# «统计学完全教程»笔记：第7章 CDF 和统计泛函的估计

$$\hat{F}_{n}(x)=\frac{\sum_{i=1}^{n} I\left(X_{i} \leqslant x\right)}{n}$$

$$I\left(X_{i} \leqslant x\right)= \begin{cases}1, & X_{i} \leqslant x \\ 0, & X_{i}>x\end{cases}$$

“在实线上”，意味着定义域是 $\mathbb{R}$ 分布函数，其实就是 CDF

1function F_n(t, X_1, X_2, ..., X_n){
2    let count = 0;
3    for (let i = 0; i < X.length; i++) {
4        if (X_i <= t) {
5            count++;
6        }
7    }
8    return count / n;
9}


Glivenko-Cantelli 定理. 使 $X_{1}, \ldots, X_{n} \sim F .$ 则

$$\sup _{x}\left|\widehat{F}_{n}(x)-F(x)\right| \stackrel{\mathrm{P}}{\longrightarrow} 0$$

$\stackrel{p}{\longrightarrow}$ 表示 依概率收敛

Dvoretzky-Kiefer-Wolfowitz (DKW) 不等式

$$\mathbb{P}\left(\sup _{x}\left|F(x)-\widehat{F}_{n}(x)\right|>\epsilon\right) \leq 2 e^{-2 n \epsilon^{2}} .$$

\begin{aligned} L(x) &=\max \left\{\widehat{F}_{n}(x)-\epsilon_{n}, 0\right\} \\ U(x) &=\min \left\{\widehat{F}_{n}(x)+\epsilon_{n}, 1\right\} \\ \text { where } \epsilon_{n} &=\sqrt{\frac{1}{2 n} \log \left(\frac{2}{\alpha}\right)} \end{aligned}

$$\mathbb{P}(L(x) \leq F(x) \leq U(x) \text { for all } x) \geq 1-\alpha .$$

## 7.2 统计泛函

$$\mu = \int_{}^{} x\ \mathrm{d} F(x)$$ $$\sigma ^2 = \int_{}^{} ( x - \mu ) ^{2} \mathrm{d}F(x)$$

$$m = \forall ^{-1} (1 / 2)$$

$\theta = T(F)$ 的**嵌入估计器（plug-in estimator）**定义为：

$$\hat{\theta }_n = T(\hat{F}_n)$$

$$T(aF+bG) = aT(F) + bT(G)$$

$$T(\hat{F}_n) = \int_{}^{} r(x) \mathrm{d} \hat{F}_n(x) = \dfrac{1}{n} \sum_{i = 1}^{n} r(X_i)$$

$$T\left(\hat{F}_{n}\right) \approx N\left(T(F), \hat{\mathrm{se}}^{2}\right)$$

$$T\left(\hat{F}_{n}\right) \pm z_{\alpha / 2} \hat{\text { se }}$$

$$\kappa=\frac{\mathbb{E}(X-\mu)^{3}}{\sigma^{3}}=\frac{\int(x-\mu)^{3} \mathrm{~d} F(x)}{\left[\int(x-\mu)^{2} \mathrm{~d} F(x)\right]^{3 / 2}} .$$

$$\hat{\kappa}=\frac{\int(x-\mu)^{3} \mathrm{~d} \hat{F}_{n}(x)}{\left[\int(x-\mu)^{2} \mathrm{~d} \hat{F}_{n}(x)\right]^{3 / 2}}=\frac{1 / n \sum_{i}\left(X_{i}-\hat{\mu}\right)^{3}}{\hat{\sigma}^{3}} .$$