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 (${\displaystyle \pi _{i}}${\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 (${\displaystyle p_{i}}${\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 ${\displaystyle \pi _{i}}${\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 ${\displaystyle \pi _{ij}}${\displaystyle \pi _{ij}}.

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

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

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

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

• Sampling techniques
• Statistics stubs

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