单样本率置信区间¶
样本率用 \(\hat{p}\) 表示，总体率用 \(p\) 表示。
渐进正态法¶
样本率 \(\hat{p}\) 在大样本时，其分布近似服从正态分布：
\[
\hat{p} \ \dot{\sim} \ N\left(p, \frac{\hat{p}(1-\hat{p})}{n}\right)
\]
使用 \(z\) 分布构建置信区间：
\[
z = \frac{\hat{p} - p}{\sqrt{\hat{p}(1-\hat{p})/n}} \sim N(0, 1)
\]
\[
\begin{align}
L & = \hat{p} - z_{1-\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \\
U & = \hat{p} + z_{1-\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}
\end{align}
\]
定义置信区间的宽度为 \(d\)，则：
\[
d = \min(U, 1) - \max(L, 0)
\]
\[
\begin{align}
L & = \hat{p} - z_{1-\alpha} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \\
U & = 1
\end{align}
\]
定义均值到置信限的距离为 \(d\)，则：
\[
d = \hat{p} - \max(L, 0)
\]
\[
\begin{align}
L & = 0 \\
U & = \hat{p} + z_{1-\alpha} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}
\end{align}
\]
定义均值到置信限的距离为 \(d\)，则：
\[
d = \min(U, 1) - \hat{p}
\]
渐进正态法（连续性校正）¶
在 渐进正态法 的基础上添加校正项 \(1/2n\)：
\[
z = \frac{\hat{p} - p \pm \frac{1}{2n}}{\sqrt{\hat{p}(1-\hat{p})/n}} \sim N(0, 1)
\]
\[
\begin{align}
L & = \hat{p} - z_{1-\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} - \frac{1}{2n} \\
U & = \hat{p} + z_{1-\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} + \frac{1}{2n}
\end{align}
\]
定义置信区间的宽度为 \(d\)，则：
\[
d = \min(U, 1) - \max(L, 0)
\]
\[
\begin{align}
L & = \hat{p} - z_{1-\alpha} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} - \frac{1}{2n} \\
U & = 1
\end{align}
\]
定义均值到置信限的距离为 \(d\)，则：
\[
d = \hat{p} - \max(L, 0)
\]
\[
\begin{align}
L & = 0 \\
U & = \hat{p} + z_{1-\alpha} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} + \frac{1}{2n}
\end{align}
\]
定义均值到置信限的距离为 \(d\)，则：
\[
d = \min(U, 1) - \hat{p}
\]
Clopper-Pearson¶
设 \(F_{x;v_1,v_2}\) 表示自由度为 \(v_1\) 和 \(v_2\) 的 \(F\) 分布的 \(x\) 分位数。
\[
\begin{align}
L & = \left[ 1 + \frac{n - n\hat{p} + 1}{n\hat{p} F_{\alpha/2;\ 2n\hat{p},\ 2(n - n\hat{p} + 1)}} \right]^{-1} \\
U & = \left[ 1 + \frac{n - n\hat{p}}{(n\hat{p} + 1) F_{1-\alpha/2;\ 2(n\hat{p} + 1), \ 2(n - n\hat{p})}} \right]^{-1}
\end{align}
\]
定义置信区间的宽度为 \(d\)，则：
\[
d = U - L
\]
\[
\begin{align}
L & = \left[ 1 + \frac{n - n\hat{p} + 1}{n\hat{p} F_{\alpha;\ 2n\hat{p},\ 2(n - n\hat{p} + 1)}} \right]^{-1} \\
U & = 1
\end{align}
\]
定义均值到置信限的距离为 \(d\)，则：
\[
d = \hat{p} - L
\]
\[
\begin{align}
L & = 0 \\
U & = \left[ 1 + \frac{n - n\hat{p}}{(n\hat{p} + 1) F_{1-\alpha;\ 2(n\hat{p} + 1), \ 2(n - n\hat{p})}} \right]^{-1}
\end{align}
\]
定义均值到置信限的距离为 \(d\)，则：
\[
d = U - \hat{p}
\]
Wilson Score¶
\[
\begin{align}
L & = \frac{\left(2n\hat{p} + z_{1-\alpha/2}^2\right) - z_{1-\alpha/2} \sqrt{z_{1-\alpha/2}^2 + 4n\hat{p}(1-\hat{p})}}{2\left(n + z_{1-\alpha/2}^2\right)} \\
U & = \frac{\left(2n\hat{p} + z_{1-\alpha/2}^2\right) + z_{1-\alpha/2} \sqrt{z_{1-\alpha/2}^2 + 4n\hat{p}(1-\hat{p})}}{2\left(n + z_{1-\alpha/2}^2\right)}
\end{align}
\]
定义置信区间的宽度为 \(d\)，则：
\[
d = U - L
\]
\[
\begin{align}
L & = 0 \\
U & = \frac{\left(2n\hat{p} + z_{1-\alpha}^2\right) + z_{1-\alpha} \sqrt{z_{1-\alpha}^2 + 4n\hat{p}(1-\hat{p})}}{2\left(n + z_{1-\alpha}^2\right)}
\end{align}
\]
定义均值到置信限的距离为 \(d\)，则：
\[
d = \hat{p} - L
\]
\[
\begin{align}
L & = \frac{\left(2n\hat{p} + z_{1-\alpha}^2\right) - z_{1-\alpha} \sqrt{z_{1-\alpha}^2 + 4n\hat{p}(1-\hat{p})}}{2\left(n + z_{1-\alpha}^2\right)} \\
U & = 1
\end{align}
\]
定义均值到置信限的距离为 \(d\)，则：
\[
d = U - \hat{p}
\]
Wilson Score 连续性校正¶
\[
\begin{align}
L & = \frac{\left(2n\hat{p} + z_{1-\alpha/2}^2 - 1\right) - z_{1-\alpha/2} \sqrt{z_{1-\alpha/2}^2 - \frac{1}{n} + 4n\hat{p}(1-\hat{p}) + 4\hat{p} - 2}}{2\left(n + z_{1-\alpha/2}^2\right)} \\
U & = \frac{\left(2n\hat{p} + z_{1-\alpha/2}^2 + 1\right) + z_{1-\alpha/2} \sqrt{z_{1-\alpha/2}^2 - \frac{1}{n} + 4n\hat{p}(1-\hat{p}) - 4\hat{p} + 2}}{2\left(n + z_{1-\alpha/2}^2\right)}
\end{align}
\]
定义置信区间的宽度为 \(d\)，则：
\[
d = U - L
\]
\[
\begin{align}
L & = 0 \\
U & = \frac{\left(2n\hat{p} + z_{1-\alpha}^2 + 1\right) + z_{1-\alpha} \sqrt{z_{1-\alpha}^2 - \frac{1}{n} + 4n\hat{p}(1-\hat{p}) - 4\hat{p} + 2}}{2\left(n + z_{1-\alpha}^2\right)}
\end{align}
\]
定义均值到置信限的距离为 \(d\)，则：
\[
d = \hat{p} - L
\]
\[
\begin{align}
L & = \frac{\left(2n\hat{p} + z_{1-\alpha}^2 - 1\right) - z_{1-\alpha} \sqrt{z_{1-\alpha}^2 - \frac{1}{n} + 4n\hat{p}(1-\hat{p}) + 4\hat{p} - 2}}{2\left(n + z_{1-\alpha}^2\right)} \\
U & = 1
\end{align}
\]
定义均值到置信限的距离为 \(d\)，则：
\[
d = U - \hat{p}
\]
Wilson Score 连续性校正置信区间宽度随样本量 \(n\) 的变化
以 \(p = 0.9\) 为例，绘制双侧 95% 置信区间宽度随样本量 \(n\) 变化的图像如下：
如果将 \(n\) 视为连续型变量，则随着 \(n\) 的增大，置信区间宽度先增大后减小，这可能会给数值求解带来一些麻烦。
若设定置信区间宽度为 \(0.8\)，则理论上存在两个数值解，实际应取较大的解作为样本量估算结果。
brentq 要求求根区间左右两端点处的函数值异号，此时可先用 minimize_scalar 求出区间内的极大值，将极大值点作为求根区间下限，再应用 brentq 进行数值求解。