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The main topic of this thesis is confounding in linear regression models. It arises when a relationship between an observed process, the covariate, and an outcome process, the response, is influenced by an unmeasured process, the confounder, associated with both. Consequently, the estimators for the regression coefficients of the measured covariates might be severely biased, less efficient and characterized by misleading interpretations. Confounding is an issue when the primary target of the work is the estimation of the regression parameters. The central point of the dissertation is the evaluation of the sampling properties of parameter estimators. This work aims to extend the spatial confounding framework to general structured settings and to understand the behaviour of confounding as a function of the data generating process structure parameters in several scenarios focusing on the joint covariate-confounder structure. In line with the spatial statistics literature, our purpose is to quantify the sampling properties of the regression coefficient estimators and, in turn, to identify the most prominent quantities depending on the generative mechanism impacting confounding. Once the sampling properties of the estimator conditionally on the covariate process are derived as ratios of dependent quadratic forms in Gaussian random variables, we provide an analytic expression of the marginal sampling properties of the estimator using Carlson’s R function. Additionally, we propose a representative quantity for the magnitude of confounding as a proxy of the bias, its first-order Laplace approximation. To conclude, we work under several frameworks considering spatial and temporal data with specific assumptions regarding the covariance and cross-covariance functions used to generate the processes involved. This study allows us to claim that the variability of the confounder-covariate interaction and of the covariate plays the most relevant role in determining the principal ma

The main topic of the thesis is the proper execution of a Bayesian inference if log-normality is assumed for data. In fact, it is known that a particular care is required in this context, since the most common prior distributions for the variance in log scale produce posteriors for the log-normal mean which do not have finite moments. Hence, classical summary measures of the posterior such as expectation and variance cannot be computed for these distributions. The thesis is aimed at proposing solutions to carry out Bayesian inference inside a mathematically coherent framework, focusing on the estimation of two quantities: log-normal quantiles (first part of the thesis) and conditioned expectations under a general log-normal linear mixed model (second part of the thesis). Moreover, in the latter section, a further investigation on a unit-level small area models is presented, considering the problem of estimating the well-known log-transformed Battese, Harter and Fuller model in the hierarchical Bayes context. Once the existence conditions for the moments of the target functionals posterior are proved, new strategies to specify prior distributions are suggested. Then, the frequentist properties of the deduced Bayes estimators and credible intervals are evaluated through accurate simulations studies: it resulted that the proposed methodologies improve the Bayesian estimates under naive prior settings and are satisfactorily competitive with the frequentist solutions available in the literature. To conclude, applications of the developed inferential strategies are illustrated on real datasets. The work is completed by the implementation of an R package named BayesLN which allows the users to easily carry out Bayesian inference for log-normal data.

The paper proposes a Bayesian hierarchical model to scale down and adjust deterministic weather model output of temperature and precipitation with meteorological observations, extending the existing literature along different directions. These non-independent data are used jointly into a stochastic calibration model that accounts for the uncertainty in the numerical model. Dependence between temperature and precipitation is introduced through spatial latent processes, at both point and grid cell resolution. Occurrence and accumulation of precipitation are considered through a two-stage spatial model due to the large number of zero measurements and the right-skewness of the distribution of positive rainfall amounts. The model is applied to data coming from the Emilia-Romagna region (Italy).

In this paper, we consider the problem of specifying priors for the variance components in the Bayesian analysis of the Besag-York-Mollié model, a model that is popular among epidemiologists for disease mapping. The model encompasses two sets of random effects: one spatially structured to model spatial autocorrelation and the other spatially unstructured to describe residual heterogeneity. In this model, prior specification for variance components is an important problem because these priors maintain their influence on the posterior distributions of relative risks when mapping rare diseases. We propose using generalised inverse Gaussian priors, a broad class of distributions that includes many distributions commonly used as priors in this context, such as inverse gammas. We discuss the prior parameter choice with the aim of balancing the prior weight of the two sets of random effects on total variation and controlling the amount of shrinkage. The suggested prior specification strategy is compared to popular alternatives using a simulation exercise and applications to real data sets.

Data fusion systems are developed to fill the gap between monitoring networks and CTMs. However, they often do not account for temporal dynamics, leading to potential inaccurate air quality assessment and forecasting. We propose a statistical data assimilation strategy for fusing the CTM output with monitoring data in order to improve air quality assessment and forecasting in the Emilia-Romagna region, Italy. We employ dynamic linear modeling to accommodate dependence across time and obtain air pollution assessment and forecasting for the current and next two days. Finally, air pollution forecast maps are provided at high spatial resolution. We apply our strategy to particulate matter (PM10) concentrations during winter 2013.