A large number of probability proportional to size designs have been suggested in the literature. Practitioners will ask themselves if the choice of design matters and, if so, to what extent. We discuss what the desirable properties of a probability proportional to size designs are. We argue the main desirable property is reliability. Other desirable properties include simplicity and, in some contexts, if the sampling design admits sample coordination. We explore the numerical properties of ten probability proportional to size designs in a simulation study. Chromy’s and Sunter’s methods are strong contenders if there is no need of sample coordination. Sunter’s method does not render a strict probability proportional to size design as small units will be drawn with simple random sampling. Should one want to at least keep the possibility of future sample coordination open, order sampling, including sequential Poisson and Pareto sampling, is the best choice.

Nowadays, highly efficient algorithms for selecting samples from the most popular distributions are available. Nevertheless, the problem of sampling from any prescribed distribution is yet to be fully resolved, especially when sampling from multidimensional distributions. A large number of methods to sample from prescribed distributions -either exactly or approximately- are available (e.g. the inverse-transform method, the acceptance-rejection method, Gibbs sampling or Metropolis-Hastings sampling), but there is no "optimal" method for all applications. We introduce an algorithm that allows for approximating a density by a mixture of easy-to-sample distributions. The resulting approximation allows for easily obtaining the desired properties of the prescribed distribution under study, e.g. the mean, variance, cumulative distribution, marginal distributions, sample selection, etc. The only requirement is the density to be approximated. This flexibility makes the algorithm a good alternative compared to well known methods that need additional knowledge before implementation. Diagnostics for measuring convergence are also proposed. The algorithm is illustrated with several examples, both univariate and multivariate. Results show that, given the stopping criteria imposed by the user, the algorithm approximates the desired distribution satisfactorily.

We consider the problem of deciding on sampling strategy, in particular sampling design. We propose a risk measure, whose minimizing value guides the choice. The method makes use of a superpopulation model and takes into account uncertainty about its parameters through a prior distribution. The method is illustrated with a real dataset, yielding satisfactory results. As a baseline, we use the strategy that couples probability proportional-to-size sampling with the difference estimator, as it is known to be optimal when the superpopulation model is fully known. We show that, even under moderate misspecifications of the model, this strategy is not robust and can be outperformed by some alternatives.

At the planning stage of a survey, statisticians have to choose the sampling strategy to implement. If auxiliary information is available, the statistician might, for example, consider using sampling with probabilities proportional to size, or, stratified sampling. With this context in mind, optimStrat, an R package, has been developed. The package assists the choice through two superpopulation models. The first one, called working model, reflects the knowledge or beliefs the statistician has about the relation between the auxiliary variables and the variable of interest. This model, however, might be misspecified. This possibility is reflected on the second model, called true model. In this way, the package allows for determining which sampling strategy is more efficient if a given working model is used when in reality the population is generated by a different model. The package includes an interactive web app that allows users not familiar with R to perform the comparisons for five sampling strategies, namely, STSI--HT, STSI--pos, STSI-reg, πps--pos and πps--reg. Additional functions allow the user to simulate study variables, stratify a given population or calculate the variance of the GREG estimator, among others.