8.4.2 一般分布的 chi2 拟合检验

对于一般总体 $X$ 的非参数检验,我们的任务是对如下的假设检验问题 (II) 作显著性检验:

其中 $F_{0}$ 为已知分布, $θ_{1}, θ_{2}, \dots, θ_{r}$ 为其参数 (一般情况下未知).

我们先作直观分析. 若样本 $(X_{1}, X_{2}, \dots, X_{n})$ 一次的观测值为 $(x_{1}, x_{2}, \dots, x_{n})$ . 根据该观测值取值的情况, 我们人为地将实轴 $(- \infty, + \infty)$ 分成 $k$ 个区间:

(- \infty, a_{1}], (a_{1}, a_{2}], (a_{2}, a_{3}], \dots (a_{k - 2}, a_{k - 1}], (a_{k - 1}, + \infty) .

且统计出 $(x_{1}, x_{2}, \dots, x_{n})$ 取值落在第 $i$ 个区间的个数为 $ν_{i} (i = 1, 2, \dots, k)$ .

若 $F_{0} (\cdot; θ_{1}, θ_{2}, \dots, θ_{r})$ 的参数已知,我们可以计算概率:

p_{1}^{0} = F_{0} (a_{1}; θ_{1}, θ_{2}, \dots, θ_{r}),

p_{2}^{0} = F_{0} (a_{2}; θ_{2}, θ_{2}, \dots, θ_{r}) - F_{0} (a_{1}; θ_{1}, θ_{2}, \dots, θ_{r}),

p_{3}^{0} = F_{0} (a_{3}; θ_{2}, θ_{2}, \dots, θ_{r}) - F_{0} (a_{2}; θ_{1}, θ_{2}, \dots, θ_{r}),

…

p_{k - 1}^{0} = F_{0} (a_{k - 1}; θ_{2}, θ_{2}, \dots, θ_{r}) - F_{0} (a_{k - 2}; θ_{1}, θ_{2}, \dots, θ_{r}),

p_{k}^{0} = 1 - F_{0} (a_{k - 1}; θ_{2}, θ_{2}, \dots, θ_{r}) .

那么, 根据 “事件的频率接近于概率” 的事实, 当 $i = 1 \sum k (\frac{ν _{i}}{n} - p_{i}^{0})^{2}$ 较大时, 应拒绝原假设 $H_{0}$ .

然而,一般情况下,参数 $(θ_{1}, θ_{2}, \dots, θ_{r})$ 是未知的. 此时,若求出参数的最大

似然估计 $(θ_{1}, θ_{2}, \dots, θ_{r})$ ,并由此估计概率:

p_{1}^{0} = F_{0} (a_{1}; θ_{1}, θ_{2}, \dots, θ_{r}),

p_{2}^{0} = F_{0} (a_{2}; θ_{2}, θ_{2}, \dots, θ_{r}) - F_{0} (a_{1}; θ_{1}, θ_{2}, \dots, θ_{r}),

p_{3}^{0} = F_{0} (a_{3}; θ_{2}, θ_{2}, \dots, θ_{r}) - F_{0} (a_{2}; θ_{1}, θ_{2}, \dots, θ_{r}),

…

p_{k - 1}^{0} = F_{0} (a_{k - 1}; θ_{2}, θ_{2}, \dots, θ_{r}) - F_{0} (a_{k - 2}; θ_{1}, θ_{2}, \dots, θ_{r}),

p_{k}^{0} = 1 - F_{0} (a_{k - 1}; θ_{2}, θ_{2}, \dots, θ_{r}) .

那么可以证明 (皮尔逊 - 费希尔 (Pearson-Fisher) 定理) 近似地有

K = i = 1 \sum k \frac{( ν _{i} - n p _{i}^{0} ) ^{2}}{n p _{i}^{0}} \sim χ^{2} (k - r - 1) . (8.4.3)

从而假设检验问题 (II) 的显著性水平为 $α$ 的拒绝域为

K = i = 1 \sum k \frac{( ν _{i} - n p _{i}^{0} ) ^{2}}{n p _{i}^{0}} \geq χ_{1 - α}^{2} (k - r - 1) . (8.4.4)

Youliang Zhong