Probability-proportional-to-size sampling

In survey methodology, probability-proportional-to-size (pps) sampling is a sampling process where each element of the population (of size N) has some (independent) chance $p_{i}$ to be selected to the sample when performing one draw. This $p_{i}$ is proportional to some known quantity $x_{i}$ so that $p_{i}={\frac {x_{i}}{\sum _{i=1}^{N}x_{i}}}$.

One of the cases this occurs in, as developed by Hanson and Hurwitz in 1943, is when we have several clusters of units, each with a different (known upfront) number of units, then each cluster can be selected with a probability that is proportional to the number of units inside it. So, for example, if we have 3 clusters with 10, 20 and 30 units each, then the chance of selecting the first cluster will be 1/6, the second would be 1/3, and the third cluster will be 1/2.

The pps sampling results in a fixed sample size n (as opposed to Poisson sampling which is similar but results in a random sample size with expectancy of n). When selecting items with replacement the selection procedure is to just draw one item at a time (like getting n draws from a multinomial distribution with N elements, each with their own $p_{i}$ selection probability). If doing a without-replacement sampling, the schema can become more complex.

Distribution and properties

If observations from some distribution F are sampled in a way that is proportional to their value, then the distribution of the values in that sample follows a Length-biased distribution, with the following density function:

$g(x)=xf(x)/E[x]$

Also: $E[g(x)]=E[x^{2}]/E[x]$

See also

• Poisson sampling
• Bernoulli sampling
• Poisson distribution
• Poisson process
• Sampling design