Are we there yet? Advances in anytime-valid methods for hypothesis testing and prediction

Statistical considerations guide the design and implementation of the experiments that scientists perform. For instance, in a clinical trial about a the efficacy of a treatment, its effect in a certain minimum amount of patients has to be observed in order to make confident assertions about the efficacy of the treatment in the general population---observing one or two patients is, as an extreme example, not enough. Sound statistical methods are required to assess the quality of these general assertions. Currently, the most flexible methods allow experimentalists to analyze the data that they gather as it is collected, and to make decisions about gathering more data, stopping an experiment or starting new one based on their findings. In short, they allow experimentalists to ask "are we there yet?" as their experiments are ongoing. This is crucial in applications such as monitoring of clinical trials, online experimentation and quality control in engineering. The statistical methods that make this degree of flexibility possible are called, in the statistical community, anytime valid; they are the main focus of this dissertation. In this work, a number of mathematical results about optimal anytime-valid methods are shown. Group-invariant models, the analysis of time-to-event data, and prediction with expert advice are investigated