Statistical inference in sampling theory (Q2740094)

The authors deal with the stochastic representation of survey data and its applications. Let us have a population \(U=\{1,2,\ldots,N\}\) and let a random vector \({\mathbf I}=(I_1,\ldots,I_{N})\) describe the sampling process in \(U\), so its outcome \(k=(k_1,\ldots,k_{N})\) identifies a sample by \(k_{i}=0\), meaning that the unit \(i\) is not sampled, and \(k_{i}>0\) meaning that the unit \(i\) is sampled \(k_{i}\) times. The pair of random vectors \(({\mathbf Y_{s},I})\) is called a stochastic representation of survey data, where vector the \({\mathbf Y_{s}}\) displays the random variables \(Y_{i}\) selected by the design vector \({\mathbf I}\).NEWLINENEWLINENEWLINEThis representation includes the design-based, model-based and model-design-based cases. The randomization process may be either with or without replacement. A general variance formula for the estimator of the sum of population means is derived. The two-phase sampling design is defined in terms of multivariate distributions. In the examples, different sampling designs for the phases are combined with each other and the probability law of the resulting selection mechanism is expressed.