定理2.5
定理2.5
簡單隨機樣本的協方差
Syx=1n?1∑i=1n(yi?yˉ)(xi?xˉ)S_{yx}=\frac{1}{n-1}\displaystyle\sum^{n}_{i=1}{(y_i-\bar{y})(x_i-\bar{x})}Syx?=n?11?i=1∑n?(yi??yˉ?)(xi??xˉ)
是總體協方差SyxS_{yx}Syx?的無偏估計
證明：
對于總體中每個單元YiY_iYi?,引入示性變量aia_iai?
ai={1若Yi入樣0若Yi不入樣a_i= \begin{cases} 1 &\text{若}Y_i\text{入樣}\\ 0 &\text{若}Y_i\text{不入樣} \end{cases} ai?={10?若Yi?入樣若Yi?不入樣?
由引理2.2可知：∑i=1NaiYi=∑i=1nyi,E(ai)=nN,E(ai2)=nN,E(aiaj)=n(n?1)N(N?1)\displaystyle\sum^{N}_{i=1}{a_iY_i}=\displaystyle\sum^{n}_{i=1}{y_i},\quad E(a_i)=\frac{n}{N},\quad E(a_i^2)=\frac{n}{N},\quad E(a_ia_j)=\frac{n(n-1)}{N(N-1)}i=1∑N?ai?Yi?=i=1∑n?yi?,E(ai?)=Nn?,E(ai2?)=Nn?,E(ai?aj?)=N(N?1)n(n?1)?
E(syx)=E(1n?1∑i=1n(xi?xˉ)(yi?yˉ))=1n?1E(∑i=1n(xiyi?xˉyi?yˉxi+xˉyˉ))=1n?1E(∑i=1nxiyi?xˉ∑i=1nyi?yˉ∑i=1nxi+∑i=1nxˉyˉ)=1n?1E(∑i=1nxiyi?nxˉyˉ?nxˉyˉ+2nxˉyˉ)=1n?1E(∑i=1nxiyi?nxˉyˉ)=1n?1E[∑i=1NXiYiai2?n(1n∑i=1nxi)(1n∑i=1nyi)]=1n?1E(∑i=1NXiYiai2?1n∑i=1nxi∑i=1nyi)=1n?1[∑i=1NXiYiai2?1n(∑i=1nxiyi+∑i≠jnxiyi)]=1n?1[∑i=1NXiYiai2?1n(∑i=1NXiYiai2+∑i≠jNXiYiaiaj)]=1n?1(E(ai2)∑i=1NXiYi?1nE(ai2)∑i=1NXiYi?1nE(aiaj)∑i≠jnXiYi)=1n?1(nN∑i=1NXiYi?1N∑i=1NXiYi?n?1N(N?1)∑i≠jnXiYi)=1n?1[n?1N∑i=1NXiYi?n?1N(N?1)(∑i=1NXi∑i=1NYi?∑i=1NXiYi)]=1n?1[(n?1N+n?1N(N?1))∑i=1NXiYi?n?1N(N?1)∑i=1NXi∑i=1NYi]=1n?1(n?1N?1∑i=1NXiYi?n?1N(N?1)NXˉ?NYˉ)=1N?1(∑i=1NXiYi?NXˉYˉ)=1N?1∑i=1N(Xi?Xˉ)(Yi?Yˉ)=Syx\begin{aligned} E(s_{yx})&=E\left(\frac{1}{n-1}\displaystyle\sum^{n}_{i=1}{(x_i-\bar{x})(y_i-\bar{y})}\right)\\ &=\frac{1}{n-1}E\left(\sum^{n}_{i=1}{(x_iy_i-\bar{x}y_i-\bar{y}x_i+\bar{x}\bar{y})}\right)\\ &=\frac{1}{n-1}E\left(\sum^{n}_{i=1}{x_iy_i}-\bar{x}\sum^{n}_{i=1}{y_i}-\bar{y}\sum^{n}_{i=1}{x_i}+\sum^{n}_{i=1}{\bar{x}\bar{y}}\right)\\ &=\frac{1}{n-1}E\left(\sum^{n}_{i=1}{x_iy_i}-n\bar{x}\bar{y}-n\bar{x}\bar{y}+2n\bar{x}\bar{y}\right)\\ &=\frac{1}{n-1}E\left(\sum^{n}_{i=1}{x_iy_i}-n\bar{x}\bar{y}\right)\\ &=\frac{1}{n-1}E\left[\sum^{N}_{i=1}{X_iY_ia_i^2}-n\left(\frac{1}{n}\sum^{n}_{i=1}{x_i}\right)\left(\frac{1}{n}\sum^{n}_{i=1}{y_i}\right)\right]\\ &=\frac{1}{n-1}E\left(\sum^{N}_{i=1}{X_iY_ia_i^2}-\frac{1}{n}\sum^{n}_{i=1}{x_i}\sum^{n}_{i=1}{y_i}\right)\\ &=\frac{1}{n-1}\left[\sum^{N}_{i=1}{X_iY_ia_i^2}-\frac{1}{n}\left(\sum^{n}_{i=1}{x_iy_i}+\sum^{n}_{i\neq j}{x_iy_i}\right)\right]\\ &=\frac{1}{n-1}\left[\sum^{N}_{i=1}{X_iY_ia_i^2}-\frac{1}{n}\left(\sum^{N}_{i=1}{X_iY_ia_i^2}+\sum^{N}_{i\neq j}{X_iY_ia_ia_j}\right)\right]\\ &=\frac{1}{n-1}\left(E(a_i^2)\sum^{N}_{i=1}{X_iY_i}-\frac{1}{n}E(a_i^2)\sum^{N}_{i=1}{X_iY_i}-\frac{1}{n}E(a_ia_j)\sum^{n}_{i\neq j}{X_iY_i}\right)\\ &=\frac{1}{n-1}\left(\frac{n}{N}\sum^{N}_{i=1}{X_iY_i}-\frac{1}{N}\sum^{N}_{i=1}{X_iY_i}-\frac{n-1}{N(N-1)}\sum^{n}_{i\neq j}{X_iY_i}\right)\\ &=\frac{1}{n-1}\left[\frac{n-1}{N}\sum^{N}_{i=1}{X_iY_i}-\frac{n-1}{N(N-1)}\left(\sum^{N}_{i=1}{X_i}\sum^{N}_{i=1}{Y_i}-\sum^{N}_{i=1}{X_iY_i}\right)\right]\\ &=\frac{1}{n-1}\left[\left(\frac{n-1}{N}+\frac{n-1}{N(N-1)}\right)\sum^{N}_{i=1}{X_iY_i}-\frac{n-1}{N(N-1)}\sum^{N}_{i=1}X_i\sum^{N}_{i=1}{Y_i}\right]\\ &=\frac{1}{n-1}\left(\frac{n-1}{N-1}\sum^{N}_{i=1}{X_iY_i}-\frac{n-1}{N(N-1)}N\bar{X}·N\bar{Y}\right)\\ &=\frac{1}{N-1}\left(\sum^{N}_{i=1}{X_iY_i}-N\bar{X}\bar{Y}\right)\\ &=\frac{1}{N-1}\sum^{N}_{i=1}{(X_i-\bar{X})(Y_i-\bar{Y})}\\ &=S_{yx}\\ \end{aligned} E(syx?)?=E(n?11?i=1∑n?(xi??xˉ)(yi??yˉ?))=n?11?E(i=1∑n?(xi?yi??xˉyi??yˉ?xi?+xˉyˉ?))=n?11?E(i=1∑n?xi?yi??xˉi=1∑n?yi??yˉ?i=1∑n?xi?+i=1∑n?xˉyˉ?)=n?11?E(i=1∑n?xi?yi??nxˉyˉ??nxˉyˉ?+2nxˉyˉ?)=n?11?E(i=1∑n?xi?yi??nxˉyˉ?)=n?11?E[i=1∑N?Xi?Yi?ai2??n(n1?i=1∑n?xi?)(n1?i=1∑n?yi?)]=n?11?E(i=1∑N?Xi?Yi?ai2??n1?i=1∑n?xi?i=1∑n?yi?)=n?11????i=1∑N?Xi?Yi?ai2??n1????i=1∑n?xi?yi?+i?=j∑n?xi?yi???????=n?11????i=1∑N?Xi?Yi?ai2??n1????i=1∑N?Xi?Yi?ai2?+i?=j∑N?Xi?Yi?ai?aj???????=n?11????E(ai2?)i=1∑N?Xi?Yi??n1?E(ai2?)i=1∑N?Xi?Yi??n1?E(ai?aj?)i?=j∑n?Xi?Yi????=n?11????Nn?i=1∑N?Xi?Yi??N1?i=1∑N?Xi?Yi??N(N?1)n?1?i?=j∑n?Xi?Yi????=n?11?[Nn?1?i=1∑N?Xi?Yi??N(N?1)n?1?(i=1∑N?Xi?i=1∑N?Yi??i=1∑N?Xi?Yi?)]=n?11?[(Nn?1?+N(N?1)n?1?)i=1∑N?Xi?Yi??N(N?1)n?1?i=1∑N?Xi?i=1∑N?Yi?]=n?11?(N?1n?1?i=1∑N?Xi?Yi??N(N?1)n?1?NXˉ?NYˉ)=N?11?(i=1∑N?Xi?Yi??NXˉYˉ)=N?11?i=1∑N?(Xi??Xˉ)(Yi??Yˉ)=Syx??
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