This thesis proposes a new class of bivariate autoregressive conditional median duration models for matched high-frequency data and develops some inferential methods for an existing univariate model as well as the bivariate models introduced here to facilitate model fitting and forecasting. During the last two decades, the autoregressive conditional mean duration (ACD) model has been playing a dominant role in analyzing irregularly spaced high-frequency financial data. Univariate ACD models have been extensively discussed in the literature. However, some major challenges remain. The existing ACD models do not provide a good distributional fit to financial durations, which are right-skewed and often exhibit unimodal hazard rates. Birnbaum-Saunders (BS) distribution is capable of modeling a wide variety of positively skewed data. Median is not only a robust measure of central tendency, but also a natural scale parameter of the BS distribution. A class of conditional median duration models, the BS-ACD and the scale-mixture BS ACD models based on the BS, BS power-exponential and Student-t BS (BSt) distributions, have been suggested in the literature to improve the quality of the model fit. The BSt-ACD model is more flexible than the BS-ACD model in terms of kurtosis and skewness. In Chapter 2, we develop the maximum likelihood estimation method for the BSt-ACD model. The estimation is performed by utilizing a hybrid of optimization algorithms. The performance of the estimates is then examined through an extensive Monte Carlo simulation study. We also carry out model discrimination using both likelihood-based method and information-based criterion. Applications to real trade durations and comparison with existing alternatives are then made. The bivariate version of the ACD model has not received attention due to non-synchronicity. Although some bivariate generalizations of the ACD model have been introduced, they do not possess enough flexibility in modeling durations since they are conditional mean-based and do not account for non-monotonic hazard rates. Recently, the bivariate BS (BVBS) distribution has been developed with many desirable properties and characteristics. It allows for unimodal shapes of marginal hazard functions. In Chapter 3, upon using this bivariate BS distribution, we propose the BVBS-ACD model as a natural bivariate extension of the BS-ACD model. It enables us to jointly analyze matched duration series, and also capture the dependence between the two series. The maximum likelihood estimation of the model parameters and associated inferential methods have been developed. A Monte Carlo simulation study is then carried out to examine the performance of the proposed inferential methods. The goodness-of-fit and predictive performance of the model are also discussed. A real bivariate duration data analysis is provided to illustrate the developed methodology. The bivariate Student-t BS (BVBSt) distribution has been introduced in the literature as a robust extension of the BVBS distribution. It provides greater flexibility in terms of the kurtosis and skewness through the inclusion of an additional shape parameter. In Chapter 4, we propose the BVBSt-ACD model as a natural extension of the BSt-ACD model to the bivariate case. We then discuss the maximum likelihood estimation of the model parameters. A simulation study is carried out to investigate the performance of these estimators. Model discrimination is then done by using information-based criterion. Methods for evaluating the goodness-of-fit and predictive ability of the model are also discussed. A simulated data example is used to illustrate the proposed model as compared to the BVBS-ACD model. Finally, in Chapter 5, some concluding comments are made and also some problems for future research are mentioned. / Thesis / Master of Science (MSc)

Survival analytic methods help to analyze lifetime data arising from medical and reliability experiments. The popular proportional hazards model, proposed by Cox (1972), is widely used in survival analysis to study the effect of risk factors on lifetimes. An important assumption in regression type analysis is that all relative risk factors should be included in the model. However, not all relative risk factors are observed due to measurement difficulty, inaccessibility, cost considerations, and so on. These unobservable risk factors can be modelled by the so-called frailty model, originally introduced by Vaupel et al. (1979). Furthermore, the frailty model is also applicable to clustered data. Cluster data possesses the feature that observations within the same cluster share similar conditions and environment, which are sometimes difficult to observe. For example, patients from the same family share similar genetics, and patients treated in the same hospital share the same group of profes- sionals and same environmental conditions. These factors are indeed hard to quantify or measure. In addition, this type of similarity introduces correlation among subjects within clusters. In this thesis, a semi-parametric frailty model is proposed to address aforementioned issues. The baseline hazards function is approximated by a piecewise constant function and the frailty distribution is assumed to be a Birnbaum-Saunders distribution. Due to the advancement in modern medical sciences, many diseases are curable, which in turn leads to the need of incorporating cure proportion in the survival model. The frailty model discussed here is further extended to a mixture cure rate frailty model by integrating the frailty model into the mixture cure rate model proposed originally by Boag (1949) and Berkson and Gage (1952). By linking the covariates to the cure proportion through logistic/logit link function and linking observable covariates and unobservable covariates to the lifetime of the uncured population through the frailty model, we obtain a flexible model to study the effect of risk factors on lifetimes. The mixture cure frailty model can be reduced to a mixture cure model if the effect of frailty term is negligible (i.e., the variance of the frailty distribution is close to 0). On the other hand, it also reduces to the usual frailty model if the cure proportion is 0. Therefore, we can use a likelihood ratio test to test whether the reduced model is adequate to model the given data. We assume the baseline hazard to be that of Weibull distribution since Weibull distribution possesses increasing, constant or decreasing hazard rate, and the frailty distribution to be Birnbaum-Saunders distribution. D ́ıaz-Garc ́ıa and Leiva-Sa ́nchez (2005) proposed a new family of life distributions, called generalized Birnbaum-Saunders distribution, including Birnbaum-Saunders distribution as a special case. It allows for various degrees of kurtosis and skewness, and also permits unimodality as well as bimodality. Therefore, integration of a generalized Birnbaum-Saunders distribution as the frailty distribution in the mixture cure frailty model results in a very flexible model. For this general model, parameter estimation is carried out using a marginal likelihood approach. One of the difficulties in the parameter estimation is that the likelihood function is intractable. The current technology in computation enables us to develop a numerical method through Monte Carlo simulation, and in this approach, the likelihood function is approximated by the Monte Carlo method and the maximum likelihood estimates and standard errors of the model parameters are then obtained numerically by maximizing this approximate likelihood function. An EM algorithm is also developed for the Birnbaum-Saunders mixture cure frailty model. The performance of this estimation method is then assessed by simulation studies for each proposed model. Model discriminations is also performed between the Birnbaum-Saunders frailty model and the generalized Birnbaum-Saunders mixture cure frailty model. Some illustrative real life examples are presented to illustrate the models and inferential methods developed here. / Thesis / Doctor of Science (PhD)

Cure rate models, introduced by Boag (1949), are very commonly used while modelling lifetime data involving long time survivors. Applications of cure rate models can be seen in biomedical science, industrial reliability, finance, manufacturing, demography and criminology. In this thesis, cure rate models are discussed under a competing cause scenario, with the assumption of proportional odds (PO) lifetime distributions for the susceptibles, and statistical inferential methods are then developed based on right-censored data. In Chapter 2, a flexible cure rate model is discussed by assuming the number of competing causes for the event of interest following the Conway-Maxwell (COM) Poisson distribution, and their corresponding lifetimes of non-cured or susceptible individuals can be described by PO model. This provides a natural extension of the work of Gu et al. (2011) who had considered a geometric number of competing causes. Under right censoring, maximum likelihood estimators (MLEs) are obtained by the use of expectation-maximization (EM) algorithm. An extensive Monte Carlo simulation study is carried out for various scenarios, and model discrimination between some well-known cure models like geometric, Poisson and Bernoulli is also examined. The goodness-of-fit and model diagnostics of the model are also discussed. A cutaneous melanoma dataset example is used to illustrate the models as well as the inferential methods. Next, in Chapter 3, the destructive cure rate models, introduced by Rodrigues et al. (2011), are discussed under the PO assumption. Here, the initial number of competing causes is modelled by a weighted Poisson distribution with special focus on exponentially weighted Poisson, length-biased Poisson and negative binomial distributions. Then, a damage distribution is introduced for the number of initial causes which do not get destroyed. An EM-type algorithm for computing the MLEs is developed. An extensive simulation study is carried out for various scenarios, and model discrimination between the three weighted Poisson distributions is also examined. All the models and methods of estimation are evaluated through a simulation study. A cutaneous melanoma dataset example is used to illustrate the models as well as the inferential methods. In Chapter 4, frailty cure rate models are discussed under a gamma frailty wherein the initial number of competing causes is described by a Conway-Maxwell (COM) Poisson distribution in which the lifetimes of non-cured individuals can be described by PO model. The detailed steps of the EM algorithm are then developed for this model and an extensive simulation study is carried out to evaluate the performance of the proposed model and the estimation method. A cutaneous melanoma dataset as well as a simulated data are used for illustrative purposes. Finally, Chapter 5 outlines the work carried out in the thesis and also suggests some problems of further research interest. / Thesis / Doctor of Philosophy (PhD)

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Boreal forests are particularly vulnerable to climate change, experiencing a much more drastic increase in temperatures and having a limited amount of more northern refugia. The trees making up these vast and important ecosystems already had to adapt previously to environmental pressures brought about by the repeated glaciations during past ice ages. Studying the patterns of adaption of these trees can thus provide valuable insights on how to mitigate future damage. This thesis presents and analyses population structure, demo- graphic history and the distribution of fitness effects (DFE) of the diploid Betula pendula and tetraploid B. pubescens across Scandinavia. Birches–being widespread in boreal forests as well as having great economical importance–constitute superb model species. The analyses of this work confirm the expectations on postglacial population expansion and diploid-tetraploid introgression. They furthermore ascertain the presence of two genetic clusters and a remarkably similar DFE for the species. This work also contributes with a transparent, reproducible and reusable pipeline which facilitates running similar analyses for related species.