3.3.2 比例分配
比例分配指的是按各层单元数占总单元数的比例,也就是按各层的层权进行分配。此时:
$$
\frac{n_h}{n} = \frac{N_h}{N} = W_h \text{ 或 } f_h = \frac{n_h}{N_h} = \frac{n}{N} = f
\tag{3.19}
$$
对于分层随机抽样,总体均值 $\overline{Y}$ 的估计为:
$$
\hat{\overline{Y}} = \overline{y}{\text{prop}} = \sum{h=1}^{L} W_h \overline{y}h = \sum{h=1}^{L} \frac{N_h}{N} \overline{y}h = \sum{h=1}^{L} \frac{n_h}{n} \sum_{i=1}^{n_h} \frac{y_{hi}}{n_h} = \frac{1}{n} \sum_{h=1}^{L} \sum_{i=1}^{n_h} y_{hi} = \frac{1}{n} \sum_{i=1}^{n} y_i = \overline{y}
\tag{3.20}
$$
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