Topics on mathematical statistics for medical applications : summary measures and exact simulation of diffusions

The first part of this thesis deals with exact simulation of multidimensional diffusion processes. The main contribution is the development of an exact rejection algorithm for sampling coupled Wright-Fisher diffusions. The algorithm’s output provides a skeleton from the diffusion sampled at a random number of time points. To complete the simulation scheme, an exact simulation strategy for sampling from the corresponding multidimensional Wright-Fisher bridges is also presented. Besides the aforementioned results, which have interest on their own, sampling strategies for coupled Wright-Fisher diffusions are of importance to assess inferential methods that have applications to the estimation of evolutionary param-eters such as selection or mutation of genetic traits over time. In particular, the coupled Wright-Fisher model tracks pairwise allele interactions across different loci over time. This model has applications in population genetics, for instance, to the analysis of interactions of networks of loci such as those encountered in the study of antibiotic resistance. The second part of this thesis presents contributions in statistical methodology for summarizing probability distributions and dealing with commonly found problems in survival analysis settings. First, a novel summary measure for probability distributions is presented, along with a general estimation strategy based on quantile function estimators that allows for inclusion of covariates in a regression framework. Consistency and asymptotic normality results are also provided. This general framework allows for extension of the use of the measure in several scenarios such as life expectancy estimation, where observed variables are often censored. Results concerning the use of the measure in combination with the Cox proportional hazards and the accelerated failure time models are also provided