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 ([math]\displaystyle{ \pi_i }[/math]). 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 ([math]\displaystyle{ p_i }[/math]). 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.
A mathematical consequence of Poisson sampling
Mathematically, the first-order inclusion probability of the ith element of the population is denoted by the symbol [math]\displaystyle{ \pi_i }[/math] 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 [math]\displaystyle{ \pi_{ij} }[/math].
The following relation is valid during Poisson sampling when [math]\displaystyle{ i\neq j }[/math]:
- [math]\displaystyle{ \pi_{ij} = \pi_{i} \times \pi_{j}. }[/math]
[math]\displaystyle{ \pi_{ii} }[/math] is defined to be [math]\displaystyle{ \pi_i }[/math].
See also
- Bernoulli sampling
- Poisson distribution
- Poisson process
- Sampling design
References
- ↑ 1.0 1.1 Carl-Erik Sarndal; Bengt Swensson; Jan Wretman (1992). Model Assisted Survey Sampling. ISBN 978-0-387-97528-3.
- ↑ Ghosh, Dhiren, and Andrew Vogt. "Sampling methods related to Bernoulli and Poisson Sampling." Proceedings of the Joint Statistical Meetings. American Statistical Association Alexandria, VA, 2002. (pdf)
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