Parametric distributions of complex survey data under informative probability sampling

The sample distribution is defined as the distribution of the sample measurements given the selected sample. Under informative sampling, this distribution is different from the corresponding population distribution, although for several examples the two distributions are shown to be in the same family and only differ in some or all the parameters. A general approach of approximating the marginal sample distribution for a given population distribution and first order sample selection probabilities is discussed and illustrated. Theoretical and simulation results indicate that under common sampling methods of selection with unequal probabilities, when the population measurements are independently drawn from some distribution (superpopulation), the sample measurements are asymptotically independent as the population size increases. This asymptotic independence combined with the approximation of the marginal sample distribution permits the use of standard methods such as direct likelihood inference or residual analysis for inference on the population distribution.