Considerations and Possible Solutions to the Problem of Estimating the Population Minimum with Applications to Earthquake Data - Saldanha, Matheus Henrique Junqueira (2024)
A myriad of physical, biological and other phenomena are better modeled with semi-infinite distribution families, in which case not knowing the populational minimum becomes a hassle when performing parametric inference. This problem has not been directly discussed in the literature thus far, but it is straightforward to devise a maximum likelihood solution (denoted hereafter as pure MLE), and endpoint estimators proposed in the literature could also be used. Although endpoint estimators are usually evaluated according to their bias and variance, in this project we argue that these are not adequate metrics, so we discuss and use alternatives. We then propose some solutions of our own, some of them aiming to achieve simplicity in terms of their computational cost, and one method (what we call maximum likelihood estimation with parameter-dependent support, or MLEPDS) where we estimate the population minimum indirectly, by maximizing a modified likelihood function L(⋅ ∣ θ) that shifts the sample by a certain amount depending on θ. Experiments demonstrate that the proposed MLEPDS method outperforms both the pure MLE method as well as the approaches that use endpoint estimators proposed in the literature. In particular, our method offers significantly better results in smaller samples, which will surely be of use to many practitioners out there who have to work with limited data. The dissertation is concluded with an application of the proposed MLEPDS method to predict the maximum magnitude of earthquakes. The probability distribution of earthquake magnitudes is subject to a lot of discussion in the literature; Kko (2004) describes a few options, which we modify appropriately for use in the MLEPDS method, with which we estimate maximum magnitudes. The regions of Japan, New Zealand, Balkan peninsula and worldwide are analyzed. Experiments show that our method overall gives higher estimates for the maximum magnitude than two other methods inspired by the literature, and also displays an apparent sensitivity in the year-by-year analysis, indicating that it manages to better capture and understand the underlying changes in seismic activity.