Sampling is a core stage in every survey. A sampling design carefully elaborated may imply not only a more accurate estimation of the parameters of interest, but also a reduction in the required sample size in a study. In this thesis we consider two particular but connected subjects. On the one hand, the selection of samples with probabilities proportional to some prescribed values. The first two papers are devoted to this topic. On the other hand, the choice of sampling design to implement in a given survey, which is a topic to which the last two papers are devoted.

Probability proportional to size sampling designs, often referred to as πps designs, are of practical interest due to their potential efficiency. In the literature we can find many of these designs, all having different characteristics. In the first paper we describe and compare ten πps designs with respect to several desired properties. The results suggest that the so called order sampling methods, as well as those proposed by Sunter and Chromy may be considered as good options from a practitioner's point of view.

In the second paper we introduce an algorithm for approximating a target distribution by a mixture distribution. Being a mixture, most of its properties are easy to calculate. We illustrate the use of the algorithm with several examples, both univariate and multivariate. The results indicate that the algorithm succeeds in approximating the target distribution.

The strategy that couples πps designs with the generalized regression estimator is optimal under a given superpopulation model. However, this optimality assumes that the model is correct and some of its parameters are known, which are assumptions that are hardly satisfied in practice. In the third paper we introduce a method that allows for incorporating uncertainty about the model parameters into the choice of the sampling design and then quantifying this uncertainty with a risk measure. The method is illustrated with a real dataset. The results show that the method allowed us to correctly choose the sampling design. The risk measure -as well as other functions that are useful at the planning stage of a survey- is implemented in the package optimStrat developed for R. The fourth paper in this thesis describes the functions in this package.