Confidence Interval Estimation 设总体 含有一个待估计参数 给定常数 , 找出统计量 使得：
P\left\{\theta_{1}\leq\theta\leq\theta_{2} \right\}=1-\alpha\quad 0<\alpha<1 \end{align}$$ 则称 $[\theta_{1},\theta_{2}]$ 为参数 $\theta$ 的置信度为 $1-\alpha$ 的置信区间 - $1-\alpha$ 置信度/置信水平/置信概率 区间以 $1-\alpha$ 的[[概率\|概率]]包含参数 $\theta$ - $\alpha$ 显著性水平/置信水平 表示该区间不包含真值 $\theta$ 的可能性 （估计出错的概率） $$\begin{align} P\left\{\left\lvert \hat{\theta}-\theta \right\rvert \leq \delta\right\}=1-\alpha \end{align}$$ - $\delta$ 为估计量与参数真值之间的误差限 - 不等式 $\hat{\theta}-\delta\leq \theta\leq\hat{\theta}+\delta$ 为**置信区间** 置信区间的长度 $\delta$ 越小，估计的精度越高 ### 区间估计的基本步骤 1. 将待估计的参数的分布转化为已知的分布，构造随机变量 由[[总体和样本\|样本]]采用[[点估计\|点估计]]得到未知参数的估计 参考[[正态总体统计量的分布特征\|正态总体统计量的分布特征]]，“对谁估计，就构造谁的分布” 2. 对给定的置信度 $1-\alpha$ 根据构造的随机变量的分布确定分位点 [[上分位点\|分位点]] 常采用概率对称的点 3. 从区间的不等式中解出待估计参数的范围，也即置信区间 置信区间是用已知量与样本的统计量表示出来的 4. 最终，置信区间以 $1-\alpha$ 的概率包含未知参数，完成区间估计 <svg xmlns="http://www.w3.org/2000/svg" version="1.1" viewBox="0 0 1530.4197185554856 536.3543131019264" width="1530.4197185554856" height="536.3543131019264"> <!-- svg-source:excalidraw --> <defs> 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-145.24, 103.27 -261.55, 142.96 -250.81 C182.66 -240.07, 228.44 -79.02, 281.89 -38.97 C335.34 1.08, 433.17 -15.35, 463.65 -10.52" stroke="#1971c2" stroke-width="2.5" fill="none" stroke-dasharray="1.5 8"></path></g></g><mask></mask><g stroke-linecap="round"><g transform="translate(357.72235966222297 327.8488844581193) rotate(0 237.14177900921072 -93.10004364067413)"><path d="M0.84 1.1 C11.58 -15.75, 37.83 -69.75, 64.85 -100.84 C91.88 -131.93, 127.5 -197.96, 162.98 -185.43 C198.46 -172.91, 225.66 -56.14, 277.72 -25.7 C329.78 4.73, 442.42 -6.59, 475.33 -2.82" stroke="#2f9e44" stroke-width="2.5" fill="none" stroke-dasharray="8 10"></path></g></g><mask></mask><g transform="translate(477.7023462557072 174.5378658697323) rotate(0 33.739990234375 9.702115764123619)"><text x="0" y="15.523714778160485" font-family="Helvetica, Segoe UI Emoji" font-size="16.873244807171574px" fill="#2f9e44" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic">卡方分布</text></g><g transform="translate(541.2175129645398 82.82260155366316) rotate(0 22.335731506347656 9.843511934839057)"><text x="0" y="15.749953454164222" font-family="Helvetica, Segoe UI Emoji" font-size="17.119151191024482px" fill="#1971c2" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic">F分布</text></g></svg> 对于对称分布的**正态分布**和 **t 分布**: $Z_{1-\alpha }=-Z_{\alpha}$ 所以：$Z_{1-\alpha/ 2 }=-Z_{\alpha / 2}$ ### 单正态总体均值与方差的区间估计 正态总体： $X\sim N(\mu,\sigma^{2})$ $\overline{X}\sim N(\mu, \dfrac{\sigma^{2}}{n})$ #### 1.方差 $\sigma^{2}$ 已知，均值 $\mu$ 的区间估计 $$\begin{align} Z=U=\dfrac{\overline{X}-\mu }{\dfrac{\sigma}{\sqrt{ n }}}\sim N(0,1) \end{align}$$ $$\begin{align} P\left\{ \left\lvert \dfrac{\overline{X} -\mu}{\sigma / \sqrt{ n }} \right\rvert \leq z_{\frac{\alpha}{2}}\right\} & = 1-\alpha \\ \end{align}$$ 分位点 $z_{\alpha / 2}$ 置信度为 $1-\alpha$ 的参数 $\mu$ 的置信区间 $$\begin{align} \left[\overline{X} - \dfrac{\delta}{\sqrt{ n }} z_{\frac{\alpha}{2}},\overline{X} + \dfrac{\delta}{\sqrt{ n }} z_{\frac{\alpha}{2}}\right] \end{align}$$ 置信区间长度 $\delta= \dfrac{\sigma}{\sqrt{ n }}z_{\frac{\alpha}{2}}$ #### 2.方差 $\sigma^{2}$ 未知，均值 $\mu$ 的区间估计 [[t分布\|t分布]] $$\begin{align} T= \dfrac{\overline{X}-\mu }{\dfrac{S}{\sqrt{ n }}} \sim t(n-1) \end{align}$$ 分位点 $t_{\alpha /2}(n-1)$ $$\begin{align} P\left\{\left\lvert \dfrac{\overline{X} -\mu}{S / \sqrt{ n }} \right\rvert \leq t_{\alpha /2} (n-1)\right\}=1-\alpha \end{align}$$ 置信区间 $$\begin{align} \left[\overline{X} - \dfrac{S}{\sqrt{ n }}t_{\alpha / 2}(n-1),\overline{X} + \dfrac{S}{\sqrt{ n }}t_{\alpha / 2}(n-1)\right] \end{align}$$ #### 3.均值 $\mu$ 已知，方差 $\sigma^{2}$ 的区间估计 [[卡方分布\|卡方分布]] $$\begin{align} \chi^{2}&=\dfrac{ \sum\limits_{i=1}^{n}(X_{i}-\mu)^{2} }{\sigma^{2}}\sim \chi^{2}(n) \end{align}$$ 分位数： $\chi^{2}_{1- \alpha/ 2}$ $\chi^{2}_{\alpha /2}$ $$\begin{align} P\left\{\chi^{2}_{ 1-\alpha/ 2} (n)\leq \chi^{2}\leq \chi^{2}_{\alpha /2 }(n)\right\}=1-\alpha \end{align}$$ $$\begin{align} P\left\{ \dfrac{\sum\limits_{i=1}^{n}(X_{i}-\mu_{0})^{2}}{\chi^{2}_{ \alpha /2}(n)} \leq \sigma^{2}\leq \dfrac{\sum\limits_{i=1}^{n}(X_{i}-\mu_{0})^{2}}{\chi^{2}_{ 1- \alpha /2}(n)}\right\}=1-\alpha \end{align}$$ #### 4.均值 $\mu$ 未知，方差 $\sigma^{2}$ 的区间估计 $$\begin{align} \dfrac{(n-1)S^{2}}{\sigma^{2}}\sim \chi^{2}(n-1) \end{align}$$ 分位数 $\chi^{2}_{1- \alpha /2}(n-1)$ $\chi^{2}_{\alpha/ 2}(n-1)$ $$\begin{align} P\left\{ \chi^{2}_{1- \alpha /2}(n-1)\leq \dfrac{(n-1)S^{2}}{\sigma^{2}}\leq\chi^{2}_{\alpha/ 2}(n-1)\right\}=1-\alpha \end{align}$$ $$\begin{align} P\left\{\dfrac{(n-1)S^{2}}{\chi^{2}_{\alpha / 2}(n-1)}\leq \sigma^{2} \leq \dfrac{(n-1)S^{2}}{\chi^{2}_{1-\alpha /2}(n-1)}\right\} \end{align}$$ ### 双正态总体的均值差的区间估计 在实际工作生活中，要根据样本分析两个不同[[总体和样本\|正态总体]] 之间的差异 $X\sim N(\mu_{1},\sigma_{1}^{2}),Y\sim N(\mu_{2},\sigma_{2}^{2})$ $\overline{X}= \dfrac{1}{n_{1}}\sum\limits_{i=1}^{n_{1}}X_{i}\quad \overline{Y}=\dfrac{1}{n_{2}}\sum\limits_{i=1}^{n_{2}}Y_{i}$ $$\begin{align} S_{1}^{2}&=\dfrac{1}{n_{1}-1}\sum\limits_{i=1}^{n_{1}}(X_{i}-\overline{X} )^{2} \\ S_{2}^{2}&=\dfrac{1}{n_{2}-1}\sum\limits_{i=1}^{n_{2}}(Y_{i}-\overline{Y} )^{2} \end{align}$$ #### 1. $\sigma^{2}_{1}=\sigma^{2}_{1}$ 均已知，$\mu_{1}-\mu_{2}$ 的区间估计 $$\begin{align} U&=\dfrac{(\overline{X} -\overline{Y} )-(\mu_{1}-\mu_{2}) }{\sqrt{ \dfrac{\sigma_{1}^{2}}{n_{1}}+\dfrac{\sigma^{2}_{2}}{n_{2}} }} \\ &\sim N(0,1) \end{align}$$ $$\begin{align} P\left\{\left\lvert U \right\rvert\leq u_{\alpha / 2} \right\}=1-\alpha \end{align}$$ $$\begin{align} \left[(\overline{X} -\overline{Y} )-u_{\alpha / 2}\sqrt{ \dfrac{\sigma_{1}^{2}}{n_{1}} +\dfrac{\sigma_{2}^{2}}{n_{2}}},(\overline{X} -\overline{Y} )+u_{\alpha / 2}\sqrt{ \dfrac{\sigma_{1}^{2}}{n_{1}} +\dfrac{\sigma_{2}^{2}}{n_{2}} }\right] \end{align}$$ #### 2. $\sigma^{2}_{1}=\sigma^{2}_{2}$ 均未知，$\mu_{1}-\mu_{2}$ 的区间估计 $$\begin{align} T&= \dfrac{\overline{X} -\overline{Y} -(\mu_{1}-\mu_{2})}{S_{w}\sqrt{ \dfrac{1}{n_{1}}+ \dfrac{1}{n_{2}} }} \\ &\sim t(n_{1}+n_{2}-2) \end{align}$$ $$\begin{align} S_{w}^{2}=\dfrac{(n_{1}-1)S_{1}^{2}+(n_{2}-1)S_{2}^{2}}{n_{1}+n_{2}-2} \end{align}$$ $$\begin{align} P\left\{\left\lvert T \right\rvert\leq t_{\alpha / 2}(n_{1}+n_{2}-2) \right\}=1-\alpha \end{align}$$ #### 3 .方差之比的区间估计 $X\sim N(\mu_{1},\sigma_{1}^{2}),Y\sim N(\mu_{2},\sigma_{2}^{2})$ 均值方差均未知 $$\begin{align} F=\dfrac{S_{1}^{2} / S_{2}^{2}}{\sigma_{1}^{2} / \sigma_{2}^{2}}\sim F(n_{1}-1,n_{2}-1) \end{align}$$ 分位数 $F_{1- \alpha /2}(n_{1}-1,n_{2}-1)$ $F_{\alpha/ 2}(n_{1}-1,n_{2}-1)$ 置信区间： $$\begin{align} \left[\dfrac{S_{1}^{2} / S_{2}^{2}}{F_{\alpha/ 2}(n_{1}-1,n_{2}-1)},\dfrac{S_{1}^{2} / S_{2}^{2}}{F_{1- \alpha /2}(n_{1}-1,n_{2}-1)}\right] \end{align}$$
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