Available methods are "Hajek", "HartleyRao", "Tille",
"Brewer1","Brewer2","Brewer3", and "Brewer4".
Note that these methods were derived for high-entropy sampling designs,
therefore they could have low performance under different designs.
Hájek (1964) approximation [method="Hajek"] is derived under Maximum Entropy sampling design
and is given by
$$\tilde{\pi}_{ij} = \pi_i\pi_j \frac{1 - (1-\pi_i)(1-\pi_j)}{d} $$
where \(d = \sum_{i\in U} \pi_i(1-\pi_i) \)
Hartley and Rao (1962) proposed the following approximation under
randomised systematic sampling [method="HartleyRao"]:
$$\tilde{\pi}_{ij} = \frac{n-1}{n} \pi_i\pi_j + \frac{n-1}{n^2} (\pi_i^2 \pi_j + \pi_i \pi_j^2)
- \frac{n-1}{n^3}\pi_i\pi_j \sum_{i\in U} \pi_j^2$$
$$ + \frac{2(n-1)}{n^3} (\pi_i^3 \pi_j + \pi_i\pi_j^3 + \pi_i^2 \pi_j^2)
- \frac{3(n-1)}{n^4} (\pi_i^2 \pi_j + \pi_i\pi_j^2) \sum_{i \in U}\pi_i^2$$
$$+ \frac{3(n-1)}{n^5} \pi_i\pi_j \biggl( \sum_{i\in U} \pi_i^2 \biggr)^2
- \frac{2(n-1)}{n^4} \pi_i\pi_j \sum_{i \in U} \pi_j^3 $$
Tillé (1996) proposed the approximation \(\tilde{\pi}_{ij} = \beta_i\beta_j\),
where the coefficients \(\beta_i\) are computed iteratively through the
following procedure [method="Tille"]:
\(\beta_i^{(0)} = \pi_i, \,\, \forall i\in U\)
\( \beta_i^{(2k-1)} = \frac{(n-1)\pi_i}{\beta^{(2k-2)} - \beta_i^{(2k-2)}} \)
\(\beta_i^{2k} = \beta_i^{(2k-1)}
\Biggl( \frac{n(n-1)}{(\beta^(2k-1))^2 - \sum_{i\in U} (\beta_k^{(2k-1)})^2 } \Biggr)^(1/2) \)
with \(\beta^{(k)} = \sum_{i\in U} \beta_i^{i}, \,\, k=1,2,3, \dots \)
Finally, Brewer (2002) and Brewer and Donadio (2003) proposed four approximations,
which are defined by the general form
$$\tilde{\pi}_{ij} = \pi_i\pi_j (c_i + c_j)/2 $$
where the \(c_i\) determine the approximation used:
Equation (9) [method="Brewer1"]:
$$c_i = (n-1) / (n-\pi_i)$$
Equation (10) [method="Brewer2"]:
$$c_i = (n-1) / \Bigl(n- n^{-1}\sum_{i\in U}\pi_i^2 \Bigr)$$
Equation (11) [method="Brewer3"]:
$$c_i = (n-1) / \Bigl(n - 2\pi_i + n^{-1}\sum_{i\in U}\pi_i^2 \Bigr)$$
Equation (18) [method="Brewer4"]:
$$c_i = (n-1) / \Bigl(n - (2n-1)(n-1)^{-1}\pi_i + (n-1)^{-1}\sum_{i\in U}\pi_i^2 \Bigr)$$