A presente dissertação é dedicada à estimação não paramétrica de uma função de probabilidade desconhecida, por intermédio do método do núcleo. Começamos por definir o estimador e exibir alguns exemplos típicos de núcleos, nomeadamente os núcleos de Dirac, de Aitchison e Aitken, de Wang e Van Ryzin, a versão simétrica do núcleo triangular clássico, bem como uma proposta mais recente baseada na distribuição de Conway-Maxwell-Poisson. Segue-se a apresentação das propriedades básicas de convergência local dos estimadores não normalizado e normalizado. No que diz respeito ao primeiro, são analisados os comportamentos assintóticos do viés, da variância e do erro quadrático médio, sendo depois estabelecida a convergência quase-certa e a normalidade assintótica do estimador. Relativamente ao estimador normalizado, a dedução dos resultados de convergência em probabilidade e quase-certa passa por analisar o comportamento assintótico da sucessão das constantes de normalização, cujo estudo é feito previamente. Posteriormente, apresentamos as propriedades globais de ambos os estimadores, com ênfase para a norma $\ell_1$. No caso do estimador normalizado, a convergência quase-certa no sentido desta norma permitirá obter um resultado do tipo Glivenko-Cantelli para a correspondente função de repartição. Ainda no âmbito das propriedades globais, é obtido o desenvolvimento assintótico do erro quadrático médio integrado para o estimador não normalizado usando a expansão de Newton, o que constitui uma abordagem original, tanto quanto sabemos. Por fim, comparamos os desempenhos dos estimadores empírico e de Aitchison e Aitken, a distância finita, através de um breve estudo de simulação.

The variogram provides an important method for measuring the dependence of attribute values between spatial locations. Suppose that the nature of the sampling process leads to the presence of clustered data; it would be advisable to use a variogram estimator that aims to adjust for clustering of samples. In this setting, the use of a non-parametric weighted estimator, obtained by considering an inverse weight to a given neighbourhood density combined with the kernel method, seems to have a satisfactory behaviour in practice. This paper pursues a theoretical study of the cluster robust estimator, by proving that it is asymptotically unbiased as well as consistent and by providing criteria for selection of the bandwidth parameter and the neighbourhood radius. Numerical studies are also included to illustrate the performance of the considered estimator and the suggested approaches.

In order to understand the burning patterns in the territory of Paragominas, Pará, in view of the strong relationship with the agricultural activities developed in the region and the negative implications that the use of fire provides to my environment, this work aims to spatialize and analyze the behavior of hot spots in the municipality, in the years 2000, 2015 and 2018. The areas affected by fires in the municipality were mapped for visual interpretation on standardized images. Qualitative and quantitative analyzes were used for the mapped data, with the aim of detailing the data regarding the panorama of the evolution of the fires. The geospatial data available for free by the Database of the Program for the Prevention and Control of Burnings and Forest Fires in the Legal Amazon (PROARCO) was handled. The Kernel method was used for the spatial distribution of the hot spots. The total number of outbreaks in the period between 2000, 2015 and 2018 was 34,323. Between the years 2015 and 2018, the municipality showed a decrease in the rate of occurrence of hot spots with 26,728 (77.87%) and 6,561 (19.11%), respectively. The year 2015 had the highest rate of fires, even after State intervention against deforestation. Therefore, it was observed that the incidence of hot spots in the municipality of Paragominas, is directly linked to agricultural activities practiced by farmers who live in the region, since the location is a reference in agricultural production in the Amazon.

The environmental contamination risk can be evaluated in a specific area by approximating the probability that the pollutant under study exceeds a critical value. This issue requires the estimation of the distribution function involved, which can be addressed by applying the indicator kriging methodology or by approximating the sill of the variogram of the underlying indicator process. These approaches demand an appropriate characterization of the indicator variogram, which in turn requires a previous specification of the trend function, if the latter is suspected to be non-constant. Since accuracy of the results will be strongly dependent on the adequate approximation of both functions, we suggest proceeding in a different way to avoid these requirements. Thus, in the current paper, two kerneltype estimators are proposed, based on first approximating the distribution at the sampled sites and then obtaining a weighted average of the resulting values, to derive a valid estimator at each (sampled or unsampled) location. Consistency of the kernel approaches is proved under rather general conditions, such as local stationarity and the existence of derivatives up to the second order of the distribution function. Numerical studies have been carried out to illustrate the performance of our proposals when compared to those procedures requiring the approximation of the indicator variogram. In a final step, the kernel-type estimation of the distribution function has been applied to map the risk of contamination by arsenic in the Central Region of Portugal. With this aim, biomonitoring data of arsenic concentrations were used to detect those zones with higher risk of arsenic accumulation, which is mainly located on the northern part of the region.

In this work, the nonparametric kernel prediction will be considered for stochastic processes, when a random design is assumed for the spatial locations. We will check that, under rather general conditions, the mean-squared prediction error tends to be negligible, as the sample size increases. However, the use of the optimal bandwidth demands the estimation of unknown quantities, whose approximation in an accurate way often turns out to be difficult in practice. Hence, alternative cross-validation approaches will be provided for the selection of both local and global bandwidths. Numerical studies were carried out in order to analyse the performance of the nonparametric predictor for both simulated and real data.