Before entering into the mathematical structure of the multi-stage sampling models, we shall have to fix upon an appropriate notation. It is shown in Table 1.
Partial expectations and variances at a given stage, keeping the stages above fixed, are suffixed by that stage number
32.\[\begin{array}{ll} \text{i.e.}, & E_{2}=E_{2}/_{1}\\ & E_{5}= E_{5}/_{1,2,3,4}\\ \text{and} & V_{5} = V_{5}/_{1,2,3,4} \end{array}\]
The following identity is used repeatedly in the mathematical derivations of the various models.
33.\[V_{s,s+1,\cdots, k} = E_{s}E_{s+1}E_{s+2}E_{s+3}E_{s+4}\cdots E_{k-1}V_{k}+E_{s}\cdots E_{k-2}V_{k-1}E_{k}+ \cdots + E_{s}V_{s+1}E_{s+2}E_{s+3}\cdots E_{k}+V_{s}E_{s+1}E_{s+2}\cdots E_{k}\ (1 \leq s \leq k)\]
The following probabilities of selection are repeatedly used in the different models because they were most suitable for our practical applications. However, in other practical situations the probabilities of selection could be different from these.
34.\[\begin{align*} P_{i_1 } i_2 \cdots i_{k - 2} i_{k - 1} &= \frac{{N_{i_1 } \cdots i_{k - 1} k}}{{\sum\limits_{i_{k - 1} = 1}^{N_{i_1 \cdots i_{k - 2} (k - 1)} } {N_{i_1 \cdots i_{k - 1} k} } }} \\ P_{i_1 \cdots i_{k - 2} } &= \frac{{\sum\limits_{i_{k - 1} = 1}^{N_{i_1 \cdots i_{k - 2} (k - 1)} } {N_{i_1 \cdots i_{k - 1} k} } }}{{\sum\limits_{i_{k - 2} = 1}^{N_{i_1 \cdots i_{k - {\rm{s}}} (k - 2)} } {\sum\limits_{i_{k - 1} = 1}^{N_{i_1 \cdots i_{k - 2} (k - 1)} } {N_{i_1 \cdots i_{k - 1} k} } } }} \\ P_{i_1 \cdots i_{k - s} } &= \frac{{\sum\limits_{i_{k - s + 1} = 1}^{N_{i_1 \cdots i_{k - s} (k - s + 1)} } {\sum\limits_{i_{k - 1} = 1}^{N_{i_1 \cdots i_{k - 2} (k - 1)} } {N_{i_1 \cdots i_{k - 1} k} } } }}{{\sum\limits_{i_{k - s} = 1}^{N_{i_1 \cdots i_{k - s - 1} (k - s)} } {\sum\limits_{i_{k - 1} = 1}^{N_{i_1 \cdots i_{k - 2} (k - 1)} } {N_{i_1 \cdots i_{k - 1} k} } } }}(s = 2, \cdots ,k - 1) \\ P_{i_1 } &= \frac{{\sum\limits_{i_2 = 1}^{N_{i_1 2} } \cdots \sum\limits_{i_{k - 1} = 1}^{N_{i_1 \cdots i_{k - 2} (k - 1)} } {N_{i_1 \cdots i_{k - 1} k} } }}{N} \end{align*} \]${P_{{i_1} \cdots {i_k}}}$is any...