Sampling – How to Apply Brewer’s Method for Sampling with Unequal Probabilities When n>2

samplingweighted-sampling

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.

Best Answer

My reading of Brewer's procedure is as follows.

  1. 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 $I_1$.
  2. 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 $I_2$.
  3. 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.

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