Chap 2 简单随机抽样

2.1 概念，表示以及基本知识

抽样技术有限总体的方差定义与数理统计的不同！
$$\begin{gathered} {\sigma }^2=\frac{1}{N}\sum _{i=1}^N(Y_i-\bar{Y}{)}^2 \\ {S}^2=\frac{1}{N-1}\sum _{i=1}^N(Y_i-\bar{Y}{)}^2 \end{gathered}$$
SRSWOR下，关于${\alpha }_i$ 的常识
$$\begin{array}{rl}E({\alpha }_i)& ={\pi }_i=f=\frac{n}{N}\\ V({\alpha }_i)& =f(1-f)=\frac{n}{N}(1-\frac{n}{N})\\ E({\alpha }_i{\alpha }_j)& =1\ast {\pi }_{ij}+0=\frac{n(n-1)}{N(N-1)}\\ Cov({\alpha }_i,{\alpha }_j)& =E({\alpha }_i{\alpha }_j)-E({\alpha }_i)E({\alpha }_j)\\ & =\frac{-f(1-f)}{N-1}\end{array}$$

2.2 估计

2.2.1 总体均值的估计

总体均值的估计：样本均值$\bar{y}={\sum }_{i=1}^n{y}_i$

• 估计的无偏性
  $$\begin{array}{rl}E(\bar{y})& =\sum _{i=1}^{C_N^n}{\bar{y}}_{i}P(S_i)\\ & =\sum _{i=1}^{C_N^n}{\bar{y}}_i\frac{1}{C_N^n}\\ & =\sum _{i=1}^{C_N^n}(Y_{i1}+Y_{i2}+...+Y_{in})\frac{1}{n{C}_N^n}\\ & =\frac{1}{n{C}_N^n}(\sum _{i=1}^N{C}_1^1{C}_{N-1}^{n-1}{Y}_i)\\ & =\frac{\frac{(N-1)!}{(n-1)!(N-n)!}}{\frac{nN!}{n!(N-n)!}}\sum _{i=1}^N{Y}_i\\ & =\frac{1}{N}\sum _{i=1}^N{Y}_i\\ & =\bar{Y}\end{array}$$
• 估计的方差
  $$\begin{array}{rl}V(\bar{y})& =V(\frac{1}{n}\sum _{i=1}^n{y}_i)\\ & =V(\frac{1}{n}\sum _{i=1}^N{\alpha }_i{Y}_i)\\ & =\frac{1}{n^2}(\sum _{i=1}^N{Y}_i^{2}V({\alpha }_i)+\sum _{i\ne j}{Y}_i{Y}_{j}Cov({\alpha }_i,{\alpha }_j))\\ & =\frac{f(1-f)}{n^2}(\sum _{i=1}^N{Y}_i^2-\sum _{i\ne j}{Y}_i{Y}_j\frac{1}{N-1})\\ & =\frac{(1-f)}{nN(N-1)}(\frac{N}{N-1}\sum _{i=1}^N{Y}_i^2-\frac{1}{N-1}(\sum _{i=1}^N{Y}_i^2+\sum _{i\ne j}{Y}_i{Y}_j))\\ & =\frac{(1-f)}{nN(N-1)}(N\sum _{i=1}^N{Y}_i^2-(\sum _{i=1}^N{Y}_i{)}^2)\\ & =\frac{(1-f)}{nN(N-1)}(N\sum _{i=1}^N{Y}_i^2-(N\bar{Y}{)}^2)\\ & =\frac{(1-f)}{n}\frac{1}{N-1}(\sum _{i=1}^N{Y}_i^2-N{\bar{Y}}^2)\\ & =\frac{(1-f)}{n}\frac{1}{N-1}(\sum _{i=1}^N{Y}_i^2-2\sum _{i=1}^N{Y}_i\bar{Y}+\sum _{i=1}^N{\bar{Y}}^2)\\ & =\frac{(1-f)}{n}\frac{1}{N-1}\sum _{i=1}^N(Y_i-\bar{Y}{)}^2\\ & =\frac{1-f}{n}{S}^2\end{array}$$
• 方差的样本估计的无偏性
  $$E(v(\bar{y}))=E(\frac{1-f}{n}{s}^2)=\frac{1-f}{n}{S}^2=E(V(\bar{y}))$$
  只需要证明（证明的关键还是要把$y_i$找到合适的形式转到${\alpha }_i{Y}_i$）
  $$\begin{array}{rl}\because E(s^2)& =E(\frac{1}{n-1}\sum _{i=1}^n(y_i-\bar{y}{)}^2)\\ & =\frac{1}{n-1}E(\sum _{i=1}^n{y}_i^2-n{\bar{y}}^2)\\ & =\frac{1}{n-1}[E(\sum _{i=1}^N{\alpha }_i{Y}_i^2)-nE{\bar{y}}^2]\\ & =\frac{1}{n-1}[\sum _{i=1}^{N}E({\alpha }_i)Y_i^2-n(V(\bar{y})+E(\bar{y}{)}^2)]\\ & =\frac{1}{n-1}[\frac{n}{N}\sum _{i=1}^N{Y}_i^2-n(\frac{1-f}{n}{S}^2+{\bar{Y}}^2)]\\ & =\frac{n}{N(n-1)}[\sum _{i=1}^N{Y}_i^2-N{\bar{Y}}^2-\frac{N}{n}(1-\frac{n}{N})S^2)]\\ & =\frac{n}{N(n-1)}[\sum _{i=1}^N(Y_i-\bar{Y}{)}^2-\frac{N}{n}{S}^2+S^2)]\\ & =\frac{n}{N(n-1)}(N-1+1-\frac{N}{n})S^2\\ & =\frac{n}{N(n-1)}(\frac{Nn-N}{n})S^2=S^2\\ \therefore E(v(\bar{y}))& =E(\frac{1-f}{n}{s}^2)=\frac{1-f}{n}{S}^2=E(V(\bar{y}))\end{array}$$

2.2.2 总体总量的估计

总体总量的估计：总体均值的乘上N即可

2.2.3 总体比例的估计

总体比例的估计：$Y_i=\{{}_{0,otherwise}^{1,第i个总体单元含有某考虑的特征}i=1,2,...N$只是总体均值估计的特殊情况

比例$P=\frac{A}{N}=\frac{{\sum }_{i=1}^N{Y}_i}{N}$

2.2.4 两个估计的协方差

$$cov(\bar{y},\bar{x})=E(\bar{y}-\bar{Y})(\bar{x}-\bar{X})$$
可以证明
$$\begin{gathered} cov(\bar{y},\bar{x})=\frac{1-f}{n}{S}_{yx} \\ 其中，S_{yx}=\frac{1}{N-1}\sum _{i=1}^N(Y_i-\bar{Y})(X_i-\bar{X})为总体协方差 \end{gathered}$$

此外，正如 2.2.1-2.2.3节中涉及到的估计量方差及其方差估计一样，$\bar{y}$与$\bar{x}$之间的协方差也有无偏估计——简单随机样本协方差。

$$\begin{gathered} s_{yx}=\frac{1}{n-1}\sum _{i=1}^n(y_i-\bar{y})(x_i-\bar{x}) \\ satisfies,E(s_{yx})=S_{yx} \end{gathered}$$

2.3 比率估计

2.4 回归估计

2.5 Carry out a SRS

2.5.1 样本量的确定