Computing inclusion probabilities for order sampling

Rosèn [1997. J. Statist. Plann. Inference 62, 159–191] introduced order sampling schemes of fixed shape which have inclusion probabilities roughly proportional to given size measures (ππps schemes). Three particular cases where the fixed shape distributions are Pareto, exponential and uniform, respectively, are specially treated. In this paper, we give general algorithms for computing the first- and second-order inclusion probabilities for a general fixed shape order sampling scheme and explicit formulae for the three special cases. Identities are given that can be used to check the accuracy of the numerical results. Examples are included as well as some comments on improving the computational efficiency and accuracy of the algorithms.