Statistical assessment of extreme values of boron and arsenic in water of Arica and Parinacota / Ludy Mireya Núñez Soza.

Doctor en Estadística.

The prediction of events that by their magnitude do not happen frequently and that have an impact on the environment is of great interest in various elds. Many researchers have focused on calculating with a high level of certainty the possible occurrence of these events, which at present have happened a little more often than expected. In many data analysis studies, \extreme events" are considered as outliers and therefore are ignored. Within the Theory of Probability and Statistics these unusual or rare events are known as extreme values, which, when plotted on a histogram are located in the tails of the distributions. At present it has been found that for a large number of data sets that these situations are heavier than classical distributions predict. Therefore, if we want to better model these heavy tails data series, we must have an appropriate mathematical method that explains the distribution with which such events occur. The answer to this problem is the Extreme Value Theory. The extreme values theory is responsible for modeling the behavior of the maximum and/or minimum values of a data set, trying to nd the shape of the limit distribution by which these values can be approximated. The limit distributions to which these values can converge are known as extreme values distributions. This asymptotic theory is similar to the Central Limit Theorem, because the extreme value theory focuses on the behavior of a sample of extremes. According to Frechet (1927), Nicolas Bernoulli was the rst to introduce in 1709 the discussion of larger mean distances from the origin. Later in 1927, the French mathematician Maurice Frechet obtained the rst asymptotic distribution for maximum order statistics. Tippet and Fisher in 1928 established the theorem that perhaps is the most important in the extreme values theory. In 1936 Von Mises presented some simple and useful sucient conditions for the convergence in distribution of the larger order statistic. The work done by these authors was the basis of the univariate theory of extreme values. Later in 1943 Boris Gnedenko was in charge of classifying the asymptotic distributions of extreme values and of giving necessary and sucient conditions under which such asymptotic distributions are valid. Subsequently De Haan's doctoral work in 1970 contributed to the development of this theory and the results of Pickans in 1975 were the rst contribution to the multivariate extreme value theory. At present advances in the extreme values theory focus ii mainly on the development of models and methods for extreme values of spatial phenomena and other more complex structures...