Computational aspects of order πpsπps sampling schemes

In an order sampling a finite population of size N has its units ordered by a ranking variable and then, a sample of the first n   units is drawn. For order πpsπps sampling, the target inclusion probabilities λ=(λk)k=1N are computed using a measure of size which is correlated with a variable of interest. The quantities λkλk, however, are different from the true inclusion probabilities πkπk. Firstly, a new, simple method to compute πkπk from λkλk is presented, and it is used to compute the inclusion probabilities of order πpsπps sampling schemes (uniform, exponential and Pareto). Secondly, given two positively co-ordinated samples drawn with order πpsπps sampling, the joint inclusion probability of a unit in both samples is approximated. This approximation can be used to derive the expected overlap or to construct an estimate of the covariance on these two samples. All presented methods use numerical integration.