Brewer's method for sampling with unequal probabilities with n>2 I'm trying to use Brewer's method to sample 12 units out of a population of 73.
I read on Brewer and Hanif's "Sampling with unequal probabilities" that the probability must be proportional to  $$\frac{P_i (1 - P_i)}{(1-rP_i)}$$. 
I know that for $n = 2$ the first unit is selected with probability $$\frac{P_i (1 - P_i)}{D  (1-2P_i)}$$ where $$D = \sum_{i=1}^N\frac{P_i(1-P_i)}{(1-2P_i)}$$ and the second unit is selected with probability $$\frac{P_j}{1-P_i}$$.
I would like to know if this selection probabilities are the same for $n >  2$ or if I have to change the probabilities for the first unit according to the sample size.
 A: My reading of Brewer's procedure is as follows.


*

*To sample the first unit, set $r=n$ and compute 
$$
D_1 = \sum_{i=1}^N\frac{P_i(1-P_i)}{(1-nP_i)}
$$
Then sample the first unit with probability 
$$P^{(1)}_i = \frac{P_i (1 - P_i)}{D_1  (1-nP_i)}$$Let the index of the first sampled unit be $I_1$.

*To sample the second unit, set $r=n-1$ and compute
$$
D_2 = \sum_{i\in\{1,\ldots,N\}, i\notin \{ I_1 \} }\frac{P_i(1-P_i)}{(1-(n-1)P_i)}
$$
Then sample the second unit with probability 
$$P^{(2)}_i = \frac{P_i (1 - P_i)}{D_2  (1-(n-1)P_i)}$$ Let the index of the first sampled unit be $I_2$.

*etc.


To sample the $k$-th unit, set $r=n-k+1$ and compute
$$
D_k = \sum_{i\in\{1,\ldots,N\}, i\notin \{ I_1, \ldots, I_{k-1} \} }\frac{P_i(1-P_i)}{(1-(n-k+1)P_i)}
$$
Then sample the $k$-th unit with probability 
$$P^{(k)}_i = \frac{P_i (1 - P_i)}{D_k  (1-(n-k+1)P_i)}$$
B&H 83 refers to Brewer (1975) in Australian J of Statistics which I don't see any way of getting.
A: Update: August 19
I've now had a look at both Brewer's 1963 and 1975 articles.  The 1975 method is indeed the generalization for (n>2) as Stas stated, with the same draw probabilities for n = 1 & 2.
This has been a very informative topic. I wondered why there were two versions of "Brewer's Method" in Roberto's question.} 
The first formula Roberto gave is from "Brewer's Procedure" (Procedure 8) in Brewer and Hanif (1982). As Stas points out, it is based on Brewer (1975).  However the second version that Roberto shows is that which Cochran (1977, p. 261) and the SAS manual  also call "Brewer's Method" for n = 2. This turns out to be from Brewer (1963). It doesn't appear in Brewer & Hanif (1982) It does, as Procedure 8.
Durbin (1967, p.54) created another  method for n = 2 (Method I). It is Procedure 9  in B & H. Durbin was unaware of Brewer (1963), and prior to publication, JNK Rao pointed out to Durbin that the pairwise inclusion probabilities are identical to those in Brewer's 1963 method.
Sampford (1967) extended Brewer's 1963 method to n>2.  He gave several suggestions for implementation. That used by SAS is what Sampford called "method c" (p. 502). It is Procedure 11 in B & H, who call it the "Rao-Sampford Rejective" method. It is interesting because all draws are made with replacement; if a duplicate appears, the sample is rejected. 
Sampford's method is easy to write down. Define
$$
\lambda_i = p_i/(1-n p_i)
$$


*

*Draw the first unit with probability $p_i$.

*Draw all others with probability $\lambda_j/\sum_k^N \lambda_k$, with replacement

*If a drawing duplicates any previously selected item, reject the sample and begin again with Step 1.
The joint probabilities  $\pi_{ij}$ of inclusion for elements $i$ and $j$ are needed to compute a proper variance estimate for the Horvitz-Thompson ($p_i$ weighted) estimator..  Sampford gives the formula for  $\pi_{ij}$  on page 503 of his article and some examples of its calculation in his section 5. See also the SAS SURVEYSELECT manual, p. 8493.
References
Brewer, Kenneth RW 1963. A Model of Systematic Sampling with Unequal Probabilities. Australian Journal of Statistics 5, no. 1: 5-13.
Brewer, Kenneth RW (1975). A Simple Procedure For Sampling $\pi$pswor1. Australian Journal of Statistics, 17(3), 166-172.
Brewer, Kenneth RW, and Muhammad Hanif. 1982. Sampling with unequal probabilities (lecture notes in statistics). 
Cochran, William G. 1977. Sampling Techniques. New York: Wiley.
Durbin, J. 1967. Design of multi-stage surveys for the estimation of sampling errors. Applied Statistics 16, no. 2: 152-164.
Sampford, MR. 1967. On sampling without replacement with unequal probabilities of selection. Biometrika 54, no. 3-4: 499-513.
SAS SURVEYSELECT MANUAL http://support.sas.com/documentation/onlinedoc/stat/131/surveyselect.pdf
