Vanishing auxiliary variables in PPS sampling - with applications in microscopy

Recently, non-uniform sampling has been suggested in microscopy to increase eciency. More precisely, sampling proportional to size (PPS) has been introduced where the probability of sampling a unit in the population is proportional to the value of an auxiliary variable. Unfortunately, vanishing auxiliary variables are a common phenomenon in microscopy and, accordingly, part of the population is not accessible, using PPS sampling. We propose a modification of the design, for which an optimal solution can be found, using a model assisted approach. The optimal design has independent interest in sampling theory. We verify robustness of the new approach by numerical results, and we use real data to illustrate the applicability.