In practice there is a long tradition of actuaries calculating reserve estimates according to deterministic methods without explicit reference to a stochastic model. For instance, the chain-ladder was originally a deterministic reserving method. Moreover, the actuaries often make ad hoc adjustments of the methods, for example, smoothing of the chain-ladder development factors, in order to fit the data set under analysis. However, stochastic models are needed in order to assess the variability of the claims reserve. The standard statistical approach would be to first specify a model, then find an estimate of the outstanding claims under that model, typically by maximum likelihood, and finally the model could be used to find the precision of the estimate. As a compromise between this approach and the actuary's way of working without reference to a model the object of the research area has often been to first construct a model and a method that produces the actuary's estimate and then use this model in order to assess the uncertainty of the estimate. A drawback of this approach is that the suggested models have been constructed to give a measure of the precision of the reserve estimate without the possibility of changing the estimate itself. The starting point of this thesis is the inconsistency between the deterministic approaches used in practice and the stochastic ones suggested in the literature. On one hand, the purpose of Paper I is to develop a bootstrap technique which easily enables the actuary to use other development factor methods than the pure chain-ladder relying on as few model assumptions as possible. This bootstrap technique is then extended and applied to the separation method in Paper II. On the other hand, the purpose of Paper III is to create a stochastic framework which imitates the ad hoc deterministic smoothing of chain-ladder development factors which is frequently used in practice.

In many industrial contexts, such as aerospace applications or cars design, numerical prediction techniquesbecome more and more useful. They restrict the use of real prototypes to a minimum and make easier thedesign phase. In such industries and in the specific for acoustic, engineers are interested in computing theresponses of systems on frequency bands. In order to predict the vibration behavior of systems overfrequency bands, standard numerical techniques usually involve many frequency-fixed computations, atmany different frequencies. Although it is a straightforward and natural mean to answer to the posed problem,such a strategy can easily lead to huge computations, and the amount of data to store often increasessignificantly. This is particularly true in the context of medium frequency bands, where these responses havea strong sensitivity to the frequency. In this work PGD (Proper Generalized Decomposition), in a first time, isapplied to found a separate functional representation over frequency and space of the unknown amplitude ofVTCR (Variational Theory of Complex Rays) formulation on a reduced frequency space. This allows tocalculate an high quality mid-frequency response over a wide band without a fine frequency discretization,saving computational resources. Moreover the PGD representation of the solution allows to save a hugeamount of space in term of stored data. In a second time, PGD technique as been applied to extend itspeculiarity to mid-frequency wide band with uncertainty.

A tool path generation method for sculptured surfaces defined by triangular meshes is presented in this thesis along with an algorithm that helps determine the best type of cutter geometry to machine a specific surface. Existing tool path planning methods for sculptured surfaces defined by triangular meshes require extensive computer processing power and result in long processing times mainly since surface topology for triangular meshes is not provided. The method presented in this thesis avoids this problem by offsetting each triangular facet individually. The combination of all the individual offsets make up a cutter location surface. A single triangle offsetting results in many more triangles; many of these are redundant, increasing the time required for data handling in subsequent steps. To avoid the large number of triangles, the proposed method creates a bounding space to which the offset surface is limited. The original surface mesh describes the bounding surface of a solid, thus it is continuous with no gaps. Therefore, the resulting bounding spaces are also continuous and without gaps. Applying the boundary space limits the size of the offset surface resulting in a reduction in the number of triangular surfaces generated. The offset surface generation may result in unwanted intersecting triangles. The tool path planning strategy addresses this issue by applying hidden-surface removal algorithms. The cutter locations from the offset surface are obtained using the depth buffer. The simulation and machining results show that the tool paths generated by this process are correct. Furthermore, the time required to generate tool paths is less than the time required by other methods. The second part of this thesis presents a method for selecting an optimal cutter type. Extensive research has been carried out to determine the best cutter size for a given machining operation. However, cutter type selection has not been studied in-depth. This work presents a method for selecting the best cutter type based on the amount of material removed. By comparing the amount of material removed by two cutters at a given cutter location the best cutter can be selected. The results show that the optimal cutter is highly dependent on the surface geometry. For most complex surfaces it was found that a combination of cutters provides the best results.

In this thesis, we consider a generalization of the classical Gerber-Shiu function in various risk models. The generalization involves introduction of two new variables in the original penalty function including the surplus prior to ruin and the deficit at ruin. These new variables are the minimum surplus level before ruin occurs and the surplus immediately after the second last claim before ruin occurs. Although these quantities can not be observed until ruin occurs, we can still identify their distributions in advance because they do not functionally depend on the time of ruin, but only depend on known quantities including the initial surplus allocated to the business. Therefore, some ruin related quantities obtained by incorporating four variables in the generalized Gerber-Shiu function can help our understanding of the analysis of the random walk and the resultant risk management. In Chapter 2, we demonstrate the generalized Gerber-Shiu functions satisfy the defective renewal equation in terms of the compound geometric distribution in the ordinary Sparre Andersen renewal risk models (continuous time). As a result, forms of joint and marginal distributions associated with the variables in the generalized penalty function are derived for an arbitrary distribution of interclaim/interarrival times. Because the identification of the compound geometric components is difficult without any specific conditions on the interclaim times, in Chapter 3 we consider the special case when the interclaim time distribution is from the Coxian class of distribution, as well as the classical compound Poisson models. Note that the analysis of the generalized Gerber-Shiu function involving three (the classical two variables and the surplus after the second last claim) is sufficient to study of four variable. It is shown to be true even in the cases where the interclaim of the first event is assumed to be different from the subsequent interclaims (i.e. delayed renewal risk models) in Chapter 4 or the counting (the number of claims) process is defined in the discrete time (i.e. discrete renewal risk models) in Chapter 5. In Chapter 6 the two-sided bounds for a renewal equation are studied. These results may be used in many cases related to the various ruin quantities from the generalized Gerber-Shiu function analyzed in previous chapters. Note that the larger number of iterations of computing the bound produces the closer result to the exact value. However, for the nonexponential bound the form of bound contains the convolution involving usually heavy-tailed distribution (e.g. heavy-tailed claims, extreme events), we need to find the alternative method to reinforce the convolution computation in this case.

Correlated data frequently arise from epidemiological studies, especially familial and longitudinal studies. Longitudinal design has been used by researchers to investigate the changes of certain characteristics over time at the individual level as well as how potential factors influence the changes. Familial studies are often designed to investigate the dependence of health conditions among family members. Various models have been developed for this type of multivariate data, and a wide variety of estimation techniques have been proposed. However, data collected from observational studies are often far from perfect, as measurement error may arise from different sources such as defective measuring systems, diagnostic tests without gold references, and self-reports. Under such scenarios only rough surrogate variables are measured. Measurement error in covariates in various regression models has been discussed extensively in the literature. It is well known that naive approaches ignoring covariate error often lead to inconsistent estimators for model parameters. In this thesis, we develop inferential procedures for analyzing correlated data with response measurement error. We consider three scenarios: (i) likelihood-based inferences for generalized linear mixed models when the continuous response is subject to nonlinear measurement errors; (ii) estimating equations methods for binary responses with misclassifications; and (iii) estimating equations methods for ordinal responses when the response variable and categorical/ordinal covariates are subject to misclassifications. The first problem arises when the continuous response variable is difficult to measure. When the true response is defined as the long-term average of measurements, a single measurement is considered as an error-contaminated surrogate. We focus on generalized linear mixed models with nonlinear response error and study the induced bias in naive estimates. We propose likelihood-based methods that can yield consistent and efficient estimators for both fixed-effects and variance parameters. Results of simulation studies and analysis of a data set from the Framingham Heart Study are presented. Marginal models have been widely used for correlated binary, categorical, and ordinal data. The regression parameters characterize the marginal mean of a single outcome, without conditioning on other outcomes or unobserved random effects. The generalized estimating equations (GEE) approach, introduced by Liang and Zeger (1986), only models the first two moments of the responses with associations being treated as nuisance characteristics. For some clustered studies especially familial studies, however, the association structure may be of scientific interest. With binary data Prentice (1988) proposed additional estimating equations that allow one to model pairwise correlations. We consider marginal models for correlated binary data with misclassified responses. We develop “corrected” estimating equations approaches that can yield consistent estimators for both mean and association parameters. The idea is related to Nakamura (1990) that is originally developed for correcting bias induced by additive covariate measurement error under generalized linear models. Our approaches can also handle correlated misclassifications rather than a simple misclassification process as considered by Neuhaus (2002) for clustered binary data under generalized linear mixed models. We extend our methods and further develop marginal approaches for analysis of longitudinal ordinal data with misclassification in both responses and categorical covariates. Simulation studies show that our proposed methods perform very well under a variety of scenarios. Results from application of the proposed methods to real data are presented. Measurement error can be coupled with many other features in the data, e.g., complex survey designs, that can complicate inferential procedures. We explore combining survey weights and misclassification in ordinal covariates in logistic regression analyses. We propose an approach that incorporates survey weights into estimating equations to yield design-based unbiased estimators. In the final part of the thesis we outline some directions for future work, such as transition models and semiparametric models for longitudinal data with both incomplete observations and measurement error. Missing data is another common feature in applications. Developing novel statistical techniques for dealing with both missing data and measurement error can be beneficial.

In the field of education, decisions are influenced by the results of various high stakes measures. Investigating the presence of differential item functioning (DIF) in a set of items ensures that results from these measures are valid. For example, if an item measuring math self-efficacy is identified as having DIF then this indicates that some other characteristic (e.g. gender) other than the latent trait of interest may be affecting an examinee’s score on that particular item. The use of hierarchical generalized linear modeling (HGLM) enables the modeling of items nested within examinees, with person-level predictors added at level-2 for DIF detection. Unlike traditional DIF detection methods that require a reference and focal group, HGLM allows the modeling of a continuous person-level predictor. This means that instead of dichotomizing a continuous variable associated with DIF into a focal and reference group, the continuous variable can be added at level-2. Further benefits of HGLM are discussed in this study. This study is an extension of work done by Williams and Beretvas (2006) where the use of HGLM with polytomous items (PHGLM) for detection of DIF was illustrated. In the Williams and Beretvas study, the PHGLM was compared with the generalized Mantel-Haenszel (GMH), for DIF detection and it was found that the two performed similarly. A Monte Carlo simulation study was conducted to evaluate HGLM’s power to detect DIF and its associated Type 1 error rates using the constrained form of Muraki’s Rating Scale Model (Muraki, 1990) as the generating model. The two methods were compared when DIF was associated with a continuous variable which was dichotomized for the GMH and used as a continuous person-level predictor with PHGLM. Of additional interest in this study was the comparison of HGLM’s performance with that of the GMH under a variety of DIF and sample size conditions. Results showed that sample size, sample size ratio and DIF magnitude substantially influenced the power performance for both GMH and HGLM. Furthermore, the power performance associated with the GMH was comparable to HGLM for conditions with large sample sizes. The mean performance for both DIF detection methods showed good Type I error control.

It is an important step in clinical practice to discriminate real diseased patients from healthy persons. It would be great to get such discrimination from some common information like personal information, life style, and the contact with diseased patient. In this study, a score is calculated for each patient based on a survey through generalized linear model, and then the diseased status is decided according to previous sexually transmitted diseases (STDs) records. This study will facilitate clinics in grouping patients into real diseased or healthy, which in turn will affect the method clinics take to screen patients: complete screening for possible diseased patient and some common screening for potentially healthy persons.

Lorenz curves are extensively used in economics to analyze income inequality metrics. In this thesis, we discuss confidence interval estimation methods for generalized Lorenz curve. We first obtain normal approximation (NA) and empirical likelihood (EL) based confidence intervals for generalized Lorenz curves. Then we perform simulation studies to compare coverage probabilities and lengths of the proposed EL-based confidence interval with the NA-based confidence interval for generalized Lorenz curve. Simulation results show that the EL-based confidence intervals have better coverage probabilities and shorter lengths than the NA-based intervals at 100p-th percentiles when p is greater than 0.50. Finally, two real examples on income are used to evaluate the applicability of these methods: the first example is the 2001 income data from the Panel Study of Income Dynamics (PSID) and the second example makes use of households’ median income for the USA by counties for the years 1999 and 2006

As genomic sequencing technologies continue to advance, researchers are furthering their understanding of the relationships between genetic variants and expressed traits (Hirschhorn and Daly, 2005). However, missing data can significantly limit the power of a genetic study. Here, the use of a regularized generalized linear model, denoted GLMNET is proposed to impute missing genotypes. The method aimed to address certain limitations of earlier regression approaches in regards to genotype imputation, particularly multicollinearity among predictors. The performance of GLMNET-based method is compared to the performance of the phase-based method fastPHASE. Two simulation settings were evaluated: a sparse-missing model, and a small-panel expan- sion model. The sparse-missing model simulated a scenario where SNPs were missing in a random fashion across the genome. In the small-panel expansion model, a set of test individuals that were only genotyped at a small subset of the SNPs of the large panel. Each imputation method was tested in the context of two data-sets: Canadian Holstein cattle data and human HapMap CEU data. Although the proposed method was able to perform with high accuracy (>90% in all simulations), fastPHASE per- formed with higher accuracy (>94%). However, the new method, which was coded in R, was able to impute genotypes with better time efficiency than fastPHASE and this could be further improved by optimizing in a compiled language.

Multivariate generalized linear mixed models (MGLMM) are used for jointly modeling the clustered mixed outcomes obtained when there are two or more responses repeatedly measured on each individual in scientific studies. The relationship among these responses is often of interest. In the clustered mixed data, the correlation could be present between repeated measurements either within the same observer or between different observers on the same subjects. This study proposes a series of in- dices, namely, intra, inter and total correlation coefficients, to measure the correlation under various circumstances of observations from a multivariate generalized linear model, especially for joint modeling of clustered count and continuous outcomes. Bayesian methods are widely used techniques for analyzing MGLMM. The need for noninformative priors arises when there is insufficient prior information on the model parameters. Another aim of this study is to propose an approximate uniform shrinkage prior for the random effect variance components in the Bayesian analysis for the MGLMM. This prior is an extension of the approximate uniform shrinkage prior. This prior is easy to apply and is shown to possess several nice properties. The methods are illustrated in terms of both a simulation study and a case example.