## Sampling from Inadequate Frames

Suppose a finite population of N units is divisible into a number of groups. If the groups are mutually exclusive and, together, they exhaust the population, the number of units belonging to each group is known and it is also possible to identify at the start of the survey which individual univocally belongs to which group, then one may undertake standard procedures of sample selection and estimation of parameters of interest. For example, one may have stratified sampling if from each group with a known composition a predetermined number t(≥1) of units is sampled. If instead, only some, but not all, the groups are decided to be sampled with preassigned selection probabilities, we have cluster sampling. The groups are called strata in case of stratified sampling where each stratum is represented in the sample with probability 1. The same groups are called clusters in case of cluster sampling when the groups are given positive selection probabilities less than 1. If the selected clusters are not fully surveyed, but only samples of individuals of the selected clusters are surveyed, then we have

two-stage sampling and the clusters are called the first-stage units or primary sampling units (fsu or psu).