Poisson sampling

In survey methodology, Poisson sampling (sometimes denoted as PO sampling^[1]: 61) is a sampling process where each element of the population is subjected to an independent Bernoulli trial which determines whether the element becomes part of the sample.^{[1]: 85 [2]}

Each element of the population may have a different probability of being included in the sample ( π i {\displaystyle \pi _{i}} ). The probability of being included in a sample during the drawing of a single sample is denoted as the first-order inclusion probability of that element ( p i {\displaystyle p_{i}} ). If all first-order inclusion probabilities are equal, Poisson sampling becomes equivalent to Bernoulli sampling, which can therefore be considered to be a special case of Poisson sampling.

The name conveys that the number of samples leads to a Poisson binomial distribution, which can approximate the Poisson distribution (via Le Cam's theorem).^[3]

Mathematically, the first-order inclusion probability of the ith element of the population is denoted by the symbol π i {\displaystyle \pi _{i}} and the second-order inclusion probability that a pair consisting of the ith and jth element of the population that is sampled is included in a sample during the drawing of a single sample is denoted by π i j {\displaystyle \pi _{ij}} .

The following relation is valid during Poisson sampling when i ≠ j {\displaystyle i\neq j} (i.e., Independence):

π i j = π i × π j . {\displaystyle \pi _{ij}=\pi _{i}\times \pi _{j}.}

π i i {\displaystyle \pi _{ii}} is defined to be π i {\displaystyle \pi _{i}} .

• Bernoulli sampling
• Poisson distribution
• Poisson process
• Sampling design
• Probability-proportional-to-size sampling

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