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The Gaussian distribution is the most fundamental distribution in statistics. However, many applications of Gaussian random fields (GRFs) are limited by the computational complexity associated to the evaluation of probability density functions. Particularly, large datasets with N irregularly sited spatial (or spatio-temporal) locations are difficult to handle for several applications of GRF such as maximum likelihood estimation (MLE) and kriging prediction. This is due to the fact that computation of the inverse of the dense covariance function requires a computational complexity of O(N^3) floating points operations in spatial or spatio-temporal context. For relatively large N the exact computation becomes unfeasible and alternative methods are necessary. Several approaches have been proposed to tackle this problem. Most assume a specific form for the spatial(-temporal) covariance function and use different methods to approximate the resulting covariance matrix. We aim at approximating covariance functions in a format that facilitates the computation of MLE and kriging prediction with very large spatial and spatio-temporal datasets. For a sufficiently general class of spatial and specific class of spatio-temporal covariance functions, a methodology is developed using a hierarchical matrix approach. Since this method was originally created for the approximation of dense matrices coming from partial differential and integral equations, a theoretical framework is formulated in terms of Stochastic Partial Differential equations (SPDEs). The application of this technique is detailed for covariance functions of GRFs obtained as solutions to SPDEs. The approximation of the covariance matrix in such a low-rank format allows for computation of the matrix-vector products and matrix factorisations in a log-linear computational cost followed by an efficient MLE and kriging prediction. The numerical studies are provided for based on spatial and spatio-temporal datasets and the H-matrix approach is compared with the other methods in terms of computational and statistical efficiency.​

Efficient classification techniques are nowadays in huge demand in observational astronomy, which is probably the oldest area of application of statistical science to the study of nature. The rapid evolution through technology has determined a paradigm shift where rich and/or massive data sets are becoming the dominant source of information. A typical sky survey collects terabytes of data each night on millions of possible objects (stars, galaxies, etc.) which themselves may encompass many observed properties. Some success stories are the Sloan Digital Sky Survey whose whelm of very high quality photometric data allowed astronomers to pinpoint the position of over one million of galaxies, and NASA's Fermi space telescope, thanks to which we are still deepening our knowledge of high-energy phenomena. Discoveries in these fields are of utmost relevance as they contain a wealth of information about the history of our Galaxy, and impact on the understanding of our own Solar System. However, the simple collection of these data sets is just the beginning of the process. A key step in astronomical breakthrough research is the meaningful analysis of the collected information. The complex structure of the data, combined with the availability of a tremendous number of observations, represents a non-negligible statistical challenge from both the theoretical and computational viewpoint. Statisticians must work at the forefront of this area. This thesis aims to enlarge the set of clustering and classification techniques available for astronomy, providing new flexible solutions to cope with the requirements imposed by the vast diversity of celestial objects. In the first part, we propose an innovative approach based on Bayesian nonparametric methods to the signal extraction of high-energy astronomical sources immersed in a strong background contamination. Our model simultaneously clusters the photons and gives an estimate of the number of sources using a Dirichlet Process (DP) mixture, while separating them from the irregularly shaped background, that is reconstructed using a novel Bayesian nonparametric technique based on B-spline functions. The resultant is then a hierarchical model in the class of mixtures of DP mixtures. We provide a suitable Markov Chain Monte Carlo algorithm to conduct the inference, and a post-processing procedure to quantify the information coming from the discovered clusters. We finally test the capacity of the model in locating and extracting the signal of the sources using several artificial datasets, and a further application on the Fermi LAT map is proposed. In the second part, we propose a novel statistical approach to separate the different levels of brightness in the emission activity of high-energy sources. The method analyses the variation of flux in time and clusters the observations to infer on the latent states of variability that correspond to distinct physical mechanisms of the source. We model the transition among the latent states with a continuous-time Markov chain, and the flux measurements in each state with an Ornstein-Uhlenbeck (OU) process. The resultant technique belongs to the class of continuous-time hidden Markov models (HMMs) and can be fitted via maximum likelihood estimation using an efficient EM algorithm. In addition, we assess the properties of the model with a proper bootstrap algorithm. We finally illustrate the efficiency of the method on a light curve from a blazar discovered by the Fermi LAT.

In recent times, the rise of `big data' has brought along major computational challenges in all the main disciplines of scientific research, including the field of spatial statistics. Some of these challenges include parametric estimation and quantification of estimation uncertainty that, when building statistical models using big data, pose an important computational load. Many methods have been proposed to address these challenges such as dimension reduction, approximation by Markov random fields, tapering of the covariance matrix, and subsampling based approaches. In this thesis a new \textit{divide-and-conquer} approach is proposed that we call \texttt{farmer} for providing effect size and standard error estimates in spatial models of big data. According to the proposed approach, all observations are divided into blocks that are mutually exclusive according to their position. For each block, the model parameters are estimated and recombined using a fixed or random meta-model to take into account the (possible) spatial dependence. This generalized method can be applied to a wide range of spatial models. For example, consider a linear Gaussian spatial model. In a simulation study, the \texttt{farmer} estimators were compared with estimators based on methods with similar sampling ideas. In the context of the Gaussian model, two applications with real data are presented. The proposed method appears computationally efficient compared to equivalent methods and has lower bias in the estimates. Furthermore, the proposed approach provides a more realistic estimate of standard errors. Finally, we propose an application of the method to generalized linear spatial models for simulated and real counting data.

Thanks to the advances in bioinformatics and high-throughput methodologies of the last decades, a large unprecedented amount of biological data coming from various experiments in metabolomics, genomics and proteomics is available. This has lead the researchers to conduct more and more comprehensive molecular proling of biological samples through different multiple aspects of genomic activities, thus introducing new challenges in the developments of statistical tools to integrate and model multi-omics data. The main research objective of this thesis is to develop a statistical framework for modelling the interactions between genes when their activity is measured on different domains; to do so, our approach relies on the concept of multilayer network, and how structures of this type can be combined with graphical models for mixed data, i.e., data comprising variables of different nature (e.g., continuous, categorical, skewed, to name a few). We further develop an algorithm for learning the structure of the undirected multilayer networks underlying the proposed models, showing its promising results through empirical analyses on cancer data, which was downloaded from the public TCGA consortium.

Ordered categorical data are frequently encountered in many fields of research, such as, sociology, psychology, quality control, medical studies, and so forth. Especially in medical research, it is inevitable to meet a lot of problems containing ordered categorical data. Our specific interest is to find convincing solutions to some of the testing problems which include restrictions in the set of alternatives, such as testing for stochastic dominance and testing for monotonic stochastic ordering while using such a kind of data. When the number of nuisance parameters of underlying distributions or that of observed variables is small, there are some likelihood-based solutions. Our interest, however, is for cases where such numbers are not small. In these cases likelihood-based methods do not work, thus our interest is to proceed nonparametrically within permutation methods. Permutation methods are conditional on the pooled set of observed data which, in turn, are typically a set of sufficient statistics under the null hypothesis for the underlying distribution. Moreover, due to the evident complexity of such problems, according to Roy (1953), we also must use their Union-Intersection representation consisting on an equivalent break-down of the hypothesis under testing into a set of simpler sub-hypotheses for each of which a permutation test is available and such tests are jointly considered. So we must stay within the nonparametric combination of several dependent permutation tests. In the thesis, guided by two medical examples from the literature, we propose suitable solutions that are proved to be admissible combinations of optimal conditional partial tests and so enjoying good asymptotic properties.

Complex dependence structures characterising modern data are routinely encountered in a large variety of research fields. Medicine, biology, psychology and social sciences are enriched by intricate architectures such as networks, tensors and more generally high-dimensional dependent data. Rich dependence structures stimulate challenging research questions and open wide methodological avenues in different areas of statistical research, providing an exciting atmosphere to develop innovative tools. A primary interest in statistical modelling of complex data is on adequately extracting information to conduct meaningful inference, providing reliable results in terms of uncertainty quantification and generalisability into future samples. These aims require ad-hoc statistical methodologies to appropriately characterize the dependence structures defining complex data as such, further improving the understanding of the mechanisms underlying the observed configurations. The focus of the thesis is on Bayesian modelling of complex dependence structures via latent variable constructs. This strategy characterises the dependence structure in an unobservable latent space, specifying the observed quantities as conditionally independent given a set of latent attributes, facilitating tractable posterior inference and an eloquent interpretation. The thesis is organized into three main parts, illustrating case studies from different fields of application and focused on studying modern challenges in neuroscience, psychology and criminal justice. Bayesian modelling of the complex data arising in these domains via latent features effectively provides valuable insights on different aspects of such structures, addressing the questions of interest and contributing to the scientific understanding.

The density-based formulation aims at recasting the clustering problem to a mathematically sound framework, by linking the groups to some features of the density assumed to underlie the data. Even if early probabilistic approaches to cluster analysis can be traced back to fifty years ago, the topic has recently found a renewed and vibrant interest in the scientific community. This may be motivated both by the computational advancements witnessed in the last years and by more conceptual and substantial reasons. The increased availability of mixed type and complex structured data has indeed required a rigorous formalization of the clustering problem. Stemming from the same roots, density-based clustering has been developed following two distinct paths. In its parametric formulation, a connection among groups and unimodal components of a mixture model is drawn. On the other hand, according to the nonparametric paradigm, clusters are seen as the domains of attraction of the modes of the density. Revolving around this approach to clustering, the thesis explores both the formulations by highlighting at the same time contact points and dissimilarities. Moreover differently structured data are considered ranging from the unidimensional setting to more complex three-way structure. Three main contributions can be highlighted. In the first one we derive some asymptotic results to address nonparametric density estimation from a clustering-oriented perspective. In the second contribution we propose an ensemble approach for density-based clustering, which inherits the strengths from both the parametric and the nonparametric formulation, and possibly enhances the robustness and the stability of the partitions. The third contribution addresses the problem of clustering complex multivariate time-dependent data by adopting a parametric approach and proposing a flexible modification of the Latent Block Model.

Bayesian demography developments, global trends for substituting traditional censuses with cheaper methods able to use available information, and new technologies require investigating and providing new models to answer new requirements. In Italy in particular, during the last years Istat worked for launching in October 2018 the ``permanent census of population and housing''. After a first discussion on censuses, changes recommended by organisations such as the UN and the European Union to the National Statistical Institutes, and on new demographic models for population size estimation, the model proposed by Bryant and Graham (2013) is analysed. The model allows for integration of different data sources, for demographic series estimation, and it is very flexible and complex at the same time. Applications of this model to the Italian population are performed, highlighting its advantages and limits. Data for the period considered (2006-2015) and metadata come from Istat. Data are not always consistent, confirming the need of statistical methods able to integrate sources and reconstruct demographic series. As expected, census data and migration flows estimation caused most of the problems. The method still needs further experimentations, therefore applications aim to compare results when varying initial assumptions and to identify their pros and cons rather than provide actual results on the Italian population. Eventually a model extension, along with the first results of its application, is proposed using the Conway-Maxwell Poisson distribution Conway and Maxwell (1962), a flexible two parameters version of the Poisson distribution.

The diffusion of an innovation in a social system is characterised as a dynamic that typically gives rise to a nite lifecycle. The pioneering approach by Bass (1969), called the Bass model (BM), has been widely used because it considers the internal rules of the social system. The model has been extended in several directions. The generalised Bass model (GBM), by Bass et al. (1994), is the most popular extension describing the exogenous interventions acting on the timing of the diusion process. Another important extension is the Guseo and Guidolin (2009) model GGM), which defines the dynamic nature of the latent market potential. However, these univariate time series models do not consider the competitive environment, in which two or more concurrent innovations enter the market, possibly at dierent times, and target the main set of potential adopters or subgroups of potential adopters with possible interaction effects. The model proposed by Guseo and Mortarino (2014) allows a exible behaviour in describing the diffusion of two products under competition, where interaction effects, namely word-of-mouth (WOM), are split into within-product and cross-product effects. This thesis proposes an extended diffusion model for three products in a competitive framework that may be able to improve the description of mutual interactions (either competition or cooperation), along with dening specific features of each product. To examine the improvement of the proposed three-competitor model (3CM) compared with the bivariate model (2CM), both models are applied to energy data to describe competition among energy sources. The intervention functions (e.g. the external shocks) are incorporated in the models when necessary to estimate the effect of incentives and policy measures, which are crucial tools in the expansion of renewables and nonrenewable energy technologies. Without incorporating intervention functions, models based on the assumption of a constant market potential are virtually unable to capture the wide variety of shapes of products in a diusion process. Following the model proposed by Guseo and Mortarino (2015), this thesis further proposes a diusion model for three competing products that are suffciently similar to share a common market potential, where the size increases over time. Models 2CM and 3CM with dynamic market potential (DMP) are also applied to the same energy data studied in the preceding part of this work. The obtained results highlight the ecacy of the models with DMP over similar models with xed market potential. Overall, the proposed models (3CM), either with a xed market potential, m, or DMP, prove to be useful in applied contexts, for example, in describing the lifecycle of products and evaluating predictions.

Test equating is a statistical procedure to ensure that scores from different test forms are comparable and can be used interchangeably. There are several methodologies available to perform equating, some of which are based on the Classical Test Theory (CTT) framework and others are based on the Item Response Theory (IRT) framework. After a short overview of latent trait models and test equating on Chapter 1, Chapter 2 of this thesis proposes a procedure to compare equating transformations originated from different frameworks. As example, we have compared Item Response Theory Observed-Score Equating (IRTOSE), Kernel Equating (KE) and IRT Kernel Equating (IRTKE) under different scenarios. Our results suggest that IRT methods tend to provide better results than KE even when the data are not generated from IRT processes. KE can provide satisfactory results if a proper pre-smoothing solution can be found, while also being much faster than IRT methods. For daily applications, we recommend observing the sensibility of the results to the equating method, minding the importance of good model fit and meeting the assumptions of the framework. Within the IRT framework, if the statistical modeling of the scores from each test form is performed independently, their respective parameters will be on different scales and thus incomparable. Equating solves this problem by transforming item parameters so they are all on the same scale. Popular IRT methods for equating pairs of test forms include the mean-sigma, mean-mean, Stocking–Lord and Haebara. For multiple forms, it might be necessary to employ more elaborate methods which take into account all the relationships between the forms. Chapter 3 addresses this issue, as we propose a new statistical methodology that simultaneously equates a large number of test forms. Our proposal differentiates itself from the current state of the art by using the likelihood function of the true item parameters and the equating coefficients to perform the simultaneous estimation of all equating coefficients and by taking into account the heteroskedasticity of the item parameter estimates as well as the correlations between those estimates on each test form. Such innovations give this new method the potential to yield equating coefficient estimates which are more efficient than what is currently available in the literature, albeit at a computational cost due to its increased complexity. This is indeed what has been observed in some of the simulations performed. Greater estimation efficiency is especially important in situations involving item parameters with extreme values.

Multivariate categorical data are routinely collected in several applications, including epidemiology, biology, and sociology, among many others. Popular models dealing with these variables include log-linear and tensor factorization models, with these lasts having the advantage of flexibly characterizing the dependence structure underlying the data. Under such framework, this Thesis aims to provide novel approaches to define compact representations of the dependence structures and to introduce new inference possibilities in tensor factorization approaches. We introduce a new class of GROuped Tensor (GROT) factorizations, which have superior performance in terms of data compression if compared to standard Parafac approach, using relatively few components to represent the joint probability mass function of the data. While popular Parafac factorizations rely on mixing together independent components, GROT mixes together grouped factorizations, equivalent to replacing vector arms in Parafac with low-dimensional tensor arms. We consider a Bayesian approach to inference with Dirichlet priors on the mixing weights and arm components, to obtain a combined low-rank and sparse structure, while facilitating efficient posterior computation via Markov chain Monte Carlo. Motivated by an application on malaria risk assessment, we also introduce a novel multivariate generalization of mixed membership models, which allows identification of correlated profiles related to different domains corresponding to separate groups of variables. We consider as a case study the Machadinho settlement project in Brazil, with the aim of defining survey based environmental and behavioral risk profiles and studying their interaction and evolution. To achieve this goal, we show that the use of correlated multiple membership vectors leads to interpretable inference requiring a lower number of profiles compared to standard formulations while inducing a more compact representation of the population level model. We propose a novel multivariate logistic normal distribution for the membership vectors, which allows easy introduction of auxiliary information in the membership profiles leveraging a multivariate latent logistic regression. A Bayesian approach to inference, relying on Pólya gamma data augmentation, facilitates efficient posterior computation via Markov chain Monte Carlo. The proposed approach is shown to outperform the classical mixed membership model in simulations, and the malaria diffusion application.

Variational approximations are approximate inference techniques for complex statisticalmodels providing fast, deterministic alternatives to conventional methods that,however accurate, take much longer to run. We extend recent work concerning variationalapproximations developing and assessing some variational tools for likelihoodbased and Bayesian inference. In particular, the first part of this thesis employs a Gaussian variational approximation strategy to handle frequentist generalized linear mixedmodels with general design random effects matrices such as those including spline basisfunctions. This method involves approximation to the distributions of random effectsvectors, conditional on the responses, via a Gaussian density. The second thread isconcerned with a particular class of variational approximations, known as mean fieldvariational Bayes, which is based upon a nonparametric product density restriction on the approximating density. Algorithms for inference and fitting for models with elaborateresponses and structures are developed adopting the variational message passingperspective. The modularity of variational message passing is such that extensions tomodels with more involved likelihood structures and scalability to big datasets are relatively simple. We also derive algorithms for models containing higher level randomeffects and non-normal responses, which are streamlined in support of computationalefficiency. Numerical studies and illustrations are provided, including comparisons witha Markov chain Monte Carlo benchmark.

There is a very rich literature proposing Bayesian approaches for clustering starting with a prior probability distribution on partitions. Most approaches assume exchangeability, leading to simple representations of such prior in terms of an Exchangeable Partition Probability Function (EPPF). Gibbs-type priors encompass a broad class of such cases, including Dirichlet and Pitman-Yor processes. Even though there have been some proposals to relax the exchangeability assumption, allowing covariate-dependence and partial exchangeability, limited consideration has been given on how to include concrete prior knowledge on the partition. Our motivation is drawn from an epidemiological application, in which we wish to cluster birth defects into groups and we have a prior knowledge of an initial clustering provided by experts. The underlying assumption is that birth defects in the same group may have similar coefficients in logistic regression analysis relating different exposures to risk of developing the defect. As a general approach for including such prior knowledge, we propose a Centered Partition (CP) process that modifies a base EPPF to favor partitions in a convenient distance neighborhood of the initial clustering. This thesis focus on providing characterization of such new class, along with properties and general algorithms for posterior computation. We illustrate the methodology through simulation examples and an application to the motivating epidemiology study of birth defects.

Our work has developed from a real epidemiological problem. The carcinogenic effect of cigarette smoking on head and neck cancer has been extensively studied in the literature highlighting a non linear dose-response relationship. Recently, the use of linear regression splines, within semiparametric models framework, has allowed an improvement in the evaluation of the association between smoking habits and head and neck cancer. Our work focuses on the development of a methodology able to improve several aspects of the estimation of the aforementioned relationship. In particular, the approximation of the spline function, represented by truncated linear basis, has been refined by addressing the problem of estimating two key quantities for the definition of a spline function: the number and position of the knots. The proposed methodology uses a Bayesian approach. We then focused on developing a streamlined methodology applicable to generalised linear models for cross-classified data. In particular, the steps necessary to calculate the covariance matrix are optimised with respect to one of the two random effects, allowing a computational gain both in terms of time and memory usage. The proposed algorithms are applied in the context of the inferential variational methods in detail to the mean field variational Bayes.

New advances in global positioning systems (GPS) and geographical information systems (GIS) translated in an explosion of spatial data collection. In this thesis we tackle two well known problems of spatial data. The first one regards the quality of the geographical coordinates used as input for many spatial models. This aspect is often neglected and we provide a general framework to deal with the uncertainty present in the spatial locations. The second problem takes the name of change of support and is related to the analysis of spatial data at a scale of aggregation that is different than the one at which they are observed. Also in this case we develop a theoretical framework to figure out the inference problems inherent to this instance. For both problems the results obtained are really promising.

Functional magnetic resonance images (fMRI) are brain scan images by MRI machine which are taken functionally cross the time. Several studies have investigated methods analyzing such images (or actually the drawn data from them) and is interestingly growing up. For examples models can predict the behaviours and actions of people based on their brain pattern, which can be useful in many fields. We do the classification study and prediction of fMRI data and we develop some approaches and some modifications on them which have not been used in such classification problems. The proposed approaches were assessed by comparing the classification error rates in a real fMRI data study. In addition, many programming codes for reading from fMRI scans and codes for using classification approaches are provided to manipulate fMRI data in practice. The codes, can be gathered later as a package in R. Also, there is a steadily growing interest in analyzing functional data which can often exploit Riemannian geometry. As a prototypical example of these kind of data, we will consider the functional data rising from an electroencephalography (EEG) signal in Brain-Computer interface (BCI) which translates the brain signals to the commands in the machine. It can be used for people with physical inability and movement problems or even in video games, which has had increased interest. To do that, a classification study on EEG signals has been proposed, while, the data in hand to be classified are matrices. A multiplicative algorithm (MPM), which is a fast and efficient algorithm, was developed to compute the power means for matrices which is the crucial step in our proposed approaches for classification. In addition, some simulation studies were used to examine the performance of MPM against existing algorithms. We will compare the behavior of different power means in terms of accuracy in our classifications, which had not been discovered previously. We will show that it is hard to have a guess to find the optimal power mean to have higher accuracy depending on the multivariate distribution of data available. Then, we also develop an approach, combination of power means, to have the benefit of all to improve the classification performance. All the codes related to the fast MPM algorithms and the codes for manipulating EEG signals in classification are written in MATLAB and can be developed later as a package.

Biological processes underlying the basic functions of a cell involve complex interactions between genes. From a technical point of view, these interactions can be represented through a graph where genes and their connections are, respectively, nodes and edges. The main research objective of this thesis is to develop a statistical framework for modelling the interactions between genes when the activity of genes is measured on a discrete scale. We propose several algorithms. First, we define an algorithm for learning the structure of a undirected graph, proving its theoretical consistence in the limit of infinite observations. Next, we tackle structure learning of directed acyclic graphs (DAGs), adopting a model specification proved to guarantee identifiability of the models. Then, we develop new algorithms for both guided and unguided structure learning of DAGs. All proposed algorithms show promising results when applied to simulated data as well as to real data.

In the recent years the challenge for new prior specifications and for complex hierarchical models became even more relevant in Bayesian inference. The advent of the Markov Chain Monte Carlo techniques, along with new probabilistic programming languages and new algorithms, extended the boundaries of the field, both in theoretical and applied directions. In the present thesis, we address theoretical and applied tasks. In the first part we propose a new class of prior distributions which might depend on the data and specified as a mixture between a noninformative and an informative prior. The generic prior belonging to this class provides less information than an informative prior and is more likely to not dominate the inference when the data size is small or moderate. Such a distribution is well suited for robustness tasks, especially in case of informative prior misspecification. Simulation studies within the conjugate models show that this proposal may be convenient for reducing the mean squared errors and improving the frequentist coverage. Furthermore, under mild conditions this class of distributions yields some other nice theoretical properties. In the second part of the thesis we use hierarchical Bayesian models for predicting some soccer quantities and we extend the usual match goals’ modeling strategy by including the bookmakers’ information directly in the model. Posterior predictive checks on in-sample and out-of sample data show an excellent model fit, a good model calibration and, ultimately, the possibility for building efficient betting strategies.

UN has generated projections for demographic components for all countries of the world deterministically before revising the methodology and recently started to generate probabilistic projections at national level , but regional/sub national level trajectories are yet to be generated. In this thesis, probabilistic projections for total fertility rate (TFR) , life expectancy at birth for males and females and population totals at national as well as regional level have been generated based on Bayesian Hierarchical modeling approach. The trajectories were also generated for variable number of countries and it was observed that continent based trajectories were enough to model the future pattern instead of going for entire globe data. This in result saves time and decreases the influence of developed world demographic pattern on developing world and vise versa. TFR results were compared based on different values of $\mu$, (i.e., values less than and more than 2.1, which is the ultimate level of replacement) and with the trajectories generated based on Bayesian Hierarchical modeling approach for phase III. For values greater than 2.0, the trajectories were taking long time to converge to replacement level than for smaller values. On the other hand trajectories of Life Expectancy have revealed significant gap between male and female projections at national as well as regional level. It was also observed that Balochistan region has lower life expectancy at birth than the rest of the regions of Pakistan. The trajectories for Population totals were generated based on the probabilistic TFR and life expectancy trajectories. The results revealed a total population of over 207 million at national level which nearly coincides with the recently released provisional figures of census. Furthermore, the results are significantly different from previously adopted deterministic results and could be a good substitute over classical deterministic approaches.

Dependent data arise frequently in applied research. When quantile regression is the statistical method of choice, several approaches have been proposed that can accommodate dependence among observations. Cluster bootstrap is one of the most popular among them. While practical, this method is generally inefficient and computationally demanding, especially when the number of clusters is large. When the primary interest is on marginal quantiles, estimating equations have been proposed that model the association between the sign of the regression residuals with the Pearson's correlation coefficient. The latter, however, is an inadequate measure of dependence between binary variables because of its range depends on their marginal probabilities. Instead, we propose to model a working association matrix through odds ratios, which are popular measures of association of binary outcomes. Different working structures can be easily estimated by suitable logistic regression models. These structures can be parameterized and may depend on covariates and clusters. Simulations demonstrated that the efficiency of the estimator increases as the working correlation structure approaches the true one. We extend the proposed method to penalized estimating equations, they have increasingly been used to reduce model complexity in several applications. We focus on penalized smoothly clipped absolute deviation models for feature selection and reduced-rank penalized smoothing splines. Simulations showed that the proposed methods potentially improve the performance of the marginal quantile regression estimator. When the correlation structure is correctly specified the estimator's efficiency increases, similarly to what happens in the non-penalized case tackled in the first part of the thesis. We applied the proposed methods to data from a study on cognitive behavior and treatment in patients with obsessive compulsive disorder. To show the full potential of the methods, we modified the original data in some of the analyses.

We propose a new parametric model for the life table distribution of deaths. The model is a mixture of three distributions: for infant and child mortality, for accidental and premature mortality, and for adult mortality. This permits to study the characteristics of the different components analyzing the changes of the shape of their distributions. In particular we focused on the features of accidental and premature mortality, since this component, for several reasons, is not often taken into account. The model offers the advantage of computing, in explicit form, the three component's contributions to life expectancy. This is useful to understand the transformations of the distribution of deaths. Moreover, all parameters in the model have a demographic interpretation, so it is easy to study trends and compare different populations. The mixture model was tested using Swedish raw data from the Human Mortality Database and then fitted for different countries. Our results about infant, childhood and adult mortality confirm what we generally know about the tendencies of these components: decrease of deaths at age 0, disappearance of childhood mortality, compression and shifting of adult distribution. For what regards premature mortality, we observed that, over time, its function becomes flatter and more symmetric, and its mode shifts progressively. This implies that almost entirely the accidental mortality has disappeared, while the premature mortality continuous to exist. We also noted that its contribution to explain life expectancy decreased during the last century, but in recent years, for some countries, especially France, it started to grow. This means that the shape of the distribution of deaths is changed: after a compression around the adult mode, an increase of the number of deaths outside the adult distribution is registered. The model does not perform satisfactory estimates when the interval of the last open age class is too wide, in particular when the late mode of the distribution is not clearly recognizable in the death distribution. To improve the estimates we employed EM algorithm, which proved to be a good method to solve this problem. EM algorithm can also be applied to reconstruct incomplete birth cohort data. To compute confidence intervals bootstrap resampling was adopted.

The use of diagnostic tests to discriminate between disease classes is becoming more and more popular in medicine, which leads to the urgent need for assessing accuracy of diagnostic tests before their implementation. To do that, a common tool is receiver operating characteristic (ROC) analysis. More precisely, the ROC curve and the area under the ROC curve (AUC) are commonly employed when two disease classes (typically, non-diseased and diseased) are considered, whereas the ROC surface and the volume under the ROC surface (VUS) are frequently used when the disease status has three categories (e.g., non-diseased, intermediate and diseased). In estimating such parameters, we assume that the true disease status of each patient can be determined by means of a gold standard test. In practice, unfortunately, the true disease status could be unavailable for all study subjects, due to the expensiveness or invasiveness of the gold standard test. Thus, often only a subset of patients undergoes disease verification. Statistical evaluations of diagnostic accuracy of a test based only on data from subjects with verified disease status are typically biased. This bias is known as verification bias. Various methods have been developed to adjust for verification bias in estimation of the ROC curve and its area for tests with binary or ordinal or continuous results. For the ROC surface and its volume, verification bias correction methods exist for tests with ordinal responses, but not for continuous tests. In this thesis, we propose several bias--corrected methods for estimating the ROC surface and the VUS of continuous diagnostic tests in presence of verification bias. In particular, these methods are constructed based on imputation and re--weighting techniques, and work well when the missingness mechanism of the true disease status is missing at random or missing not at random. The asymptotic behaviors of the estimators are also studied. To illustrate how to use the methods in real applications, two datasets dealing with epithelial ovarian cancer are considered. To support researchers in carrying out the ROC surface analysis in presence of verification bias, an R package and the corresponding Shiny web application have been created.

The existence of bias in inferential procedures based on the likelihood function has given rise to a great deal of research in the statistical literature. The magnitude of such bias plays a crucial role in estimation: if large, misleading conclusions on the quantities of interest are likely to be drawn. This is a matter of serious concern when the available sample size is small to moderate or when the model under study does not meet the regularity conditions for usually reliable maximum likelihood inference. In the present thesis, we attempt to reduce the impact of bias in both these circumstances, by following distinct paths. For finite-sample problems, we propose a convenient way to refine Wald-type inference in regression settings through asymptotic bias correction of the $z$-statistic. Such approach stems from the intuition of seeing that pivot as the estimator of a model reparametrization. For non-regular problems, with special focus on scenarios characterized by the presence of incidental parameters, we suggest a strategy to extend the current range of applications of the modified profile likelihood. This solution, founded on Monte Carlo simulation, is versatile enough to cope with several nonstandard modeling frameworks for grouped data.

Despite a long tradition in the study of graphs and relational data, for decades the analysis of complex networks was limited by difficulties in data collection and computational burdens. The advent of new technologies in life sciences, as well as in our daily life, has suddenly shed light on the many interconnections that our world features, from friendships and collaborations between individuals or organizations, to functional couplings between cellular molecules. This has highly facilitated the collection of relational data, fostering an unprecedented interest in network science. Understanding relations encoded in complex networks, however, still represents a challenging task, and statistical methods that can help to summarize and simplify complex networks are needed. In this thesis we show that often one can gain a deep insight of a network by focusing their attention on communities, i.e. on clusters of nodes, and on the relations that exist between them. We begin by presenting NEAT, a network-based test that allows to assess relations between gene sets in a gene interaction network. NEAT extends traditional gene enrichment analysis tests by incorporating information on interactions between genes and it overcomes some limitations of existing network enrichment analysis approaches. Then, we propose two extended stochastic blockmodels that allow to infer the relations that exist between communities from relations between pairs of individuals in a social network. We advocate the use of penalized inference to estimate these models, with the aim of deriving a sparse reduced graph between communities. Application of these models to bill cosponsorship networks in the Italian Chamber of Deputies allows us to reconstruct the pattern of collaborations between Italian political parties from 2001 to 2015. Finally, we propose a novel clustering strategy for sequences of graphs, based on mixtures of generalized linear models. We show that the proposed clustering method not only is capable to retrieve subpopulations of networks within a cross-sectional or longitudinal sequence of networks, but it also allows to directly characterize them by considering each of the components that form the mixture model.

Given a big set of data with several variables, the aim is the evaluation of each unit with a method that produces a synthetic measure describing a complex and non-observable concept; this goal is achieved respecting the characteristics of the variables, specially the measurement scale. The information gathered with partially ordered sets (poset) reflects this aim, because posets depends only on the order relations among the observations, and allows to handle ordinal and dichotomous variables fairly. In this setting, the vector of variables observed on a unit is handled as a unique entity called profile and not as a group of different variables that need to be aggregated. Starting from recent developments in poset theory, this thesis is organized in three parts. The first proposes to obtain a unique indicator combining the values given by the severity measures for evaluation, derived from the fuzzy identification method. The second contribution is the HOGS (Height Of Groups by Sampling) procedure, which is aimed to estimate the average rank of groups of units of a big population. HOGS is a step forward the statistical estimation of the average rank of a profile; furthermore it allows the estimation of the effect of external variables on the synthetic measure. The last results are two new R functions: the first computes the approximated average rank for large data sets overcoming the usual sample sizes considered by the software usually used until now, the second implements the information given by the frequency of profiles in the computation of approximated average rank, making its use more profitable for social sciences.

In the framework of functional data analysis we propose two Bayesian Nonparametric models. In the first model, motivated by an application in neuroimaging, functions are assumed to be spatially correlated and clustered together by an underlying Functional Dependent Dirichlet process which encodes a conditional autoregressive dependence structure to guide the spatial selection. Spatial symmetries of the functional responses in the brain can be appropriately accounted for in our framework. Motivated by the Italian natural gas balancing platform, in the second model time dependence are induced in the weights of the underlying Functional Dependent Dirichlet process through a dynamic linear model defined over a partitioned function space. Typical shape characteristics of the functions are modeled by flexible spline-based curve estimates as atoms of the process. In both applications Bayesian variable selection techniques are used to select significant sets of bases coefficients in each cluster. Gibbs sampling algorithms are developed for posterior computation, simulation studies and application to real data assess the performance of our approaches.

In deregulated electricity markets, accurate modeling and forecasting of different variables, e.g. demand, prices, production etc. have obtained increasing importance in recent years. As in most electricity markets, the daily demand and prices are determined the day before the physical delivery by means of (semi-) hourly concurrent auctions, accurate forecasts are necessary for the efficient management of power systems. However, it is well known that electricity (demand/price) data exhibit some specific features, among which, daily, weekly and annual periodic patterns as well as non-constant mean and variance, jumps and dependency on calendar effects. Modeling and forecasting, thus, is a challenging task. This thesis tackles these two issues, and to do this, two approaches are followed. In the first case, we address the issue of modeling and out-of-sample forecasting electricity demand and price time series. For this purpose, an additive component model was considered that includes some deterministic and a stochastic residual components. The deterministic components include a long-term dynamics, annual and weekly periodicities and calendar effects. The first three components were estimated using splines while the calendar effects were modeled using dummy variables. The residual component is instead treated as stochastic and different univariate and multivariate models have been considered with increasing level of complexity. In both cases, linear parametric and nonlinear nonparametric models, as well as functional based models, have been estimated and compared in a one day-ahead out-of-sample forecast framework. The class of univariate models includes parametric autoregressive models (AR), nonparametric and nonlinear regression models based on splines (NPAR) and scalar-response functional models, that in turns can be formulated parametrically (FAR) or non parametrically (NPFAR). The multivariate models are vector autoregressive models (VAR) and functionalresponse, parametric (FFAR) and nonparametric (NPFFAR), models. For this issue, five different electricity markets, namely, British electricity market (APX Power UK), Nord Pool electricity market (NP), Italian electricity market (IPEX), Pennsylvania-New Jersey-Maryland electricity market (PJM) and Portuguese electricity market (OMIE(Po)) were considered for the period 2009 to 2014. The first five years were used for model estimation while the year 2014 was left for one-day-ahead forecasts. Predictive performances are first evaluated by means of descriptive indicators and then through a test to assess the significance of the differences. The analyses suggest that the multivariate approach leads to better results than the univariate one and that, within the multivariate framework, functional models are the most accurate, with VAR being a competitive model in some cases. The results also lead to another important finding concerning to the performance of parametric and nonparametric approach that showed strong linkage with underlying process. Finally the obtained results were compared with other works in the literature that suggest our forecasting errors are smaller compared with the state-of-art prediction techniques used in the literature. In the second part of this thesis the issue of electricity price forecasting is revisited following a completely different approach. The main idea of this approach is that of modeling the daily supply and demand curves, predicting them and finding the intersection of the predicted curves in order to find the predicted market clearing price and volume. In this approach, the raw bids/offers data for demand and supply, corresponding to each (half-) hour is first aggregated in a specific order. The functional approach converts the resulted piece wise curves into smooth functions. For this issue, parametric functional model (FFAR) and the nonlinear nonparametric counterpart (NPFFAR) were considered. As benchmark, an ARIMA model was fitted to the scalar time series corresponding to the market clearing prices obtained from the crossing points of supply and demand curves. Data from Italian electricity market were used for this issue and the results are summarized by different descriptive indicators. As in the first case, results show superior forecasting performance of our functional approach compare to ARIMA. Among different models, the nonparametric functional model produces better results compared to parametric models. Apart from the improvement in forecasting accuracy, it is important to stress that this approach can be used for optimizing bidding strategies. As forecasting the whole curves gives deep insight into the market, our analysis showed that this strategy can significantly improve bidding strategies and maximize traders profit.

Network data representing relationship structures among a set of nodes are available in many fields of applications covering social science, neuroscience, business intelligence and broader relational settings. Although early probability models for networks date back almost sixty years, this field of research is still an object of intense and dynamic interest. A primary reason for the recent growth of statistical methodologies in modeling of networks is that the routine collection of such data is a recent development. Online social networks, novel neuroimaging technologies, improved business intelligence analyses and sophisticated computer algorithms monitoring world news media, currently provide increasingly complex network data sets along with novel motivating applications and new methodological questions. A challenging issue in such settings is that data are available via multiple network observations and hence the rich literature in modeling of a single network falls far short of the goal of providing flexible inference in this scenario. Statistical modeling of replicated network data is still on its infancy and several questions remain about coherence of inference, flexibility, computational tractability and other key issues. Motivated by complex applications from different domains, this thesis aims to take a sizable step towards addressing these issues via Bayesian nonparametric modeling. The thesis is organized in two main frameworks, further divided in different topics. The first thread develops flexible and computationally tractable stochastic processes for modeling dynamic networks, which incorporate temporal dependence and exploit latent network structures. The second focuses on defining a provably flexible representation for the probabilistic generative mechanism underlying a network-valued random variable, which is able to provide valuable insights both on shared and subject -- or phenotype -- specific sources of variability in the network structure

High-throughput assays are transforming the study of biology, and are generating a rich, complex and diverse collection of high-dimensional data sets. Building systematic knowledge from this data is a cumulative process, which requires analyses that integrate multiple sources, studies, and technologies. The increased availability of ensembles of studies on related clinical populations, assaying technologies, and genomic features poses two categories of very important multi-study statistical components: 1) common factors shared across multiple studies; 2) study-specific factors. To capture these two different quantities, in this thesis we propose a novel class of factor analysis models, both under a frequentist and Bayesian approach. In the frequentist approach an ECM algorithm is provided to obtain the maximum likelihood estimates. Moreover, we propose a Bayesian approach to apply the method to settings with more variables than subjects. In modeling dependencies among many variables, a sparse structure underlying the associations among genes is assumed. Both methods allow to perform joint analysis of multiple high-throughput studies. The results are helpful for combining multiple studies, identifying reproducible biology across studies and interesting study-specific components, and removing idiosyncratic variation that lacks cross-study reproducibility.

In the recent years, quantile regression methods have attracted relevant interest in the statistical and econometric literature. This phenomenon is due to the advantages arising from the quantile regression approach, mainly the robustness of the results and the possibility to analyse several quantiles of a given random variable. Such as features are particularly appealing in the context of economic and financial data, where extreme events assume critical importance. The present thesis is based on quantile regression, with focus on the economic and financial environment. First of all, we propose new approaches in developing asset allocation strategies on the basis of quantile regression and regularization techniques. It is well known that quantile regression model minimizes the portfolio extreme risk, whenever the attention is placed on the estimation of the response variable left quantiles. We show that, by considering the entire conditional distribution of the dependent variable, it is possible to optimize different risk and performance indicators. In particular, we introduce a risk-adjusted profitability measure, useful in evaluating financial portfolios under a pessimistic perspective, since the reward contribution is net of the most favorable outcomes. Moreover, as we consider large portfolios, we also cope with the dimensionality issue by introducing an l1-norm penalty on the assets weights. Secondly, we focus on the determinants of equity risk and their forecasting implications. Several market and macro-level variables influence the evolution of equity risk in addition to the well-known volatility persistence. However, the impact of those covariates might change depending on the risk level, being different between low and high volatility states. By combining equity risk estimates, obtained from the Realized Range Volatility, corrected for microstructure noise and jumps, and quantile regression methods, we evaluate, in a forecasting perspective, the impact of the equity risk determinants in different volatility states and, without distributional assumptions on the realized range innovations, we recover both the points and the conditional distribution forecasts. In addition, we analyse how the relationships among the involved variables evolve over time, through a rolling window procedure. The results show evidence of the selected variables' relevant impacts and, particularly during periods of market stress, highlight heterogeneous effects across quantiles. Finally, we study the dynamic impact of uncertainty in causing and forecasting the distribution of oil returns and risk. We analyse the relevance of recently developed news-based measures of economic policy uncertainty and equity market uncertainty in causing and predicting the conditional quantiles and distribution of the crude oil variations, defined both as returns and squared returns. For this purpose, on the one hand, we study the causality relations in quantiles through a non-parametric testing method; on the other hand, we forecast the conditional distribution on the basis of the quantile regression approach and the predictive accuracy is evaluated by means of several suitable tests. Given the presence of structural breaks over time, we implement a rolling window procedure to capture the dynamic relations among the variables.

Regression Discontinuity Design (RDD) is one of the most popular designs in the field of causal inference in nonexperimental settings. It is based on the idea that the treatment is (totally or partially) determined by a threshold point of an observed continuous variable. When the treatment is just partially determined by that variable, it is usu- ally defined fuzzy RDD. In this setting, given a certain outcome, the only effect that one is able to identify is the Average Treatment Effect (ATE) for the subpopulation of the Compliers at the threshold point. The ATE could be obtained by the ratio between the discontinuity at the threshold point in the average of the observed outcome divided by the discontinuity in the treatment probability. This thesis explores, from a methodological and empirical perspective, how the change of slope at the threshold point is informative for the estimate of the parameter of interest. Starting from the changes of the eligibility criterion for retirement that took place in Italy in the ’90s we propose an alternative estimator, based on Instrumental Variables, that is a combination of the discontinuity and the change of slope. Furthermore we provide a simulation study to compare the efficiency of the different estimators. Then we analyze the effects of retirement on the subjective well–being. Finally we generalize the results using the Two Sample Instrumental Variable estimator, in order to improve the efficiency of estimates based on administrative data and to con- struct delayed outcomes for the same cohorts.

The adverse health effect of exposure to high pollutant concentration has been the focus of many recent studies. This is particularly true for ground level ozone which is considered in the present thesis. The effect has been estimated at different geographic locations, and it has been shown that it may be spatially heterogeneous. Within such widely accepted studies, two major issues arise which are the focus of this thesis: how to best measure daily individual exposure to a pollutant and how the health effect of the exposure is affected by geographic location both in strength and shape. The first issue is related to the fact that the concentration of ozone varies widely during the day, producing a distinctive daily pattern. Traditionally, the daily pattern of the pollutant is collapsed to a single summary figure which is then taken to represent daily individual exposure. In this thesis, we propose a more accurate approaches to measure pollutant exposure which address the limitations in the use of the standard exposure measure. The methods are based on principle of functional data analysis, which treats the daily pattern of concentration as a function to account for temporal variation of the pollutant. The predictive efficiency of our approach is superior to that of models based on the standard exposure measures. We propose a functional hierarchical approach to model data which are coming from multiple geographic locations, and estimate pollutant exposure effect allowing daily variation and spatial heterogeneity of the effect at once. The approach is general and can also be considered as the analogue of the multilevel models to the case in which the predictor is functional and the response is scalar.

Biological pathways underlie the basic functions of a living cell. They are complex diagrams featuring genes, proteins and other small molecules, showing how they work together to achieve a particular biological effect. From a technical point of view, they are networks represented through a graph where genes and their connections are, respectively, nodes and edges of a graph. The main research objective of this thesis is to develop a framework for simulating effects of gene silencing. To this end, we propose a three step approach. First, we refine the structure of a pathway via our CK2 algorithm. Next, we assess the uncertainty in the refined structure. Finally, we simulate gene silencing through intervention analysis in causal graphical models. The proposed approach showed promising results when applied to the problem of predicting the effect of the knockdown of the nkd gene in Drosophila Melanogaster.

In last decades researchers have focused on the analysis of SEM models with nonlinear effects involving exogenous variables, i.e. which are not linearly dependent on other variables. The main problems studied are the estimation process, the choice of the indicators for nonlinear terms, when the variables are unobserved and the possibility of distinguishing interaction models from curvilinear models, while the causal analysis is not taken into account. In this thesis I introduce nonlinear terms involving endogenous variables in SEM model with 2 mediators. I focus my attention on the interaction and curvilinear effects as its particular case. This analysis is made both with observed variables and with unobserved or latent variables. To address causal analysis, I propose two different approaches and I compare them using simulated data with different sample sizes and different covariances between the 2 mediators. I find that my model with Pearl's (2012) causal theory and exogenous interaction, i.e. which does not depend linearly on other variables, is preferable for its simplicity and because it requires a smaller sample size. Pearl's theory can be applied to very general models and for this reason it has problems when the mediators are correlated given the mediated variable. Then I propose a modified formula to apply this theory. This approach has good performances both for interaction models and for curvilinear models and I propose a procedure to recognize the true model. Finally from a managerial perspective using the exogenous interaction model with Pearl's modified causal theory proposed by me, I show that, in a customer satisfaction context, positive emotions and negative emotions influence "jointly" future behavior. As emotions are in turn influenced by the design of the restaurant, managers can use it to enhance customers' loyalty both directly and indirectly by jointly inducing more positive emotions and less negative ones. This way a model with interacting mediators may help to better understand customers' behavior

The rather generic title of this Thesis is due to the fact that several aspects of biological phenomena have been investigated. Most of this work was addressed at the investigation of the limitations of one of the essential tools for analyzing gene expression data: cluster analysis. With several hundred of clustering methods in existence, there is clearly no shortage of clustering algorithms but, at the same time, satisfactory answers to some basic questions are still to come. In particular, we present a novel algorithm for the clustering of static data and a new strategy for the clustering of short-length time-course data. Finally, we analyzed data coming from Cap Analysis Gene Expression, a relatively new technology useful for the genome-wide promoter analysis and still mostly unexplored.

Spatiotemporal phenomena related to the rainfall measurements can be characterised by statistical models grounded on physical concepts instead of being identified by spatiotemporal patterns based on standard correlations and related analytical tools. This perspective is useful in understanding if the relationships among neighbouring zones and consecutive years are attributable to latent physical mechanisms. Satellite data are used to examine this theory and provide evidence on empirical basis. A recent hydrological theory, which is based on the concept of self-organisation, consists of simplified physical mechanisms that are essential for the explanation of local data relationships. The regression models inspired by the diffusion of innovation can approximate the evolution of the rainfall process within a year through a more straightforward perspective. However, the multitude of collected data requires innovative techniques of data management and advanced analytical solutions, in order to achieve optimal results in reasonable time. Indeed, the nonlinear least squares and nonlinear quantile regression are considered to make inference on the response variable given some covariates. A new quantile regression technique is developed in order to provide simultaneous estimates that do not violate the monotonicity property of quantiles. The nonlinear least squares highlight strong connections among rainfall and the salient features of the measurements areas. Furthermore, the quantile regression analyses quantify the intrinsic variability of the data.

Genome-wide association studies, GWAS, typically contain hundreds of thousands single nucleotide polymorphisms, SNPs, genotyped for few numbers of samples. The aim of these studies is to identify regions harboring SNPs or to predict the outcomes of interest. Since the number of predictors in the GWAS far exceeds the number of samples, it is impossible to analyze the data with classical statistical methods. In the current GWAS, the widely applied methods are based on single marker analysis that does assess association of each SNP with the complex traits independently. Because of the low power of this analysis for detecting true association, simultaneous analysis has recently received more attention. The new statistical methods for simultaneous analysis in high dimensional settings have a limitation of disparity between the number of predictors and the number of samples. Therefore, reducing the dimensionality of the set of SNPs is required. This thesis reviews single marker analysis and simultaneous analysis with a focus on Bayesian methods. It addresses the weaknesses of these approaches with reference to recent literature and illustrating simulation studies. To bypass these problems, we first attempt to reduce dimension of the set of SNPs with random projection technique. Since this method does not improve the predictive performance of the model, we present a new two-stage approach that is a hybrid method of single and simultaneous analyses. This full Bayesian approach selects the most promising SNPs in the first stage by evaluating the impact of each marker independently. In the second stage, we develop a hierarchical Bayesian model to analyze the impact of selected markers simultaneously. The model that accounts for related samples places the local-global shrinkage prior on marker effects in order to shrink small effects to zero while keeping large effects relatively large. The prior specification on marker effects, which is hierarchical representation of generalized double Pareto, improves the predictive performance. Finally, we represent the result of real SNP-data analysis through single-maker study and the new two-stage approach.

Diffusion of innovation is an attempt to study the behaviour of agents in the complex network structure, contagion of information and the related consequences. Pioneering approach by Bass (1969) is further developed with numerous research works. Further considerations bring to the introduction of heterogeneity effect and marketing mix variables, that has been studied later. Empirical results show that the basic Bass model and its generalisations (GBM) can be studied as a modified form of the basic Logistic model. Recent work by Bemmaor (1994) explains the diffusion dynamics as a mixture of probability distributions obtained from individual level heterogeneity. Therefore, the basis of this thesis is to formulate heterogeneous agent based diffusion modelling at the aggregate level, in order to predict the future behaviour of the diffusion path, based on its past behaviour. This thesis extends the contribution of heterogeneous agents' behaviour and tries to put some probabilistic assumption at the individual adoption attitude (adoption propensity) and obtains the diffusion dynamics as an aggregate level observed phenomenon (consequences). Special emphasis is given to the contribution of heterogeneous group of imitators towards the diffusion process. The existing Generalised Bass model (GBM) has been updated with additional information of the imitators' participation that results a mixture of GBM with Bemmaor heterogeneity effect, which is important to describe the social contagion of information. To accommodate the effect of heterogeneity among the involved agents, both in innovators and imitators subgroup, this thesis suggests an extension of Bemmaor model, named modified Bemmaor model (MBM), which is further modified considering GBM with heterogeneity. Proposed models are applied to real data sets under technological diffusion of innovation perspectives. Obtained results highlight efficacy of the proposed models with additional heterogeneity parameters and strong agreements with the research outputs conducted with similar objectives. Proposed models are very useful in applied context and could be extended further for multiple innovations, simultaneous innovation or multiple regimes of innovations diffusion contexts

With the advent of globalization and the recent financial turmoil, the interest for the analysis of dependencies between financial time series has significantly increased. Risk measures such as value-at-risk are heavily affected by the joint extreme comovements of associated risk factors. This thesis suggests some copula-based statistical tools which can be useful in order to have more insights into the nature of the association between random variables in the tail of their distributions. Preliminarily, an overview of important definitions and properties in copula theory is given, and some known measures of tail dependence based on the notion of tail dependence coefficients and rank correlations are introduced. A first proposal consists of a graphical tool based on the so-called tail concentration function, in order to distinguish different families of copulas in a 2D configuration. This can be used as a copula selection tool in practical fitting problems, when one wants to choose one or more copulas to model the dependence structure in the data, highlighting the information contained in the tail. The thesis mainly deals with financial time series applications, where copula functions and the related concepts of tail copula and tail dependence coefficients are used to characterize the dependence structure of asset returns. Classical cluster analysis tools are revisited by introducing suitable copula-based tail dependence measures, which are exploited in the identification of similarities or dissimilarities between the variables of interest and, in particular, between financial time series. Such an approach is designed to investigate the joint behaviour of pairs of time series when they are taking on extremely low values. Either the asymptotic and the finite behaviour are assessed. The proposed methodology is based on a suitable copula-based time series model(GARCH-copula model), in order to model the marginal behaviour of each time series separately from the dependence pattern. Moreover, non-parametric estimation procedures are adopted for describing the pairwise dependencies, thus avoiding any model assumption. Simulation studies are conducted in order to check the performances of the proposed procedures and applications to financial data are presented showing their practical implementation. The information coming from the output of the introduced clustering techniques can be exploited for automatic portfolio selection procedures in order to hedge the risk of a portfolio, by taking into account the occurrence of joint losses. A two-stage portfolio diversification strategy is proposed and empirical analysis are provided. Results show how the suggested approach to the clustering of financial time series can be used by an investor to have more insights into the relationships among different assets in crisis periods. Moreover, the application to portfolio selection framework suggests a cautious usage of standard procedures that may not work when the markets are expected to experience periods of high volatility.

There is a high potential of information available in Behaviour Risk Factor Surveillance (BRFS) data, and especially for studying trends, as these data collect information in an ongoing and almost continuous manner for long periods of time. In order to account for the complex and dynamic relationships between the variables and avoid the aggregation of measures so as not to lose information in variability, the use of varying coefficient models with non-parametric techniques have been studied. These models allow the study of the trends and inter-relationships in the effects of the variables on the outcome of interest either over time or space, therefore providing valuable information for health policy interventions. A comparison of the possible estimation techniques, using the Italian surveillance data, has resulted in the selection of P-splines for estimation due to the flexibility in their use and the faster computation times. This estimation method was applied for a time varying coefficient model for a smoking status outcome variable using Italian surveillance data, and a time varying coefficient model for an obesity status outcome variable using U.S.A. surveillance data. The results of these models provide coefficient plots in which one can observe which subgroups of the population have an effect on the outcome which is changing over time. A spatial varying coefficient model was also studied for one point in time using smoothing spline estimation with tensor product smooths, and the maps produced from this model were able to show how the probabilities of the outcome variable (obesity) are changing across the counties of a U.S. state within each population subgroup. The strengths and limitations of these methods are discussed, as well as recommendations for further research such as the study of a spatial-temporal model using health surveillance data. Notwithstanding few limitations, the varying coefficient model represents an effective approach proving to produce interesting results (not accessible with the usual standard epidemiological approach) in this particular field of application and with BRFS data.

Environmental computer models are deterministic models devoted to predict several environmental phenomena such as air pollution or meteorological events. Numerical model output is given in terms of averages over grid cells, usually at high spatial and temporal resolution. However, these outputs are often biased with unknown calibration and not equipped with any information about the associated uncertainty. Conversely, data collected at monitoring stations is more accurate since they essentially provide the true levels. Due the leading role played by numerical models, it now important to compare model output with observations. Statistical methods developed to combine numerical model output and station data are usually referred to as data fusion. In this work, we first combine ozone monitoring data with ozone predictions from the Eta-CMAQ air quality model in order to forecast real-time current 8-hour average ozone level defined as the average of the previous four hours, current hour, and predictions for the next three hours. We propose a Bayesian downscaler model based on first differences with a flexible coefficient structure and an efficient computational strategy to fit model parameters. Model validation for the eastern United States shows consequential improvement of our fully inferential approach compared with the current real-time forecasting system. Furthermore, we consider the introduction of temperature data from a weather forecast model into the downscaler, showing improved real-time ozone predictions. Finally, we introduce a hierarchical model to obtain spatially varying uncertainty associated with numerical model output. We show how we can learn about such uncertainty through suitable stochastic data fusion modeling using some external validation data. We illustrate our Bayesian model by providing the uncertainty map associated with a temperature output over the northeastern United States.

The Bayesian approach to statistical inference in fundamentally probabilistic. Exploiting the internal consistency of the probability framework, the posterior distribution extracts the relevant information in the data, and provides a complete and coherent summary of post data uncertainty. However, summarising the posterior distribution often requires the calculation of awkward multidimensional integrals. A further complication with the Bayesian approach arises when the likelihood functions is unavailable. In this respect, promising advances have been made by theory of Approximate Bayesian Computations (ABC). This thesis focuses on computational methods for the approximation of posterior distributions, and it discusses six original contributions. The first contribution concerns the approximation of marginal posterior distributions for scalar parameters. By combining higher-order tail area approximation with the inverse transform sampling, we define the HOTA algorithm which draws independent random sample from the approximate marginal posterior. The second discusses the HOTA algorithm with pseudo-posterior distributions, \eg, posterior distributions obtained by the combination of a pseudo-likelihood with a prior within Bayes' rule. The third contribution extends the use of tail-area approximations to contexts with multidimensional parameters, and proposes a method which gives approximate Bayesian credible regions with good sampling coverage properties. The forth presents an improved Laplace approximation which can be used for computing marginal likelihoods. The fifth contribution discusses a model-based procedure for choosing good summary statistics for ABC, by using composite score functions. Lastly, the sixth contribution discusses the choice of a default proposal distribution for ABC that is based on the notion of quasi-likelihood.

Genomic studies aim at identifying genes’ location and function across the genome of living organisms of interest. In the last decades, this field of research has been the object of lively interest, motivated by the introduction of microarray and sequencing technologies, capable of providing huge amounts of data concern- ing several aspects of the genome. Statistical tools have proven to be necessary in this context, to support and sometimes lead biological investigation, impractical or impossible to be conducted over the whole set of data provided by the above men- tioned technologies. In this Thesis, we introduce novel statistical tools for dealing with well-known problems in the genomic field, such as identifying differential expression in microarray data, and evaluating differential binding in the context of ChIP-Seq data. Our specific interest will be inference on quantiles, motivated by their interpretability, even for irregularly shaped distributions of the data, and by the fact that they allow to compare different aspects of the whole distribution of the data. We propose Studentized and pseudo-Studentized statistics, whose struc- ture resembles closely that of a classic t-test, and evaluate their performances via simulated studies and application to real data.

This Ph.D. thesis is devoted to Statistical Process Control (SPC) methods for monitoring over time the stability of a relation between two variables (profile). Very often in literature the functional form of the relation is assumed to be known, whereas in this work we concentrated on generic and unknown relations which have to be estimated with the usual nonparametric regression techniques. The original contributes are two, resented in chapters 2 and 3 respectively. In Chapter 1 we make a brief overview on the topic in order to make you become familiar with these specific problems of Statistical Process Control (SPC) applications and we introduce you to the original parts of this work. In Chapter 2 we envelope and compare five new control charts for monitoring on-line unknown general, and not only linear, relations among variables over time under the assumption of the normality of the errors; these charts combine in an original way the following techniques: self-starting methods, useful to drop the distinction between Phase I and Phase II of the analysis; very known multivariate charting schemes as MEWMA and CUSCORE; nonparametric testing techniques as wavelet methods and kernel linear smoothing. In Chapter 3, instead, we construct a test statistic useful to check with a completely nonparametric procedure the stability of a process retrospectively, thus off-line. Both second and third chapters are structured in the following way: brief literature review; framework and model considered in our study; simulation study; a section with some useful complements on the topics and relative research carried out; conclusion and suggestions for future research.

Proportional hazards models are among the most popular regression models in survival analysis. Multi-state models generalise them in the sense of jointly considering different types of events along with their interrelations, whereas frailty models introduce random effects to account for unobserved risk factors, possibly shared by groups of subjects. The integration of frailty and multi-state methodology is interesting to control for unobserved heterogeneity in presence of complex event history structures, particularly appealing in multicenter clinical trials applications. In the present thesis we propose the incorporation of nested frailties in the transition-specific hazard function; then, we develop and evaluate both parametric and semi-parametric inference. Simulation studies, performed thanks to an innovative method for generating dependent multi-state survival data, show that parametric inference is correct but extremely imprecise, whilst semiparametric methods are very competitive to evaluate the effect of covariates. Two case studies are presented, relative to cancer multicenter clinical trials. The multi-state nature of the models allows to study the treatment effect taking into account intermediate events, while the presence of frailties reduces the attenuation effect due to clustering. Finally, we present two new software tools, one to fit parametric frailty models with up to twenty possible combinations of baseline and frailty distributions, and one implementing semiparametric inference for multilevel frailty models, essential to fit the new nested frailty multi-state models.

We concentrate on the statistical methods used in Biogeography for modelling the spatial distribution of bird species. Due to the difficulty of specifying a joint multivariate spatial covariance structure in environmental processes, we factor such a joint distribution into a series of conditional models linked together in a hierarchical framework. We have a process that corresponds to an unobservable map with the actual information about a bird species, and the data correspond to the observations that are connected to that process. Markov chain Monte Carlo (MCMC) simulation approaches are used for models involving multiple levels incorporating dependence structures. We use a Bayesian algorithm for drawing samples from the posterior distribution in order to obtain estimates of the parameters and reconstruct the true map based on data. We present different methods to overcome the problem of calculating the distribution of the Markov random field that is used in the MCMC algorithm. During the analysis it is desirable to delete some of the predictors from the model and only use a subset of covariates in the estimation procedure. We use the method by Kuo & Mallick (1998) (KM) for variable selection and combine it with multiple independent chains which successfully improves the mixing behaviour. In simulation studies we show the better performance of the pseudolikelihood over other likelihood approximation methods, and the good performance of the KM method with this type of data. We illustrate the application of the methods with the complete analysis of the spatial distribution of two bird species (Sturnella magna and Anas rubripes) based on a real data set. We show the advantages of using the hidden structure and the spatial interaction parameter in the spatial hidden Markov model over other simpler models, like the ordinary logistic model or the autologistic model without observation errors.

Composite likelihood is a particular pseudo-likelihood built by adequately combining likelihoods based on lower dimensional events. It appears to be a very appealing alternative to the standard likelihood when the latter is too time-consuming to evaluate or unavailable due to a complex, and possibly unknown, structure of dependence in the data. After the brief ntroduction in the first chapter, Chapter 2 gives notation and basic definitions, but also states a condition for full efficiency of the maximum composite likelihood estimator in exponential families. The core of the thesis is Chapter 3, where we explore a linear combination of two types of composite likelihood which leads to a new objective function that depends on a constant to be chosen. In particular, this new combined composite likelihood uses both bivariate margins and univariate margins. Exact and asymptotic properties are explored. The exact properties lead to the identification of a possible strategy for finding the range of admissible values for the constant. The resulting estimator enjoys desirable asymptotic properties such as consistency and asymptotic normality. Two examples are analyzed in details, also through simulation studies. Chapter 4 studies a weighted independence likelihood in a prediction framework. The aim of this chapter is to determine the weights in order to get an improved prediction of a component of interest of the data vector. In particular, the weights are calculated by means of a delete-one approach in a cross-validation procedure. Through simulation studies, situations in which the weighted independence likelihood works well with respect to the standard independence likelihood are highlighted

In this paper, we tackle the study of the relationship between daily non accidental deaths and air pollution in the city of Philadelphia in the years 1974 -1988. For modelling the data, we propose to make use of dynamic generalized linear models. These models allow to deal with the serial dependence and time-varying effects of the covariates. Inference is performed by using extended Kaiman filter and smoother.

This thesis addresses different aspects of observation-driven time series modeling. The main contributions concern the reliability of likelihood-based inference and the specification of dynamic models to capture complex behaviors observed in time series data. As concerns inference, the main focus of the thesis is on invertibility conditions for observation-driven time series models. Invertibility plays a key role in ensuring the consistency of likelihood-based estimators. However, the invertibilty conditions typically employed in the literature are often unfeasible to be checked. Therefore, the reliability of inference fails to be guaranteed in practice. This thesis contributes to the literature by deriving feasible conditions that ensure the consistency of the maximum likelihood estimator for a wide class of models. One of the most appealing features of our consistency results is that they hold for both correctly specified and misspecified models. Several empirical examples covering different observation-driven models are presented. These examples highlight the practical relevance of the theoretical results. As concerns model specification, we cover two lines of research. The first is related to integer-valued time series data. We propose an extension to the class of Integer-valued Autoregressive models that allows the survival probability to vary over time. We show how our model can be easily estimated by maximum likelihood and we prove the consistency of the estimator. The flexibility of the proposed approach is shown through a simulation experiment and an application to a real time series of crime reports. Finally, the second line of research on model specification is an extension of the Generalized Autoregressive Score framework. We propose a class of models that updates time-varying parameters at different speeds in different time periods. The new updating equation can be employed to describe time series where the amount of information contained in the data is changing over time. This peculiarity is highlighted through a simulation study and we provide theoretical foundations for the proposed approach. Furthermore, two empirical applications to S&P 500 stock returns and US inflation illustrate how our method can be useful in practice.