Thesis (Ph.D. (Statistics)) -- University of Limpopo, 2022 / This thesis is based on application of frailty and copula models to under five child mortality data set in South Africa. The main purpose of the study was to apply sample splitting techniques in a survival analysis setting and compare clustered survival models considering left truncation to the under five child mortality data set in South Africa. The major contributions of this thesis is in the application of the shared frailty model and a class of Archimedean copulas in particular, Clayton-Oakes copula with completely monotone generator, and introduction of sample splitting techniques in a survival analysis setting. The findings based on shared frailty model show that clustering effect was sig nificant for modelling the determinants of time to death of under five children, and revealed the importance of accounting for clustering effect. The conclusion based on Clayton-Oakes model showed association between survival times of children from the same mother. It was found that the parameter estimates for the shared frailty and the Clayton-Oakes models were quite different and that the two models cannot be comparable. Gender, province, year, birth order and whether a child is part of twin or not were found to be significant factors affect ing under five child mortality in South Africa. / NRF-TDG Flemish Interuniversity Council Institutional corporation (VLIR-IUC) VLIR-IUC Programme of the University of Limpopo

Ocean wave energy is considered to be one of the important potential renewable energy resources for sustainable development. Various wave energy converter technologies have been proposed to harvest the energy from ocean waves. This thesis is based on the linear generator wave energy converter developed at Uppsala University. The research in this thesis focuses on the foundation optimization and the power absorption optimization of the wave energy converters and on the wave climate modelling at the Lysekil wave converter test site. The foundation optimization study of the gravity-based foundation of the linear wave energy converter is based on statistical analysis of wave climate data measured at the Lysekil test site. The 25 years return extreme significant wave height and its associated mean zero-crossing period are chosen as the maximum wave for the maximum heave and surge forces evaluation. The power absorption optimization study on the linear generator wave energy converter is based on the wave climate at the Lysekil test site. A frequency-domain simplified numerical model is used with the power take-off damping coefficient chosen as the control parameter for optimizing the power absorption. The results show a large improvement with an optimized power take-off damping coefficient adjusted to the characteristics of the wave climate at the test site. The wave climate modelling studies are based on the wave climate data measured at the Lysekil test site. A new mixed distribution method is proposed for modelling the significant wave height. This method gives impressive goodness of fit with the measured wave data. A copula method is applied to the bivariate joint distribution of the significant wave height and the wave period. The results show an excellent goodness of fit for the Gumbel model. The general applicability of the proposed mixed-distribution method and the copula method are illustrated with wave climate data from four other sites. The results confirm the good performance of the mixed-distribution and the Gumbel copula model for the modelling of significant wave height and bivariate wave climate.

Em análise de sobrevivência, usualmente o interesse esté em estudar o tempo até a ocorrência de um evento. Quando as observações estão sujeitas a mais de um tipo de evento (por exemplo, diferentes causas de óbito) e a ocorrência de um evento impede a ocorrência dos demais, tem-se uma estrutura de riscos competitivos. Em algumas situações, no entanto, o interesse está em estudar dois eventos, sendo que um deles (evento terminal) impede a ocorrência do outro (evento intermediário), mas não vice-versa. Essa estrutura é conhecida como riscos semicompetitivos e foi definida por Fine et al.(2001). Neste trabalho são consideradas duas abordagens para análise de dados com essa estrutura. Uma delas é baseada na construção da função de sobrevivência bivariada por meio de cópulas da família Arquimediana e estimadores para funções de sobrevivência são obtidos. A segunda abordagem é baseada em um processo de três estados, conhecido como processo doença-morte, que pode ser especificado pelas funções de intensidade de transição ou funções de risco. Neste caso, considera-se a inclusão de covariáveis e a possível dependência entre os dois tempos observados é incorporada por meio de uma fragilidade compartilhada. Estas metodologias são aplicadas a dois conjuntos de dados reais: um de 137 pacientes com leucemia, observados no máximo sete anos após transplante de medula óssea, e outro de 1253 pacientes com doença renal crônica submetidos a diálise, que foram observados entre os anos 2009-2011. / In survival analysis, usually the interest is to study the time until the occurrence of an event. When observations are subject to more than one type of event (e.g, different causes of death) and the occurrence of an event prevents the occurrence of the other, there is a competing risks structure. In some situations, nevertheless, the main interest is to study two events, one of which (terminal event) prevents the occurrence of the other (nonterminal event) but not vice versa. This structure is known as semicompeting risks, defined initially by Fine et al. (2001). In this work, we consider two approaches for analyzing data with this structure. One approach is based on the bivariate survival function through Archimedean copulas and estimators for the survival functions are obtained. The second approach is based on a process with three states, known as Illness-Death process, which can be specified by the transition intensity functions or risk functions. In this case, the inclusion of covariates and a possible dependence between the two times is taken into account by a shared frailty. These methodologies are applied to two data sets: the first one is a study with 137 patients with leukemia that received an allogeneic marrow transplant, with maximum follow up of 7 years; the second is a data set of 1253 patientswith chronic kidney disease on dialysis treatment, followed from 2009 until 2011.

Em análise de sobrevivência, usualmente o interesse esté em estudar o tempo até a ocorrência de um evento. Quando as observações estão sujeitas a mais de um tipo de evento (por exemplo, diferentes causas de óbito) e a ocorrência de um evento impede a ocorrência dos demais, tem-se uma estrutura de riscos competitivos. Em algumas situações, no entanto, o interesse está em estudar dois eventos, sendo que um deles (evento terminal) impede a ocorrência do outro (evento intermediário), mas não vice-versa. Essa estrutura é conhecida como riscos semicompetitivos e foi definida por Fine et al.(2001). Neste trabalho são consideradas duas abordagens para análise de dados com essa estrutura. Uma delas é baseada na construção da função de sobrevivência bivariada por meio de cópulas da família Arquimediana e estimadores para funções de sobrevivência são obtidos. A segunda abordagem é baseada em um processo de três estados, conhecido como processo doença-morte, que pode ser especificado pelas funções de intensidade de transição ou funções de risco. Neste caso, considera-se a inclusão de covariáveis e a possível dependência entre os dois tempos observados é incorporada por meio de uma fragilidade compartilhada. Estas metodologias são aplicadas a dois conjuntos de dados reais: um de 137 pacientes com leucemia, observados no máximo sete anos após transplante de medula óssea, e outro de 1253 pacientes com doença renal crônica submetidos a diálise, que foram observados entre os anos 2009-2011. / In survival analysis, usually the interest is to study the time until the occurrence of an event. When observations are subject to more than one type of event (e.g, different causes of death) and the occurrence of an event prevents the occurrence of the other, there is a competing risks structure. In some situations, nevertheless, the main interest is to study two events, one of which (terminal event) prevents the occurrence of the other (nonterminal event) but not vice versa. This structure is known as semicompeting risks, defined initially by Fine et al. (2001). In this work, we consider two approaches for analyzing data with this structure. One approach is based on the bivariate survival function through Archimedean copulas and estimators for the survival functions are obtained. The second approach is based on a process with three states, known as Illness-Death process, which can be specified by the transition intensity functions or risk functions. In this case, the inclusion of covariates and a possible dependence between the two times is taken into account by a shared frailty. These methodologies are applied to two data sets: the first one is a study with 137 patients with leukemia that received an allogeneic marrow transplant, with maximum follow up of 7 years; the second is a data set of 1253 patientswith chronic kidney disease on dialysis treatment, followed from 2009 until 2011.

One basic problem in statistical sciences is to understand the relationships among multivariate outcomes. Although it remains an important tool and is widely applicable, the regression analysis is limited by the basic setup that requires to identify one dimension of the outcomes as the primary measure of interest (the "dependent" variable) and other dimensions as supporting this variable (the "explanatory" variables). There are situations where this relationship is not of primary interest. For example, in actuarial sciences, one might be interested to see the dependence between annual claim numbers of a policyholder and its impact on the premium or the dependence between the claim amounts and the expenses related to them. In such cases the normality hypothesis fails, thus Pearson's correlation or concepts based on linearity are no longer the best ones to be used. Therefore, in order to quantify the dependence between non-normal outcomes one needs different statistical tools, such as, for example, the dependence concepts and the copulas. This thesis is devoted to modelling dependence with applications in actuarial sciences and is divided in two parts: the first one concerns dependence in frequency credibility models and the second one dependence between continuous outcomes. In each part of the thesis we resort to different tools, the stochastic orderings (which arise from the dependence concepts), and copulas, respectively. During the last decade of the 20th century, the world of insurance was confronted with important developments of the a posteriori tarification, especially in the field of credibility. This was dued to the easing of insurance markets in the European Union, which gave rise to an advanced segmentation. The first important contribution is due to Dionne & Vanasse (1989), who proposed a credibility model which integrates a priori and a posteriori information on an individual basis. These authors introduced a regression component in the Poisson counting model in order to use all available information in the estimation of accident frequency. The unexplained heterogeneity was then modeled by the introduction of a latent variable representing the influence of hidden policy characteristics. The vast majority of the papers appeared in the actuarial literature considered time-independent (or static) heterogeneous models. Noticeable exceptions include the pioneering papers by Gerber & Jones (1975), Sundt (1988) and Pinquet, Guillén & Bolancé (2001, 2003). The allowance for an unknown underlying random parameter that develops over time is justified since unobservable factors influencing the driving abilities are not constant. One might consider either shocks (induced by events like divorces or nervous breakdown, for instance) or continuous modifications (e.g. due to learning effect). In the first part we study the recently introduced models in the frequency credibility theory, which can be seen as models of time series for count data, adapted to actuarial problems. More precisely we will examine the kind of dependence induced among annual claim numbers by the introduction of random effects taking unexplained heterogeneity, when these random effects are static and time-dependent. We will also make precise the effect of reporting claims on the a posteriori distribution of the random effect. This will be done by establishing some stochastic monotonicity property of the a posteriori distribution with respect to the claims history. We end this part by considering different models for the random effects and computing the a posteriori corrections of the premiums on basis of a real data set from a Spanish insurance company. Whereas dependence concepts are very useful to describe the relationship between multivariate outcomes, in practice (think for instance to the computation of reinsurance premiums) one need some statistical tool easy to implement, which incorporates the structure of the data. Such tool is the copula, which allows the construction of multivariate distributions for given marginals. Because copulas characterize the dependence structure of random vectors once the effect of the marginals has been factored out, identifying and fitting a copula to data is not an easy task. In practice, it is often preferable to restrict the search of an appropriate copula to some reasonable family, like the archimedean one. Then, it is extremely useful to have simple graphical procedures to select the best fitting model among some competing alternatives for the data at hand. In the second part of the thesis we propose a new nonparametric estimator for the generator, that takes into account the particularity of the data, namely censoring and truncation. This nonparametric estimation then serves as a benchmark to select an appropriate parametric archimedean copula. This selection procedure will be illustrated on a real data set.

Les copulas archimédiennes hiérarchiques ont récemment gagné en intérêt puisqu’elles généralisent la famille de copules archimédiennes, car elles introduisent une asymétrie partielle. Des algorithmes d’échantillonnages et des méthodes ont largement été développés pour de telles copules. Néanmoins, concernant l’estimation par maximum de vraisemblance et les tests d’adéquations, il est important d’avoir à disposition la densité de ces variables aléatoires. Ce travail remplie ce manque. Après une courte introduction aux copules et aux copules archimédiennes hiérarchiques, une équation générale sur les dérivées des noeuds et générateurs internes apparaissant dans la densité des copules archimédiennes hiérarchique. sera dérivée. Il en suit une formule tractable pour la densité des copules archimédiennes hiérarchiques. Des exemples incluant les familles archimédiennes usuelles ainsi que leur transformations sont présentés. De plus, une méthode numérique efficiente pour évaluer le logarithme des densités est présentée. / Nested Archimedean copulas recently gained interest since they generalize the well-known class of Archimedean copulas to allow for partial asymmetry. Sampling algorithms and strategies have been well investigated for nested Archimedean copulas. However, for likelihood based inference such as estimation or goodness-of-fit testing it is important to have the density. The present work fills this gap. After a short introduction on copula and nested Archimedean copulas, a general formula for the derivatives of the nodes and inner generators appearing in nested Archimedean copulas is developed. This leads to a tractable formula for the density of nested Archimedean copulas. Various examples including famous Archimedean families and transformations of such are given. Furthermore, a numerically efficient way to evaluate the log-density is presented.

Les copulas archimédiennes hiérarchiques ont récemment gagné en intérêt puisqu’elles généralisent la famille de copules archimédiennes, car elles introduisent une asymétrie partielle. Des algorithmes d’échantillonnages et des méthodes ont largement été développés pour de telles copules. Néanmoins, concernant l’estimation par maximum de vraisemblance et les tests d’adéquations, il est important d’avoir à disposition la densité de ces variables aléatoires. Ce travail remplie ce manque. Après une courte introduction aux copules et aux copules archimédiennes hiérarchiques, une équation générale sur les dérivées des noeuds et générateurs internes apparaissant dans la densité des copules archimédiennes hiérarchique. sera dérivée. Il en suit une formule tractable pour la densité des copules archimédiennes hiérarchiques. Des exemples incluant les familles archimédiennes usuelles ainsi que leur transformations sont présentés. De plus, une méthode numérique efficiente pour évaluer le logarithme des densités est présentée. / Nested Archimedean copulas recently gained interest since they generalize the well-known class of Archimedean copulas to allow for partial asymmetry. Sampling algorithms and strategies have been well investigated for nested Archimedean copulas. However, for likelihood based inference such as estimation or goodness-of-fit testing it is important to have the density. The present work fills this gap. After a short introduction on copula and nested Archimedean copulas, a general formula for the derivatives of the nodes and inner generators appearing in nested Archimedean copulas is developed. This leads to a tractable formula for the density of nested Archimedean copulas. Various examples including famous Archimedean families and transformations of such are given. Furthermore, a numerically efficient way to evaluate the log-density is presented.