n} (X+Y)=\sum_{k=1}^{n} X+\sum_{k=1}^{n} Y\\
&=&\displaystyle E\left[\sum_{k=1}^{n}\left( X_k-\mu \right)^3\right]
\displaystyle -3E\left[ \left( \overline{X}-\mu \right) \sum_{k=1}^{n} \left( X_k-\mu \right)^2 \right]
\displaystyle +3E\left[ \left( \overline{X}-\mu \right)^2 \sum_{k=1}^{n} \left( X_k-\mu \right) \right]
\displaystyle - E\left[\left( \overline{X}-\mu \right)^3 \sum_{k=1}^{n}1\right]
\displaystyle \\
&&\displaystyle\,\dotso\,E\left[X+Y\right]=E\left[X\right]+E\left[Y\right]\\
&=&\displaystyle \sum_{k=1}^{n}E\left[\left( X_k-\mu \right)^3\right]
\displaystyle -3E\left[ \left( \overline{X}-\mu \right) \sum_{k=1}^{n} \left( X_k-\mu \right)^2 \right]
\displaystyle +3E\left[ \left( \overline{X}-\mu \right)^2 \left( \sum_{k=1}^{n}X_k - \sum_{k=1}^{n}\mu \right) \right]
\displaystyle - E\left[n\left( \overline{X}-\mu \right)^3\right]
\displaystyle \\
&&\displaystyle\,\dotso\,
\displaystyle E\left[X+Y\right]=E\left[X\right]+E\left[Y\right]
,\quad \sum_{k=1}^{n} (X+Y)=\sum_{k=1}^{n} X+\sum_{k=1}^{n} Y\\
&=&\displaystyle \sum_{k=1}^{n}E\left[\left( X_k-\mu \right)^3\right]
\displaystyle -3E\left[ \left( \overline{X}-\mu \right) \sum_{k=1}^{n} \left( X_k-\mu \right)^2 \right]
\displaystyle +3E\left[ \left( \overline{X}-\mu \right)^2 \left( n\overline{X} - n\mu \right) \right]
\displaystyle - E\left[n\left( \overline{X}-\mu \right)^3\right]
\displaystyle \\
&&\displaystyle\,\dotso\,
\sum_{k=1}^{n}X_k=n\overline{X}
,\quad \sum_{k=1}^{n}\mu=n\mu\\
&=&\displaystyle n\mu_3
\displaystyle -3E\left[ \left( \overline{X}-\mu \right) \sum_{k=1}^{n} \left( X_k-\mu \right)^2\right]
\displaystyle +3E\left[ n\left( \overline{X}-\mu \right)^3\right]
\displaystyle -nE\left[\left( \overline{X}-\mu \right)^3\right]
\displaystyle \\
&&\displaystyle\,\dotso\,
\href{https://shikitenkai.blogspot.com/2019/07/mu-sigma2beta1.html}{E\left[\left( X_k-\mu \right)^3\right]=\mu_3}\\
&=&\displaystyle n\mu_3
\displaystyle -3E\left[ \left( \overline{X}-\mu \right) \sum_{k=1}^{n} \left( X_k-\mu \right)^2\right]
\displaystyle +3nE\left[ \left( \overline{X}-\mu \right)^3\right]
\displaystyle - nE\left[\left( \overline{X}-\mu \right)^3\right]
\displaystyle \\
&=&\displaystyle n\mu_3
\displaystyle -3E\left[ \left( \overline{X}-\mu \right) \sum_{k=1}^{n} \left( X_k-\mu \right)^2\right]
\displaystyle +2nE\left[ \left( \overline{X}-\mu \right)^3\right]
\displaystyle \\
&=&\displaystyle n\mu_3
\displaystyle -3E\left[ \left( \overline{X}-\mu \right) \sum_{k=1}^{n} \left( X_k-\mu \right)^2\right]
\displaystyle +2n\frac{\mu_3}{n^2}
\displaystyle \\
&&\displaystyle\,\dotso\,
\href{https://shikitenkai.blogspot.com/2019/07/overlinexmu3.html}{E\left[\left( \overline{X}-\mu \right)^3\right]=\frac{\mu_3}{n^2}}\\
&=&\displaystyle n\mu_3
\displaystyle -3E\left[ \left( \overline{X}-\mu \right) \sum_{k=1}^{n} \left( X_k-\mu \right)^2\right]
\displaystyle +2\frac{\mu_3}{n}
\displaystyle \\
&=&\displaystyle \mu_3\left( \frac{n^2+2}{n} \right)
\displaystyle -3E\left[ \left( \overline{X}-\mu \right) \sum_{k=1}^{n} \left( X_k-\mu \right)^2\right]
\displaystyle \\
&=&\displaystyle \mu_3\left( \frac{n^2+2}{n} \right)
\displaystyle -3E\left[ \left( \overline{X}-\mu \right) \sum_{k=1}^{n} \left( X_k^2-2\mu X_k+\mu^2 \right) \right]
\displaystyle \\
&=&\displaystyle \mu_3\left( \frac{n^2+2}{n} \right)
\displaystyle -3E\left[ \left( \overline{X}-\mu \right) \left( \sum_{k=1}^{n}X_k^2-2\mu \sum_{k=1}^{n}X_k+\mu^2\sum_{k=1}^{n}1 \right) \right]
\displaystyle \\
&=&\displaystyle \mu_3\left( \frac{n^2+2}{n} \right)
\displaystyle -3E\left[ \left( \overline{X}-\mu \right) \left( \sum_{k=1}^{n}X_k^2-2\mu n\overline{X}+n\mu^2 \right) \right]
\displaystyle \\
&&\displaystyle\,\dotso\,
\sum_{k=1}^{n}X_k=n\overline{X}
,\quad \sum_{k=1}^{n}1=n\\
&=&\displaystyle \mu_3\left( \frac{n^2+2}{n} \right)
\displaystyle -3E\left[
\displaystyle \overline{X}\left( \sum_{k=1}^{n}X_k^2- 2\mu n\overline{X}+ n\mu^2 \right)
\displaystyle -\mu\left( \sum_{k=1}^{n}X_k^2-2\mu n\overline{X}+ n\mu^2 \right)
\displaystyle \right]
\displaystyle \\
&=&\displaystyle \mu_3\left( \frac{n^2+2}{n} \right)
\displaystyle -3E\left[
\displaystyle \left( \overline{X}\sum_{k=1}^{n}X_k^2-2n\mu \overline{X}^2+ n\mu^2\overline{X} \right)
\displaystyle -\left( \mu\sum_{k=1}^{n}X_k^2-2\mu^2 n\overline{X}+n\mu^3 \right)
\displaystyle \right]
\displaystyle \\
&=&\displaystyle \mu_3\left( \frac{n^2+2}{n} \right)
\displaystyle -3E\left[
\displaystyle \overline{X}\sum_{k=1}^{n}X_k^2-2n\mu \overline{X}^2+ n\mu^2\overline{X}
\displaystyle -\mu\sum_{k=1}^{n}X_k^2+2\mu^2 n\overline{X}-n\mu^3
\displaystyle \right]
\displaystyle \\
&=&\displaystyle \mu_3\left( \frac{n^2+2}{n} \right)
\displaystyle -3E\left[
\displaystyle \overline{X}\sum_{k=1}^{n}X_k^2
\displaystyle -2n\mu \overline{X}^2- \mu \sum_{k=1}^{n}X_k^2
\displaystyle + n\mu^2\overline{X} +2\mu^2 n\overline{X}
\displaystyle -n\mu^3
\displaystyle \right]
\displaystyle \\
&=&\displaystyle \mu_3\left( \frac{n^2+2}{n} \right)
\displaystyle -3E\left[
\displaystyle \overline{X}\sum_{k=1}^{n}X_k^2
\displaystyle - \mu \left(2n \overline{X}^2+\sum_{k=1}^{n}X_k^2\right)
\displaystyle + n\mu^2\left( \overline{X} +2 \overline{X}\right)
\displaystyle -n\mu^3
\displaystyle \right]
\displaystyle \\
&=&\displaystyle \mu_3\left( \frac{n^2+2}{n} \right)
\displaystyle -3E\left[
\displaystyle \overline{X}\sum_{k=1}^{n}X_k^2
\displaystyle - \mu\left(2n \overline{X}^2+\sum_{k=1}^{n}X_k^2\right)
\displaystyle + 3n\mu^2\overline{X}
\displaystyle -n\mu^3
\displaystyle \right]
\displaystyle \\
&&\displaystyle\,\dotso\,
\displaystyle \href{https://shikitenkai.blogspot.com/2019/07/blog-post_41.html}{\sum_{k=1}^{n}X_k^2=\sum_{k=1}^{n}(X_k-\overline{X})^2+n\overline{X}^2}\\
&=&\displaystyle \mu_3\left( \frac{n^2+2}{n} \right)
\displaystyle -3E\left[
\displaystyle \overline{X}\,\left(\sum_{k=1}^{n}\left(X_k-\overline{X}\right)^2+n\overline{X}^2\right)
\displaystyle - \mu \left(2n \overline{X}^2+\left(\sum_{k=1}^{n}\left(X_k-\overline{X}\right)^2+n\overline{X}^2\right)\right)
\displaystyle +3n\mu^2\overline{X}
\displaystyle -n\mu^3
\displaystyle \right]
\displaystyle \\
&=&\displaystyle \mu_3\left( \frac{n^2+2}{n} \right)
\displaystyle -3E\left[
\displaystyle \overline{X}\sum_{k=1}^{n}\left(X_k-\overline{X}\right)^2
\displaystyle +n\overline{X}^3
\displaystyle -2n\mu \overline{X}^2
\displaystyle -\mu\sum_{k=1}^{n}\left(X_k-\overline{X}\right)^2
\displaystyle - n\mu \overline{X}^2
\displaystyle +3n\mu^2\overline{X}
\displaystyle -n\mu^3
\displaystyle \right]
\displaystyle \\
&=&\displaystyle \mu_3\left( \frac{n^2+2}{n} \right)
\displaystyle -3E\left[
\displaystyle \left( \overline{X}-\mu \right) \sum_{k=1}^{n}\left(X_k-\overline{X}\right)^2
\displaystyle +n\overline{X}^3
\displaystyle -3n\mu \overline{X}^2
\displaystyle +3n\mu^2\overline{X}
\displaystyle -n\mu^3
\displaystyle \right]
\displaystyle \\
&=&\displaystyle \mu_3\left( \frac{n^2+2}{n} \right)
\displaystyle -3E\left[
\displaystyle \left( \overline{X}-\mu \right) \sum_{k=1}^{n}\left(X_k-\overline{X}\right)^2
\displaystyle \right]
\displaystyle -3E\left[
\displaystyle n\overline{X}^3
\displaystyle -3n\mu \overline{X}^2
\displaystyle +3n\mu^2\overline{X}
\displaystyle -n\mu^3
\displaystyle \right]
\displaystyle \\
&=&\displaystyle \mu_3\left( \frac{n^2+2}{n} \right)
\displaystyle -3E\left[\overline{X}-\mu \right]
\displaystyle \sum_{k=1}^{n}\left(X_k-\overline{X}\right)^2
\displaystyle -3E\left[
\displaystyle n\overline{X}^3
\displaystyle -3n\mu \overline{X}^2
\displaystyle +3n\mu^2\overline{X}
\displaystyle -n\mu^3
\displaystyle \right]
\displaystyle \\
&=&\displaystyle \mu_3\left( \frac{n^2+2}{n} \right)
\displaystyle -3\left(E\left[\overline{X} \right]-\mu\right)
\displaystyle \sum_{k=1}^{n}\left(X_k-\overline{X}\right)^2
\displaystyle -3E\left[
\displaystyle n\overline{X}^3
\displaystyle -3n\mu \overline{X}^2
\displaystyle +3n\mu^2\overline{X}
\displaystyle -n\mu^3
\displaystyle \right]
\displaystyle \\
&=&\displaystyle \mu_3\left( \frac{n^2+2}{n} \right)
\displaystyle -3\left(\mu-\mu\right)
\displaystyle \sum_{k=1}^{n}\left(X_k-\overline{X}\right)^2
\displaystyle -3E\left[
\displaystyle n\overline{X}^3
\displaystyle -3n\mu \overline{X}^2
\displaystyle +3n\mu^2\overline{X}
\displaystyle -n\mu^3
\displaystyle \right]
\displaystyle \\
&=&\displaystyle \mu_3\left( \frac{n^2+2}{n} \right)
\displaystyle -3E\left[
\displaystyle n\overline{X}^3
\displaystyle -3n\mu \overline{X}^2
\displaystyle +3n\mu^2\overline{X}
\displaystyle -n\mu^3
\displaystyle \right]
\displaystyle \\
&=&\displaystyle \mu_3\left( \frac{n^2+2}{n} \right)
\displaystyle -3E\left[n
\displaystyle \left(\overline{X}^3
\displaystyle -3\mu \overline{X}^2
\displaystyle +3\mu^2\overline{X}
\displaystyle -\mu^3\right)
\displaystyle \right]
\displaystyle \\
&=&\displaystyle \mu_3\left( \frac{n^2+2}{n} \right)
\displaystyle -3E\left[
\displaystyle n\left(\overline{X}-\mu\right)^3
\displaystyle \right]
\displaystyle \\
&&\displaystyle\,\dotso\,A^3-3A^2B+3AB^2-B^3=(A-B)^3\\
&=&\displaystyle \mu_3\left( \frac{n^2+2}{n} \right)
\displaystyle -3nE\left[
\displaystyle \left(\overline{X}-\mu\right)^3
\displaystyle \right]
\displaystyle \\
&&\displaystyle\,\dotso\,E[cX]=cE[X]\\
&=&\displaystyle \mu_3\left( \frac{n^2+2}{n} \right)
\displaystyle -3n\frac{\mu_3}{n^2}
\displaystyle \\
&&\displaystyle\,\dotso\,
\href{https://shikitenkai.blogspot.com/2019/07/overlinexmu3.html}{E\left[\left( \overline{X}-\mu \right)^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\left( \frac{n^2+2}{n} -3\frac{n}{n}\right)
\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\(\overline{X}\)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\(\hat{\mu}_3\)
$$\begin{array}{rcl}
\displaystyle \hat{\mu}_3
&=&\displaystyle \frac{n}{(n-1)(n-2)}E\left[\sum_{k=1}^{n}(X_k-\overline{X})^3\right]\\
\end{array}$$
0 件のコメント:
コメントを投稿