Assessing Simplifying Hypotheses in Density Estimation

Abstract

In the classical statistical analysis of univariate random variables, most distribution approaches are focused on phenomena that are symmetrically distributed and concentrated around a single point. However, such models may fail to capture more complex underlying structures, usually present in real data. To solve this issue, several distribution proposals were designed to catch and reveal situations with asymmetry and multimodality. For more complex structures of continuous data, such as circular data, that is, samples that can be represented as points on the circumference of a unit circle, this problem can be also found. However, before applying these more flexible but complicated models, it is important to determine whether it is worth it. In this sense, the goal of this thesis is twofold. First, testing the number of modes for linear and circular data. A review of the different proposals available in the statistical literature is provided and a new method outperforming these previous proposals is presented for both settings, linear and circular. The second objective is determining if the underlying distribution of the data is (reflective) symmetric around a central direction in circular data. Regarding this goal, a new proposal is presented and it is proved that it is optimal for testing circular symmetry. The performance of all the developed tests, in the finite case, is also analysed through simulation studies and illustrated using different real data applications.