Statistics and Actuarial Science

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This is the collection for the University of Waterloo's Department of Statistics and Actuarial Science.

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A 2-class maintenance model with dynamic server behavior
(Springer, 2019-04-29) Granville, Kevin; Drekic, Steve
We analyze a 2-class maintenance system within a single-server polling model framework. There are C+f machines in the system, where C is the cap on the number of machines that can be turned on simultaneously (and hence, be at risk of failure), and the excess f machines comprise a maintenance float which can be used to replace machines that are taken down for repair. The server’s behavior is dynamic, capable of switching queues upon a machine failure or service completion depending on both queue lengths. This generalized server behavior permits the analysis of several classic service policies, including preemptive resume priority, non-preemptive priority, and exhaustive. More complicated polices can also be considered, such as threshold-based ones and a version of the Bernoulli service rule. The system is modeled as a level-dependent quasi-birth-and-death process and matrix analytic methods are used to find the steady-state joint queue length distribution, as well as the distribution for the sojourn time of a broken machine. An upper bound on the expected number of working machines as a function of C is derived, and Little’s Law is used to find the relationship between the expected number of working machines and the expected sojourn time of a failed machine when f=0 or f≥1. Several numerical examples are presented, including how one might optimize an objective function depending on the mean number of working machines, with penalty costs attributed to increasing C or f.
Actuarial Inference and Applications of Hidden Markov Models
(University of Waterloo, 2011-08-17T15:00:03Z) Till, Matthew Charles
Hidden Markov models have become a popular tool for modeling long-term investment guarantees. Many different variations of hidden Markov models have been proposed over the past decades for modeling indexes such as the S&P 500, and they capture the tail risk inherent in the market to varying degrees. However, goodness-of-fit testing, such as residual-based testing, for hidden Markov models is a relatively undeveloped area of research. This work focuses on hidden Markov model assessment, and develops a stochastic approach to deriving a residual set that is ideal for standard residual tests. This result allows hidden-state models to be tested for goodness-of-fit with the well developed testing strategies for single-state models. This work also focuses on parameter uncertainty for the popular long-term equity hidden Markov models. There is a special focus on underlying states that represent lower returns and higher volatility in the market, as these states can have the largest impact on investment guarantee valuation. A Bayesian approach for the hidden Markov models is applied to address the issue of parameter uncertainty and the impact it can have on investment guarantee models. Also in this thesis, the areas of portfolio optimization and portfolio replication under a hidden Markov model setting are further developed. Different strategies for optimization and portfolio hedging under hidden Markov models are presented and compared using real world data. The impact of parameter uncertainty, particularly with model parameters that are connected with higher market volatility, is once again a focus, and the effects of not taking parameter uncertainty into account when optimizing or hedging in a hidden Markov are demonstrated.
Actuarial Ratemaking in Agricultural Insurance
(University of Waterloo, 2015-08-06) Zhu, Wenjun
A scientific agricultural (re)insurance pricing approach is essential for maintaining sustainable and viable risk management solutions for different stakeholders including farmers, governments, insurers, and reinsurers. The major objective of this thesis is to investigate high dimensional solutions to refine the agricultural insurance and reinsurance pricing. In doing so, this thesis develops and evaluates three high dimensional approaches for constructing actuarial ratemaking framework for agricultural insurance and reinsurance, including credibility approach, high dimensional copula approach, and multivariate weighted distribution approach. This thesis comprehensively examines the ratemaking process, including reviews of different detrending methods and the generating process of the historical loss cost ratio's (LCR's, which is defined as the ratio of indemnities to liabilities). A modified credibility approach is developed based on the Erlang mixture distribution and the liability weighted LCR. In the empirical analysis, a comprehensive data set representing the entire crop insurance sector in Canada is used to show that the Erlang mixture distribution captures the tails of the data more accurately compared to conventional distributions. Further, the heterogeneous credibility premium based on the liability weighted LCR’s is more conservative, and provides a more scientific approach to enhance the reinsurance pricing. The agriculture sector relies substantially on insurance and reinsurance as a mechanism to spread loss. Climate change may lead to an increase in the frequency and severity of spatially correlated weather events, which could lead to an increase in insurance costs, or even the unavailability of crop insurance in some situations. This could have a profound impact on crop output, prices, and ultimately the ability to feed the world rowing population into the future. This thesis proposes a new reinsurance pricing framework, including a new crop yield forecasting model that integrates weather and crop production information from different risk geographically related regions, and closed form reinsurance pricing formulas. The framework is empirically analyzed, with an original weather index system we set up, and algorithms that combine screening regression (SR), cross validation (CV) and principle component analysis (PCA) to achieve efficient dimension reduction and model selection. Empirical results show that the new forecasting model has improved both in-sample and out-of-sample forecasting abilities. Based on this framework, weather risk management strategies are provided for agricultural reinsurers. Adverse weather related risk is a main source of crop production loss, and in addition to farmers, this exposure is a major concern to insurers and reinsurers who act as weather risk underwriters. To date, weather hedging has had limited success, largely due to challenges regarding basis risk. Therefore, this thesis develops and compares different weather risk hedging strategies for agricultural insurers and reinsurers, through investigating the spatial dependence and aggregation level of systemic weather risks across a country. In order to reduce basis risk and improve the efficiency of weather hedging strategies, this thesis refines the weather variable modeling by proposing a flexible time series model that assumes a general hyperbolic (GH) family for the margins to capture the heavy-tail property of the data, together with the Lévy subordinated hierarchical Archimedean copula (LSHAC) model to overcome the challenge of high-dimensionality in modeling the dependence of weather risk. Wavelet analysis is employed to study the detailed characteristics within the data from both time and frequency scales. Results show that it is of great importance of capturing the appropriate dependence structure of weather risk. Further, the results reveal significant geographical aggregation benefits in weather risk hedging, which means that more effective hedging may be achieved as the spatial aggregation level increases. It has been discussed that it is necessary to integrate auxiliary variables such as weather, soil, and other information into the ratemaking system to refine the pricing framework. In order to investigate a possible scientific way to reweight historical loss data with auxiliary variables, this thesis proposes a new premium principle based on multivariate weighted distribution. Some designable properties such as linearity and stochastic order preserving are derived for the new proposed multivariate weighted premium principle. Empirical analysis using a unique data set of the reinsurance experience in Manitoba from 2001 to 2011 compares different premium principles and shows that integrating auxiliary variables such as liability and economic factors into the pricing framework will redistribute premium rates by assigning higher loadings to more risky reinsurance contracts, and hence help reinsurers achieve more sustainable profits in the long term.
Adaptive policies and drawdown problems in insurance risk models
(University of Waterloo, 2015-08-31) Li, Shu
Ruin theory studies an insurer's solvency risk, and to quantify such a risk, a stochastic process is used to model the insurer's surplus process. In fact, research on ruin theory dates back to the pioneer works of Lundberg (1903) and Cramer (1930), where the classical compound Poisson risk model (also known as the Cramer-Lundberg model) was first introduced. The research was later extended to the Sparre Andersen risk model, the Markov arrival risk model, the Levy insurance risk model, and so on. However, in most analysis of the risk models, it is assumed that the premium rate per unit time is constant, which does not always reflect accurately the insurance environment. To better reflect the surplus cash flows of an insurance portfolio, there have been some studies (such as those related to dividend strategies and tax models) which allow the premium rate to take different values over time. Recently, Landriault et al. (2012) proposed the idea of an adaptive premium policy where the premium rate charged is based on the behaviour of the surplus process itself. Motivated by their model, the first part of the thesis focuses on risk models including certain adjustments to the premium rate to reflect the recent claim experience. In Chapter 2, we generalize the Gerber-Shiu analysis of the adaptive premium policy model of Landriault et al. (2012). Chapter 3 proposes an experience-based premium policy under the compound Poisson dynamic, where the premium rate changes are based on the increment between successive random review times. In Chapter 4, we examine a drawdown-based regime-switching Levy insurance model, where the drawdown process is used to model an insurer's level of financial distress over time, and to trigger regime-switching (or premium changes). Similarly to ruin problems which examine the first passage time of the risk process below a threshold level, drawdown problems relate to the first time that a drop in value from a historical peak exceeds a certain level (or equivalently the first passage time of the reflected process above a certain level). As such, drawdowns are fundamentally relevant from the viewpoint of risk management as they are known to be useful to detect, measure and manage extreme risks. They have various applications in many research areas, for instance, mathematical finance, applied probability and statistics. Among the common insurance surplus processes in ruin theory, drawdown episodes have been extensively studied in the class of spectrally negative Levy processes, or more recently, its Markov additive generalization. However, far less attention has been paid to the Sparre Andersen risk model, where the claim arrival process is modelled by a renewal process. The difficulty lies in the fact that such a process does not possess the strong Markov property. Therefore, in the second part of the thesis (Chapter 5), we extend the two-sided exit and drawdown analyses to a renewal risk process. In conclusion, the general focus of this thesis is to derive and analyze ruin-related and drawdown-related quantities in insurance risk models with adaptive policies, and assess their risk management impacts. Chapter 6 ends the thesis by some concluding remarks and directions for future research.
Algorithmic Analysis of a General Class of Discrete-based Insurance Risk Models
(University of Waterloo, 2013-08-28T13:59:23Z) Singer, Basil Karim
The aim of this thesis is to develop algorithmic methods for computing particular performance measures of interest for a general class of discrete-based insurance risk models. We build upon and generalize the insurance risk models considered by Drekic and Mera (2011) and Alfa and Drekic (2007), by incorporating a threshold-based dividend system in which dividends only get paid provided some period of good financial health is sustained above a pre-specified threshold level. We employ two fundamental methods for calculating the performance measures under the more general framework. The first method adopts the matrix-analytic approach originally used by Alfa and Drekic (2007) to calculate various ruin-related probabilities of interest such as the trivariate distribution of the time of ruin, the surplus prior to ruin, and the deficit at ruin. Specifically, we begin by introducing a particular trivariate Markov process and then expressing its transition probability matrix in a block-matrix form. From this characterization, we next identify an initial probability vector for the process, from which certain important conditional probability vectors are defined. For these vectors to be computed efficiently, we derive recursive expressions for each of them. Subsequently, using these probability vectors, we derive expressions which enable the calculation of conditional ruin probabilities and, from which, their unconditional counterparts naturally follow. The second method used involves the first claim conditioning approach (i.e., condition on knowing the time the first claim occurs and its size) employed in many ruin theoretic articles including Drekic and Mera (2011). We derive expressions for the finite-ruin time based Gerber-Shiu function as well as the moments of the total dividends paid by a finite time horizon or before ruin occurs, whichever happens first. It turns out that both functions can be expressed in elegant, albeit long, recursive formulas. With the algorithmic derivations obtained from the two fundamental methods, we next focus on computational aspects of the model class by comparing six different types of models belonging to this class and providing numerical calculations for several parametric examples, highlighting the robustness and versatility of our model class. Finally, we identify several potential areas for future research and possible ways to optimize numerical calculations.
Ambiguity aversion and optimal derivative-based pension investment with stochastic income and volatility
(Elsevier, 2018-03-01) Zeng, Yan; Li, Danping; Chen, Zheng; Yang, Zhou
This paper provides a derivative-based optimal investment strategy for an ambiguity-adverse pension investor who faces not only risks from time-varying income and market return volatility but also uncertain economic conditions over a long time horizon. We derive a robust dynamic derivative strategy and show that the optimal strategy under ambiguity aversion reduces the exposures to market return risk and volatility risk and that the investor holds opposite positions for the two risk exposures. In the presence of a derivative, ambiguity has distinct effects on the optimal investment strategy. More important, we demonstrate the utility improvement when considering ambiguity and exploiting derivatives and show that ambiguity aversion and derivative trading significantly improve utility when return volatility increases. This improvement becomes more significant under ambiguity aversion over a long investment horizon.
Analysis of a Threshold Strategy in a Discrete-time Sparre Andersen Model
(University of Waterloo, 2007-09-26T19:17:14Z) Mera, Ana Maria
In this thesis, it is shown that the application of a threshold on the surplus level of a particular discrete-time delayed Sparre Andersen insurance risk model results in a process that can be analyzed as a doubly infinite Markov chain with finite blocks. Two fundamental cases, encompassing all possible values of the surplus level at the time of the first claim, are explored in detail. Matrix analytic methods are employed to establish a computational algorithm for each case. The resulting procedures are then used to calculate the probability distributions associated with fundamental ruin-related quantities of interest, such as the time of ruin, the surplus immediately prior to ruin, and the deficit at ruin. The ordinary Sparre Andersen model, an important special case of the general model, with varying threshold levels is considered in a numerical illustration.
Analysis of Correlated Data with Measurement Error in Responses or Covariates
(University of Waterloo, 2010-09-30T18:17:34Z) Chen, Zhijian
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.
Analysis of duration data from longitudinal surveys subject to loss to follow-up
(University of Waterloo, 2010-09-30T21:07:06Z) Mariaca Hajducek, C. Dagmar
Data from longitudinal surveys give rise to many statistical challenges. They often come from a vast, heterogeneous population and from a complex sampling design. Further, they are usually collected retrospectively at intermittent interviews spaced over a long period of time, which gives rise to missing information and loss to follow-up. As a result, duration data from this kind of surveys are subject to dependent censoring, which needs to be taken into account to prevent biased analysis. Methods for point and variance estimation are developed using Inverse Probability of Censoring (IPC) weights. These methods account for the random nature of the IPC weights and can be applied in the analysis of duration data in survey and non-survey settings. The IPC estimation techniques are based on parametric estimating function theory and involve the estimation of dropout models. Survival distributions without covariates are estimated via a weighted Kaplan-Meier method and regression modeling through the Cox Proportional Hazards model and other models is based on weighted estimating functions. The observational frameworks from Statistics Canada's Survey of Labour and Income Dynamics (SLID) and the UK Millenium Cohort Study are used as motivation, and durations of jobless spells from SLID are analyzed as an illustration of the methodology. Issues regarding missing information from longitudinal surveys are also discussed.
Analysis of Financial Data using a Difference-Poisson Autoregressive Model
(University of Waterloo, 2011-05-17T17:40:20Z) Baroud, Hiba
Box and Jenkins methodologies have massively contributed to the analysis of time series data. However, the assumptions used in these methods impose constraints on the type of the data. As a result, difficulties arise when we apply those tools to a more generalized type of data (e.g. count, categorical or integer-valued data) rather than the classical continuous or more specifically Gaussian type. Papers in the literature proposed alternate methods to model discrete-valued time series data, among these methods is Pegram's operator (1980). We use this operator to build an AR(p) model for integer-valued time series (including both positive and negative integers). The innovations follow the differenced Poisson distribution, or Skellam distribution. While the model includes the usual AR(p) correlation structure, it can be made more general. In fact, the operator can be extended in a way where it is possible to have components which contribute to positive correlation, while at the same time have components which contribute to negative correlation. As an illustration, the process is used to model the change in a stock’s price, where three variations are presented: Variation I, Variation II and Variation III. The first model disregards outliers; however, the second and third include large price changes associated with the effect of large volume trades and market openings. Parameters of the model are estimated using Maximum Likelihood methods. We use several model selection criteria to select the best order for each variation of the model as well as to determine which is the best variation of the model. The most adequate order for all the variations of the model is $AR(3)$. While the best fit for the data is Variation II, residuals' diagnostic plots suggest that Variation III represents a better correlation structure for the model.
Analysis of interval-censored recurrent event processes subject to resolution
(Wiley, 2015-09) Cook, Richard J.; Shen, Hua
Interval-censored recurrent event data arise when the event of interest is not readily observed but the cumulative event count can be recorded at periodic assessment times. In some settings, chronic disease processes may resolve, and individuals will cease to be at risk of events at the time of disease resolution. We develop an expectation-maximization algorithm for fitting a dynamic mover-stayer model to interval-censored recurrent event data under a Markov model with a piecewise-constant baseline rate function given a latent process. The model is motivated by settings in which the event times and the resolution time of the disease process are unobserved. The likelihood and algorithm are shown to yield estimators with small empirical bias in simulation studies. Data are analyzed on the cumulative number of damaged joints in patients with psoriatic arthritis where individuals experience disease remission.
Analysis of Islamic Stock Indices
(University of Waterloo, 2009-04-29T19:14:04Z) Mohammed, Ansarullah Ridwan
In this thesis, an attempt is made to build on the quantitative research in the field of Islamic Finance. Firstly, univariate modelling using special GARCH-type models is performed on both the FTSE All World and FTSE Shari'ah All World indices. The AR(1) + APARCH(1,1) model with standardized skewed student-t innovations provided the best overall fit and was the most successful at VaR modelling for long and short trading positions. A risk assessment is done using the Conditional Tail Expectation (CTE) risk measure which concluded that in short trading positions the FTSE Shari'ah All World index was riskier than the FTSE All World index but, in long trading positions the results were not conclusive as to which is riskier. Secondly, under the Markowitz model of risk and return the performance of Islamic equity is compared to conventional equity using various Dow Jones indices. The results indicated that even though the Islamic portfolio is relatively less diversified than the conventional portfolio, due to several investment restrictions, the Shari'ah screening process excluded various industries whose absence resulted in risk reduction. As a result, the Islamic portfolio provided a basket of stocks with special and favourable risk characteristics. Lastly, copulas are used to model the dependency structure between the filtered returns of the FTSE All World and FTSE Shari'ah All World indices after fitting the AR(1) + APARCH(1,1) model with standardized skewed student-t innovations. The t copula outperformed the others and a demonstration of forecasting using the copula-extended model is done.
Analysis of Longitudinal Surveys with Missing Responses
(University of Waterloo, 2008-09-12T14:19:00Z) Carrillo Garcia, Ivan Adolfo
Longitudinal surveys have emerged in recent years as an important data collection tool for population studies where the primary interest is to examine population changes over time at the individual level. The National Longitudinal Survey of Children and Youth (NLSCY), a large scale survey with a complex sampling design and conducted by Statistics Canada, follows a large group of children and youth over time and collects measurement on various indicators related to their educational, behavioral and psychological development. One of the major objectives of the study is to explore how such development is related to or affected by familial, environmental and economical factors. The generalized estimating equation approach, sometimes better known as the GEE method, is the most popular statistical inference tool for longitudinal studies. The vast majority of existing literature on the GEE method, however, uses the method for non-survey settings; and issues related to complex sampling designs are ignored. This thesis develops methods for the analysis of longitudinal surveys when the response variable contains missing values. Our methods are built within the GEE framework, with a major focus on using the GEE method when missing responses are handled through hot-deck imputation. We first argue why, and further show how, the survey weights can be incorporated into the so-called Pseudo GEE method under a joint randomization framework. The consistency of the resulting Pseudo GEE estimators with complete responses is established under the proposed framework. The main focus of this research is to extend the proposed pseudo GEE method to cover cases where the missing responses are imputed through the hot-deck method. Both weighted and unweighted hot-deck imputation procedures are considered. The consistency of the pseudo GEE estimators under imputation for missing responses is established for both procedures. Linearization variance estimators are developed for the pseudo GEE estimators under the assumption that the finite population sampling fraction is small or negligible, a scenario often held for large scale population surveys. Finite sample performances of the proposed estimators are investigated through an extensive simulation study. The results show that the pseudo GEE estimators and the linearization variance estimators perform well under several sampling designs and for both continuous response and binary response.
Analysis of Multi-State Models with Mismeasured Covariates or Misclassified States
(University of Waterloo, 2015-05-22) He, Feng
Multi-state models provide a useful framework for estimating the rate of transitions between defined disease states and understanding the influence of covariates on transitions in studies of the disease progression. Statistical analysis of data from studies of disease progression often involves a number of challenges. A particular challenge is that the classification of the disease state may be subject to error. Another common problem is that there are many sources of heterogeneity in the data in which situation the assumption of time-homogeneous for common Markov models is not valid. In addition, it is common for discrete covariates subject to misclassification and the panel data collected from disease progression studies is time-dependence in the covariates. In Chapter 2, the progressive multi-state model with misclassification is developed to simultaneously estimate transition rates and account for potential misclassification. The performance of the maximum likelihood and pairwise likelihood estimators is evaluated by simulation studies. The proposed progressive model is illustrated on coronary allograft vasculopathy data, in which the diagnosis based on the coronary angiography is subject to error. In Chapter 3, hidden mover-stayer models are proposed to provide a solution to a type of heterogeneity where the population consists of both movers and stayers in the presence of misclassification. The likelihood inference procedure based on the EM algorithm is developed for the proposed model. The performance of the likelihood method is investigated through simulation studies. The proposed method is applied to the Waterloo Smoking Prevention Project. In Chapter 4, we propose estimation procedures for Markov models with binary covariates subject to misclassification. We show that the model is not identifiable under covariate misclassification. Consequently, we develop likelihood inference methods based on known reclassification probabilities and the main/validation study design. Simulation studies are conducted to investigate the performance of proposed methods and the consequence of the naive analysis which ignores the misclassification. In Chapter 5, we consider two-state Markov models where time-dependent surrogate covariates are available. We exploit both functional and structural inference methods to reduce or remove bias effects induced from covariate measurement error. The performance of proposed methods is investigated based on simulation studies.
Analysis of some risk models involving dependence
(University of Waterloo, 2010-08-12T15:51:27Z) Cheung, Eric C.K.
The seminal paper by Gerber and Shiu (1998) gave a huge boost to the study of risk theory by not only unifying but also generalizing the treatment and the analysis of various risk-related quantities in one single mathematical function - the Gerber-Shiu expected discounted penalty function, or Gerber-Shiu function in short. The Gerber-Shiu function is known to possess many nice properties, at least in the case of the classical compound Poisson risk model. For example, upon the introduction of a dividend barrier strategy, it was shown by Lin et al. (2003) and Gerber et al. (2006) that the Gerber-Shiu function with a barrier can be expressed in terms of the Gerber-Shiu function without a barrier and the expected value of discounted dividend payments. This result is the so-called dividends-penalty identity, and it holds true when the surplus process belongs to a class of Markov processes which are skip-free upwards. However, one stringent assumption of the model considered by the above authors is that all the interclaim times and the claim sizes are independent, which is in general not true in reality. In this thesis, we propose to analyze the Gerber-Shiu functions under various dependent structures. The main focus of the thesis is the risk model where claims follow a Markovian arrival process (MAP) (see, e.g., Latouche and Ramaswami (1999) and Neuts (1979, 1989)) in which the interclaim times and the claim sizes form a chain of dependent variables. The first part of the thesis puts emphasis on certain dividend strategies. In Chapter 2, it is shown that a matrix form of the dividends-penalty identity holds true in a MAP risk model perturbed by diffusion with the use of integro-differential equations and their solutions. Chapter 3 considers the dual MAP risk model which is a reflection of the ordinary MAP model. A threshold dividend strategy is applied to the model and various risk-related quantities are studied. Our methodology is based on an existing connection between the MAP risk model and a fluid queue (see, e.g., Asmussen et al. (2002), Badescu et al. (2005), Ramaswami (2006) and references therein). The use of fluid flow techniques to analyze risk processes opens the door for further research as to what types of risk model with dependency structure can be studied via probabilistic arguments. In Chapter 4, we propose to analyze the Gerber-Shiu function and some discounted joint densities in a risk model where each pair of the interclaim time and the resulting claim size is assumed to follow a bivariate phase-type distribution, with the pairs assumed to be independent and identically distributed (i.i.d.). To this end, a novel fluid flow process is constructed to ease the analysis. In the classical Gerber-Shiu function introduced by Gerber and Shiu (1998), the random variables incorporated into the analysis include the time of ruin, the surplus prior to ruin and the deficit at ruin. The later part of this thesis focuses on generalizing the classical Gerber-Shiu function by incorporating more random variables into the so-called penalty function. These include the surplus level immediately after the second last claim before ruin, the minimum surplus level before ruin and the maximum surplus level before ruin. In Chapter 5, the focus will be on the study of the generalized Gerber-Shiu function involving the first two new random variables in the context of a semi-Markovian risk model (see, e.g., Albrecher and Boxma (2005) and Janssen and Reinhard (1985)). It is shown that the generalized Gerber-Shiu function satisfies a matrix defective renewal equation, and some discounted joint densities involving the new variables are derived. Chapter 6 revisits the MAP risk model in which the generalized Gerber-Shiu function involving the maximum surplus before ruin is examined. In this case, the Gerber-Shiu function no longer satisfies a defective renewal equation. Instead, the generalized Gerber-Shiu function can be expressed in terms of the classical Gerber-Shiu function and the Laplace transform of a first passage time that are both readily obtainable. In a MAP risk model, the interclaim time distribution must be phase-type distributed. This leads us to propose a generalization of the MAP risk model by allowing for the interclaim time to have an arbitrary distribution. This is the subject matter of Chapter 7. Chapter 8 is concerned with the generalized Sparre Andersen risk model with surplus-dependent premium rate, and some ordering properties of certain ruin-related quantities are studied. Chapter 9 ends the thesis by some concluding remarks and directions for future research.
Analysis of Time Dependent Aggregate Claims
(University of Waterloo, 2016-07-22) Xu, Di
Estimation of aggregate claim amounts is a fundamental task in Actuarial science, based on which risk theory, ruin theory and reinsurance theory can be studied. Properties, including moments, Laplace transforms, and probability functions of aggregate claims have been extensively studied by many scholars under various models (see, e.g., Hogg and Klugman (1984)). The main classical model is the compound Poisson risk model, where the interclaim times are independent of the claim severities. Scholars started to explore this problem by considering more general counting processes, such as mixed Poisson processes (e.g., Willmot (1986)) and renewal processes (e.g., Andersen (1957)). Afterwards, the independence assumptions on multiple risk factors were gradually relaxed. Additionally, the observation times are further randomized to fit the reality better. In this thesis, we propose to analyze the aggregate claims until both randomized and deterministic time horizons by incorporating inflation and payment (reporting) delays into the analysis. Dependence between the claim occurrence times (also interclaim times) and claim severities is further considered. A comprehensive review on the study of the aggregate claims is given in Chapter 1. Chapter 2 introduces the relevant preliminary knowledge on the aggregate models and techniques used in this thesis. Chapter 3 examines the Laplace transforms of the aggregate claims under a nonhomogeneous birth process, which covers Poisson, mixed Poisson and linear contagion model. Furthermore, the claim occurrence times influence the distribution of the claim severities. Under some assumptions on the counting process, the time-dependent aggregate claims are represented as a random sum of independent and identically distributed random variables. The aggregate incurred but not reported (IBNR) claims are studied in Chapter 4 due to their essential role in reserving. A recursive formula is identified for the moments of the total discounted IBNR claims under a generalized renewal risk model where the interclaim times, claim severities and random reporting lags have an arbitrary dependence structure. The probability mass function of the number of IBNR claims is obtained under certain assumptions on the marginal distributions of the interclaim times, claim severities and reporting lags. To address the influence of the economic environment, a Markovian arrival process is introduced in Chapter 5 to analyze the IBNR claim problem. A straightforward representation and a closed-form expression are identified for the moments of the total discounted IBNR claim amount and numbers respectively without adding much difficulty to the analysis. Instead of a deterministic time horizon as considered in Chapters 3, 4 and 5, attention has also been paid to the analysis under a randomized observation time (see, e.g., Stanford et al. (2005) and Ramaswami et al. (2008)). Randomization in the time horizon usually leads to more tractable expressions for given quantities (e.g., Albrecher et al. (2011, 2013)). However, in the case of time-dependent aggregate claims, it only adds extra integration to the expressions of relevant quantities. In this thesis, instead of working with general random time horizons, we work with some specific random time horizons, i.e. two-sided exit time, in Chapter 6. The two-sided exit problem has been the subject matter of risk management analysis to better understand the dynamic of various insurance risk processes. In the two-sided exit setting, the discounted aggregate claims are investigated under a dependent renewal process (also known as dependent Sparre Andersen risk process). Utilizing Laplace transforms, we identify the fundamental solutions to a given integral equation, which will be shown to play a role similar to the scale matrix for spectrally-negative Markov-additive processes (e.g., Kyprianou and Palmowski (2008)). Explicit expressions and recursions are then identified for the two-sided probabilities and the moments of the aggregate claims respectively. Chapter 7 ends the thesis by some concluding remarks and directions for future research.
Application of Block Sieve Bootstrap to Change-Point detection in time series
(University of Waterloo, 2010-08-31T20:12:59Z) Zaman, Saad
Since the introduction of CUSUM statistic by E.S. Page (1951), detection of change or a structural break in time series has gained significant interest as its applications span across various disciplines including economics, industrial applications, and environmental data sets. However, many of the early suggested statistics, such as CUSUM or MOSUM, lose their effectiveness when applied to time series data. Either the size or power of the test statistic gets distorted, especially for higher order autoregressive moving average processes. We use the test statistic from Gombay and Serban (2009) for detecting change in the mean of an autoregressive process and show how the application of sieve bootstrap to the time series data can improve the performance of our test to detect change. The effectiveness of the proposed method is illustrated by applying it to economic data sets.
An Application of Matrix Analytic Methods to Queueing Models with Polling
(University of Waterloo, 2019-08-23) Granville, Kevin
We review what it means to model a queueing system, and highlight several components of interest which govern the behaviour of customers, as well as the server(s) who tend to them. Our primary focus is on polling systems, which involve one or more servers who must serve multiple queues of customers according to their service policy, which is made up of an overall polling order, and a service discipline defined at each queue. The most common polling orders and service disciplines are discussed, and some examples are given to demonstrate their use. Classic matrix analytic method theory is built up and illustrated on models of increasing complexity, to provide context for the analyses of later chapters. The original research contained within this thesis is divided into two halves, finite population maintenance models and infinite population cyclic polling models. In the first half, we investigate a 2-class maintenance system with a single server, expressed as a polling model. In Chapter 2, the model we study considers a total of C machines which are at risk of failing when working. Depending on the failure that a machine experiences, it is sorted into either the class-1 or class-2 queue where it awaits service among other machines suffering from similar failures. The possible service policies that are considered include exhaustive, non-preemptive priority, and preemptive resume priority. In Chapter 3, this model is generalized to allow for a maintenance float of f spare machines that can be turned on to replace a failed machine. Additionally, the possible server behaviours are greatly generalized. In both chapters, among other topics, we discuss the optimization of server behaviour as well as the limiting number of working machines as we let C go to infinity. As these are systems with a finite population (for a given C and f), their steady-state distributions can be solved for using the algorithm for level-dependent quasi-birth-and-death processes without loss of accuracy. When a class of customers are impatient, the algorithms covered in this thesis require their queue length to be truncated in order for us to approximate the steady-state distribution for all but the simplest model. In Chapter 4, we model a 2-queue polling system with impatient customers and k_i-limited service disciplines. Finite buffers are assumed for both queues, such that if a customer arrives to find their queue full then they are blocked and lost forever. Finite buffers are a way to interpret a necessary truncation level, since we can simply assume that it is impossible to observe the removed states. However, if we are interested in approximating an infinite buffer system, this inconsistency will bias the steady-state probabilities if blocking probabilities are not negligible. In Chapter 5, we introduce the Unobserved Waiting Customer approximation as a way to reduce this natural biasing that is incurred when approximating an infinite buffer system. Among the queues considered within this chapter is a N-queue system with exhaustive service and customers who may or may not be impatient. In Chapter 6, we extend this approximation to allow for reneging rates that depend on a customer's place in their queue. This is applied to a N-queue polling system which generalizes the model of Chapter 4.
Applications of claim investigation in insurance surplus and claims models
(University of Waterloo, 2018-01-03) Huynh, Mirabelle
Claim investigation is a fundamental part of an insurer's business. Queues form as claims accumulate and claims are investigated according to some queueing mechanism. The natural existence of queues in this context prompts the inclusion of a queue-based investigation mechanism to model features like congestion inherent in the claims handling process and further to assess their overall impact on an insurer's risk management program. This thesis explicitly models a queue-based claim investigation mechanism (CIM) in two classical models for insurance risk, namely, insurer surplus models (or risk models) and aggregate claim models (or loss models). Incorporating a queue-based CIM into surplus and aggregate claims models provides an additional degree of realism and as a result, can help insurers better characterize and manage risk. In surplus analysis, more accurate measures for ruin-related quantities of interest such as those relating to the time to ruin and the deficit at ruin can be developed. In aggregate claims models, more granular models of the claims handling process (e.g., by decomposing claims into those that are settled and those that have been reported but not yet settled) can help insurers target the source of inefficiencies in their processing systems and later mitigate their financial impact on the insurer. As a starting point, Chapter 2 proposes a simple CIM consisting of one server and no waiting places and superimposes this CIM onto the classical compound Poisson surplus process. An exponentially distributed investigation time is considered and then generalized to a combination of n exponentials. Standard techniques of conditioning on the first claim are used to derive a defective renewal equation (DRE) for the Gerber-Shiu discounted penalty function (or simply, the Gerber-Shiu function) m(u) and probabilistic interpretations for the DRE components are provided. The Gerber-Shiu function, introduced in Gerber and Shiu (1998), is a valuable analytical tool, serving as a unified means of risk analysis as it generates various ruin-related quantities of interest. Chapter 3 extends and generalizes the analysis in Chapter 2 by proposing a more complex CIM consisting of a single queue with n investigation units and a finite capacity of m claims. More precisely, we consider CIMs which admit a (spectrally negative) Markov Additive Process (MAP) formulation for the insurer's surplus and the analysis will heavily rely and benefit from recent developments in the fluctuation theory of MAPs. MAP formulations for four possible CIM generalizations are more specifically analyzed. Chapter 4 superimposes the more general CIM from Chapter 3 onto the aggregate claims process to obtain an "aggregate payment process". It is shown that this aggregate payment process has a Markovian Arrival Process formulation that is preserved under considerable generalizations to the CIM. A distributional analysis of the future payments due to reported but not settled claims ("RBNS payments") is then performed under various assumptions. Throughout the thesis, numerical analyses are used to illustrate the impact of variations in the CIM on the ruin probability (Chapters 2 and 3) and on the Value-at-Risk (VaR) and Tail-Value-at-Risk (TVaR) of RBNS payments (Chapter 4). Concluding remarks and avenues for further research are found in Chapter 5.
Applications of Geometry in Optimization and Statistical Estimation
(University of Waterloo, 2016-01-25) Maroufy, Vahed
Geometric properties of statistical models and their influence on statistical inference and asymptotic theory reveal the profound relationship between geometry and statistics. This thesis studies applications of convex and differential geometry to statistical inference, optimization and modelling. We, particularly, investigate how geometric understanding assists statisticians in dealing with non-standard inferential problems by developing novel theory and designing efficient computational algorithms. The thesis is organized in six chapters as it follows. Chapter 1 provides an abstract overview to a wide range of geometric tools, including affine, convex and differential geometry. It also provides the reader with a short literature review on the applications of geometry in statistical inference and exposes the geometric structure of commonly used statistical models. The contributions of this thesis are organized in the following four chapters, each of which is the focus of a submitted paper which is either accepted or under revision. Chapter 2 introduces a new parametrization to general family of mixture models of the exponential family. Despite the flexibility and popularity of mixture models, their associated parameter spaces are often difficult to represent due to fundamental identification problems. Other related problems include the difficulty of estimating the number of components, possible unboundedness and non-concavity of the log-likelihood function, non-finite Fisher information, and boundary problems giving rise to non-standard analysis. For instance, the order of a finite mixture is not well defined and often can not be estimated from a finite sample when components are not well separated, or some are not observed in the sample. We introduce a novel family of models, called the discrete mixture of local mixture models, which reparametrizes the space of general mixtures of the exponential family, in a way that the parameters are identifiable, interpretable, and, due to a tractable geometric structure, the space allows fast computational algorithms. This family also gives a well-defined characterization to the number of components problem. The component densities are flexible enough for fitting mixture models with unidentifiable components, and our proposed algorithm only includes the components for which there is enough information in the sample. Chapter 3 uses geometric concepts to characterize the parameter space of local mixture models (LMM), introduced in \cite{Marriott2002} as a local approximation to continuous mixture models. Although LMMs are shown to satisfy nice inferential properties, their parameter space is restricted by two types of boundaries, called the hard boundary and the soft boundary. The hard boundary guarantees that an LMM is a density function, while the soft boundary ensures that it behaves locally in a similar way to a mixture model. The boundaries are shown to have particular geometric structures that can be characterized by geometry of polytopes, ruled surface and developable surfaces. As working examples the LMM of a normal model and the LMM of a Poisson distribution are considered. The boundaries described in this chapter have both discrete aspects, (i.e. the ability to be approximated by polytopes), and smooth aspects (i.e. regions where the boundaries are exactly or approximately smooth). Chapter 4 uses the model space introduced in Chapter 2 for extending a prior model and defining a perturbation space in the Bayesian sensitivity analysis. This perturbation space is well-defined, tractable, and consistent with the elicited prior knowledge, the three properties that improve the methodology in \cite{Gustafson1996}. We study both local and global sensitivity in conjugate Bayesian models. In the local analysis the worst direction of sensitivity is obtained by maximizing the directional derivative of a functional between the perturbation space and the space of posterior expectations. For finding the maximum global sensitivity, however, two criteria are used; the divergence between posterior predictive distributions and the difference between posterior expectations. Both local and global analyses lead to optimization problems with a smooth boundary restriction. Chapter 5 studies Cox's proportional hazard model with an unobserved frailty for which no specific distribution is assumed. The likelihood function, which has a mixture structure with an unknown mixing distribution, is approximated by the model introduced in Chapter 2, which is always identifiable and estimable. The nuisance parameters in the approximating model, which represent the frailty distribution through its moments, lie in a convex space with a smooth boundary, characterized as a smooth manifold. Using differential geometric tools, a new algorithm is proposed for maximizing the likelihood function restricted by the smooth yet non-trivial boundary. The regression coefficients, the parameters of interest, are estimated in a two step optimization process, unlike the existed methodology in \cite{Klein1992} which assumes a gamma assumption and uses Expectation-Maximization approach. Simulation studies and data examples are also included, illustrating that the new methodology is promising as it returns small estimation bias; however, it produces larger standard deviation compared to the EM method. The larger standard deviation can be the result of using no information about the shape of the frailty model, while the EM model assumes the gamma model in advance; however, there are still ways to improve this methodology. Also, the simulation section and data analysis in this chapter is rather incomplete and more work needs to be done. Chapter 6 outlines a few topics as future directions and possible extensions to the methodologies developed in this thesis.