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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Dynamic Treatment Regimes for Within- and Between-Group Interference in Clustered and Hierarchical Datasets
(University of Waterloo, 2025-09-22) Mossman, Alexandra
Precision medicine is an interdisciplinary field that aims to tailor treatments based on an individual’s unique characteristics. Dynamic treatment regimes (DTRs) formalize this process through step-by-step decision rules that utilize patient-specific information at each stage of analysis and subsequently output recommendations for the optimal course of action. To date, much of the biostatistical literature has analyzed DTR estimation under the assumption of no interference; that is, a given individual’s outcome is not affected by the treatments received by others. However, this assumption is often violated in a variety of social and spatial networks, such as households and communities, and particularly in the context of infectious diseases or resource allocation. Although some recent developments have been made for DTR estimation for couples in households and in networks where general forms of interference are taking place, it has yet to be shown how DTRs can be estimated for individuals in clustered networks where interference may occur within and between predefined groups. Specifically, our attention shifts toward hierarchical networks, which provide a framework where interference can occur both within and between groups in the same hierarchy but not across hierarchies. This thesis contains three main projects that contribute to DTR estimation in the context of service utilization within the healthcare system and under interference networks. In Chapter 3, we show how the dynamic weighted ordinary least squares regression (dWOLS) DTR methodology can be applied to determine whether a patient would benefit from being discharged to home or admitted to a hospital using data simulated from a retrospective cohort study for patients who experienced an opioid-related overdose in British Columbia. This project was inspired from collaboration with a provincial health authority in Canada, which has resulted in the draft of a manuscript that can be submitted for publication following an ethics approval. In Chapter 4, we demonstrate an application of dWOLS to hierarchical networks where both within- and between-group interference takes place, and through a series of simulation studies, we show how incorrectly assuming that there only exists within-group interference can result in lower rates of optimal treatments assigned, particularly as the number of subgroups (and thus the potential for between-group interference) increases. Our models assess the performance of “standard” overlap weights constructed from logistic mixed models with nested random intercepts as well as network weights that are constructed from a joint propensity score. We apply our findings to the simulated opioid overdose dataset, where our goal is to illustrate how sequences of recommended dispositions for opioid overdose patients can be optimized if interference occurs within and between different facilities. Finally, in Chapter 5, we propose several sets of simulation studies to assess the extent to which interference may be taking place between individuals to warrant use of modified dWOLS methodology that accounts for interference.
Variable Selection and Prediction for Multistate Processes under Complex Observation Schemes
(University of Waterloo, 2025-09-19) Li, Xianwei
This thesis addresses variable selection and prediction in time-to-event analysis under complex observation schemes that commonly arise in biomedical studies. Such schemes may lead to right-censored data, interval-censored event times, or dual-censoring scenarios. Across three main chapters, we develop variable selection methods for multistate processes, address challenges arising from incomplete data under complex observation schemes, and investigate the implications of model misspecification, such as using simpler models in place of multistate models, and the potential risks of violating assumptions on covariate effects estimation and predictive performance. We begin with considering the problem of variable selection for progressive multistate processes under intermittent observation in Chapter 2. This study is motivated by the need to identify which among a large list of candidate markers play a role in the progression of joint damage in psoriatic arthritis (PsA) patients. We adopted a penalized log-likelihood approach and developed an innovative Expectation-Maximization (EM) algorithm such that the maximization step can exploit existing software for penalized Poisson regression thereby enabling flexible use of common penalty functions. Simulation studies show good performance in identifying important markers with different penalty functions. We applied the algorithm in the motivating application involving a cohort of patients with psoriatic arthritis with repeated assessments of joint damage, and identified human leukocyte antigen (HLA) markers which are associated with disease progression, among a large group of candidate markers. Chapter 3 extends this algorithm to more general multistate processes, and to more complex observation schemes. We consider the classical illness-death model which offers a useful framework for studying the progression of chronic disease while jointly modeling death. The exact time of disease progression is not observed directly but progression status is recorded at intermittent assessment times; the time to death is subject to right-censoring. This creates a dual observation scheme where progression times are interval-censored and survival times are subject to right censoring. A penalized observed data likelihood approach is proposed which allows for separate penalties across different intensity functions. An EM algorithm is again developed to facilitate use of different penalties for variable selection on disease progression and death through penalized Poisson regression. This adaptation retains the flexibility to exploit existing software with commonly used penalty functions. Simulation studies show good finite-sample performance in variable selection with different combination of penalty functions. We also explored how various aspects of the variable selection algorithm affect performance such as use of nonparametric baseline intensities and different ways to select the optimal tuning parameter(s). An application to data from the National Alzheimer’s Coordinating Center (NACC) demonstrates the use of our method in jointly modeling dementia progression and mortality. Chapter 4 builds on insights from Chapters 2 and 3 by investigating how simpler marginal methods targeting entry time to the absorbing state (e.g., a Cox proportional hazards model) compared to full multistate models. Here we retain use of the illness-death process as the basis of the investigation, but consider settings where transition times are only right-censored. We first study the limiting values of regression estimators from a Cox proportional hazards model when the data generating process is based on a Markov illness-death model. The potential impact of modeling the multistate processes based on a misspecified model is also investigated by considering cases where a) important covariates are omitted, or b) the Markov assumption is violated. We then examine the implications of model misspecification when the goal is prediction - this is done by evaluating the predictive performance of a misspecified Cox regression model for overall survival and a misspecified Fine-Gray model for disease progression, and comparing their respective predictive performance against that of the true illness-death model. We find that the limiting value of regression coefficients estimators obtained from Cox models and Fine-Gray models depend on several factors, including the baseline hazard ratio of death between the intermediate and initial states, the probability of moving through the intermediate state, and covariate effects on all transitions. However, the corresponding predictive accuracy is not substantially compromised despite biases in the regression coefficient estimators in most scenarios we investigated. The limiting value of regression coefficients obtained from a Markov illness-death model and the corresponding predictive accuracy are sensitive to model misspecification such as omitting important covariates and violation of the Markov assumption. The practical implications are illustrated using a dataset of patients with metastatic breast cancer in the control arm to predict overall survival and fracture risk. Chapter 5 reviews the contributions of this thesis and discusses problems warranting future research.
Insurance Pricing, and Optimal Strategy Design under Distributional Model Uncertainty
(University of Waterloo, 2025-09-05) Shi, Ziyue
The objective of this thesis is to develop theoretically sound and practically applicable solutions to two optimal reinsurance models, and to investigate the properties of the premium with background risk. Specifically, Chapter 1 introduces model uncertainty into the optimal reinsurance problem, while Chapter 2 focuses on solving the optimal reinsurance problem within the framework of a performance-based variable premium. Chapter 3 examines the properties of the indifference premium for an insurable risk in the presence of a background risk, considering both scenarios with and without model uncertainty. In Chapter 2, we explore a non-cooperative optimal reinsurance problem incorporating likelihood ratio uncertainty, aiming to minimize the worst-case risk of the total retained loss for the insurer. We establish a general relation between the optimal reinsurance strategy under the reference probability measure and the solution in the worst-case scenario. This relation can be generalized to insurance design problems quantified by tail risk measures. We provide a sufficient and necessary condition for when the problem using reference measure has at least one common optimal solution with its worst-case counterpart. As an application of this relation, optimal policies for the worst-case scenario quantified by the expectile risk measure are determined. Additionally, we explore the corresponding cooperative problem and compare its value function with that of the non-cooperative model. In the literature, insurance and reinsurance pricing is typically determined by a premium principle, characterized by a risk measure that reflects the policy seller's risk attitude. Building on the work of Meyers (1980) and Chen et al. (2016), Chapter 3 propose a new performance-based variable premium scheme for reinsurance policies, where the premium depends on both the distribution of the ceded loss and the actual realized loss. Under this scheme, the insurer and the reinsurer face a random premium at the beginning of the policy period. Based on the realized loss, the premium is adjusted into either a "reward" or "penalty" scenario, resulting in a discount or surcharge at the end of the policy period. We characterize the optimal reinsurance policy from the insurer's perspective under this new variable premium scheme. In addition, we formulate a Bowley optimization problem between the insurer and the monopoly reinsurer. Numerical examples demonstrate that, compared to the expected-value premium principle, the reinsurer prefers the variable premium scheme as it reduces the reinsurer's total risk exposure. Chapter 4 investigates the properties of the indifference premium for an insurable risk in the present of a background risk. We establish conditions that guarantee the uniqueness of the premium and provide general properties of the indifference premium without imposing specific assumptions on the dependence structure between the insurable risk and the background risk. Furthermore, we demonstrate that the dependence structure between these risks significantly affects the behavior of the indifference premium. In particular, we analyze the indifference premium under various dependence structures, including independence, stochastic monotonicity, and mutual exclusivity. We further explore the impact of model uncertainty on the indifference premium by deriving equivalent conditions for the dependence structures between the insurable and the background risks and their corresponding uniform random variables. Finally, we illustrate our findings using expected utility preference operators as representative examples.
Spatio-temporal methods for applications in biological and environmental studies
(University of Waterloo, 2025-09-02) He, Shiyu
With advances in technology over the past few decades, the availability and resolution of spatial and spatio-temporal data have increased dramatically. As a result, many research fields now face new analytical and computational challenges in effectively utilizing such complex data. This thesis develops spatio-temporal models within the Bayesian framework to address practical problems in epidemiology, engineering, and environmental science. In Chapter 2, we develop a Bayesian hierarchical model to investigate the temporal and spatial evolution of spike protein sequences of SARS-CoV-2, which serve as critical targets for vaccines and neutralizing antibodies. To reduce dimensionality and facilitate interpretation, the sequences are grouped into representative clusters based on their similarity. The robustness of the model is demonstrated through simulation studies. We then apply the model to real-world sequence data and the results uncover clear geographical differences in the spread of SARS-CoV-2 variants. In Chapter 3, we propose a Bayesian hierarchical model to address the challenge of spatial misalignment in spatio-temporal data using the INLA-SPDE approach. The research is motivated by the study of harmful algal bloom (HAB) events in environmental science, where the convention is to conduct separate analyses based on either in situ samples or satellite images. Our methodology combines the different data sources in a “fusion” model via the construction of projection matrices in both spatial and temporal domains. Simulation studies demonstrate that the proposed fusion model generally outperforms standalone models in both parameter estimation and predictive accuracy. This fusion framework thus represents an important step toward a unified characterization of bloom dynamics. In Chapter 4, we investigate the effects of the spatial arrangement of knots on the flexural properties of lumber crossarms using Bayesian methods. The proposed frameworks integrate a deterministic analysis of induced mechanical stress with a Bayesian modeling approach that captures uncertainties in the failure mechanisms. To further enhance model flexibility, we extend the framework to a Bayesian mixture model, which enables the identification and quantification of multiple underlying failure mechanisms observed in lumber testing.
Bayesian Integral-based Methods for Differential Equation Models
(University of Waterloo, 2025-08-28) Xu, Mingwei
Ordinary differential equations (ODEs) are widely considered for modeling the dynamics of complex systems across various scientific areas. When inferring the parameters of ODEs from noisy observations, most existing parameter estimation methods that bypass numerical integration tend to rely on basis functions or Gaussian processes to approximate the ODE solution and its derivatives. Due to the sensitivity of the ODE solution to its derivatives, these methods can be hindered by estimation error, especially when only sparse time-course observations are available. Furthermore, in high-dimensional sparse ODE systems, existing methods for structure identification are predominantly frequentist, and uncertainty quantification for trajectory estimation remains a significant challenge. In this thesis, we present a Bayesian collocation framework that operates on the integrated form of the ODEs and also avoids the expensive use of numerical solvers. In contrast to frequentist methods, the Bayesian framework provides better quantification of uncertainty. Specifically, we propose three methods to recover ODE systems with both known and unknown functional forms, as well as for model validation in systems with known functions. First, for ODEs with known functional forms, our methodology can handle general nonlinear systems to infer parameters from noisy observations. We then extend the approach to ODEs with unknown functional forms to identify system structure. Under the additive ODE model assumption, we develop a unified framework that combines the likelihood, integrated ODE constraints, and a group-wise sparse penalty, enabling simultaneous system identification and trajectory estimation. Finally, for cases involving several competing functional forms that exhibit similar dynamics, we incorporate model inadequacy as a function into the measurement model, allowing simultaneous model validation and parameter estimation. We demonstrate the favourable performance of our proposed methods, compared to existing ones, in terms of estimation accuracy and model validation. We also illustrate the methods with real data analyses.
Probabilistic Methods of Parameter Inference for Differential Equations
(University of Waterloo, 2025-08-26) Wu, Mo Han
Parameter estimation for differential equations is a fundamental problem in many scientific fields. This thesis is concerned with developing statistically and computationally efficient methods of parameter inference for ordinary differential equations (ODEs) and stochastic differential equations (SDEs). For ODEs, this often involves numerically solving the differential equation at each likelihood evaluation, a task traditionally handled with deterministic numerical solvers. Here we consider likelihood approximations based on probabilistic numerical solvers, which have been shown to produce more reliable parameter estimates by better accounting for numerical uncertainty. In contrast, parameter inference for SDEs requires integrating over high-dimensional latent variables in state-space models, which is computationally expensive. We propose a variational inference framework that significantly reduces this computational burden by converting the integration problem into an optimization problem. Chapter 2 is a review of several existing probabilistic ODE solvers and associated parameter inference methods. In particular, we detail the commonly-used paradigm of approximating the ODE solution with a nonlinear state-space model, and then linearizing it to easily perform relevant computations via the Kalman filtering and smoothing recursions. In the data-free setting, we extend convergence results previously established only for the forward pass of the Kalman algorithm to the backward pass as well. This is a key result for establishing the convergence of several associated parameter learning methods. We provide empirical evidence that supports our theory and demonstrate that the backward pass estimator is more accurate than using a forward pass alone. We also propose a novel variation of a probabilistic method for parameter inference. In Chapter 3, we present a novel probabilistic approximation to the ODE likelihood that reduces the parameter sensitivity inherent in the true likelihood by directly learning from noisy observations. Leveraging the efficient Kalman filter algorithm, our method scales linearly in both ODE variables and time discretization points. Furthermore, it is applicable to ODE problems with partially unobserved components and arbitrary measurement noise. Several numerical experiments demonstrate that our method produces more reliable estimates when compared to other probabilistic methods and, in extremely sensitive problems, exhibits greater robustness than the exact ODE likelihood. In Chapter 4, we introduce a scalable stochastic variational inference framework for estimating parameters in SDE models. Our method effectively captures the forward-backward information propagation in state-space models by using a recurrent neural network (RNN) to estimate the quantities needed for the forward-backward recursions. The procedure scales linearly with the number of SDE discretization steps. Experimental results show that it produces more reliable parameter posteriors than a number of competing variational and non-variational methods, particularly in high-dimensional random effects models.
Assessment, Mitigation, and Backtesting of Extreme Risks in Insurance and Finance
(University of Waterloo, 2025-08-25) Zhao, Yimiao
In this thesis, we focus on the quantitative assessment, mitigation, and backtesting of extreme risks in insurance and finance. We adopt tools from extreme value theory (EVT), dependence modeling, and statistical testing to address fundamental challenges in managing low-probability but high-impact events. The research contributes to three key aspects of extreme risk management: systemic risk assessment, catastrophe risk mitigation, and risk model evaluation. We begin with the assessment of systemic risk under extreme scenarios. Systemic events are characterized by strong dependence among individual entities and heavy-tailed risk behavior. In Chapter 3, we develop an asymptotic framework for systemic risk measures, covering both Value-at-Risk-based and expectile-based risk measures. Second-order asymptotic approximations are derived to improve the accuracy of risk quantification beyond conventional first-order results. Special attention is given to expectile-based systemic risk measures, which provide a more conservative assessment of systemic risk. The limitation of diversification for heavy-tailed risks has been well documented in the literature, especially for extremely heavy-tailed risks with infinite first moment. However, in practical insurance markets, catastrophic risks often exhibit extremely heavy tails but are also subject to truncation due to limited liability or policy design. In Chapter 4, we investigate the effectiveness of catastrophe risk pooling under these realistic constraints. Building on EVT, we characterize conditions under which diversification benefits remain achievable. The analysis incorporates tail heaviness, loss scaling, liability structure, and risk-sharing rules, providing theoretical foundations and practical guidance for catastrophe risk management. In Chapter 5, we turn to the problem of risk model evaluation through backtesting. Conventional backtesting methods often rely on strict model assumptions and may fail under model misspecification, structural change or dependence uncertainty. Building on existing model-free testing approaches using e-values and e-processes, we extend these ideas to develop a backtesting framework for identifiable and elicitable risk measures, including Value-at-Risk, Expected Shortfall, and expectiles. The proposed framework delivers valid statistical inference for both standard and comparative backtests, and supports robust risk assessment across a wide range of risk levels, not limited to extreme events. Throughout the thesis, theoretical results are complemented by extensive simulation studies and real-world applications. These findings contribute to advancing the theoretical foundations and practical methodologies for extreme risk management in modern insurance and financial systems.
Risk Sharing in Monopolistic Insurance Markets: Hidden Types and Bowley-Optimal Pricing.
(University of Waterloo, 2025-08-21) Shi, Benxuan
This thesis investigates optimal insurance design in a monopolistic market under both complete and incomplete information. The motivation stems from real-world insurance settings where insurers may either fully understand policyholders’ risk characteristics or must infer them from observable behavior. We develop theoretical frameworks to model both environments and derive analytical characterizations of optimal contracts under varying assumptions. In the first part, we analyze a classical full-information setting in which the insurer knows both the policyholder’s loss distribution and risk preferences. The policyholder is assumed to be risk-averse, modeled by an increasing and concave utility function, while the insurer is risk-neutral. In a monopolistic market, the insurer, as the sole contract provider, holds significant influence over both the structure and pricing of insurance contracts. We study the impact of contract forms—such as deductibles and coinsurance—on the insurer’s optimal pricing strategy, which we express through a \textit{loading function} drawn from a class of increasing and convex functions. A central concept introduced in this framework is the \textit{Bowley solution}, which captures the sequential nature of decision-making between the insurer and the policyholder. We relate this framework to foundational literature, particularly \cite{chan1985reinsurer}. Our analysis shows that linear loading functions (yielding expected-value premiums) are optimal under coinsurance, while piecewise linear functions (aligned with stop-loss premiums) are optimal under deductible contracts. The second part retains the full-information assumption but departs from traditional convex pricing rules. Instead, we introduce ambiguity in risk assessment by distorting the probability measure using a distortion function, reflecting subjective or behavioral risk perceptions. Symmetrically, the policyholder evaluates contracts using a distortion risk measure rather than expected utility. We retain the Bowley sequential structure but relax restrictions on the contract form, assuming only that indemnity schedules are uniformly Lipschitz continuous—an assumption that helps address moral hazard. Under this generalized framework, we find that full insurance becomes optimal when the policyholder is strictly risk-averse. If the policyholder evaluates risk using Value-at-Risk (VaR), the optimal contract becomes a policy limit contract with a sharp pricing distortion aligned with the VaR confidence level. For policyholders with inverse-S-shaped distortion functions (common in behavioral models), the optimal contract takes a deductible form, and the insurer’s distortion partially mirrors the policyholder’s up to a key threshold. These results offer insight into how non-linear transformations of risk perception shape contract design. In the third part, we consider an incomplete information setting in which the insurer cannot observe a policyholder’s risk attitude. We model heterogeneity using Yaari’s dual utility theory, parameterizing preferences via a continuum of distortion functions indexed by a type parameter $\theta$. This setup introduces adverse selection: policyholders may misreport their type to secure better terms. To address this, the insurer must design a menu of contracts—each pairing a specific indemnity schedule and premium—to ensure \textit{individual rationality} (voluntary participation) and \textit{incentive compatibility} (truthful type revelation). We formulate the insurer’s profit maximization problem subject to these constraints and apply tools from mechanism design and contract theory to characterize the optimal solution. Under suitable assumptions, we find that the optimal menu consists of layered contracts with desirable properties: the most risk-averse types receive full insurance (a property known as efficiency at the top), and both coverage and pricing increase with the degree of risk aversion. The least risk-averse type is indifferent between participating and opting out, while the insurer extracts strictly positive profit from more risk-averse individuals. We also examine how the optimal menu is affected by the introduction of a fixed participation cost. In this case, the insurer chooses to withdraw part of the menu, excluding contracts targeted at the least risk-averse individuals. Additionally, we study an alternative objective in which the insurer designs an \textit{incentive-efficient} menu—one that incorporates policyholder welfare alongside profit. We show that the layered structure remains optimal in this setting and provide a detailed characterization of the associated properties of the incentive-efficient contract menu. Overall, this thesis contributes to the theoretical foundations of insurance economics in monopolistic markets and provides insights into the design and pricing of insurance contracts under both complete and asymmetric information.
Statistical Inference in ROC Curve Analysis
(University of Waterloo, 2025-07-07) Hu, Dingding
The receiver operating characteristic (ROC) curve is a powerful statistical tool to evaluate the diagnostic abilities of a binary classifier for varied discrimination thresholds. It has been widely applied in various scientific areas. This thesis considers three inference problems in the ROC curve analysis. In Chapter 1, we introduce the basic concept of the ROC curve, along with some of its summary indices. We then provide an overview of the research problems and outline the structure of the subsequent chapters. Chapter 2 focuses on improving the ROC curve analysis with a single biomarker by incorporating the assumption that higher biomarker values indicate greater disease severity or likelihood. We interpret “greater severity” as a higher probability of disease, which corresponds to the likelihood ratio ordering between diseased and healthy individuals. Under this assumption, we propose a Bernstein polynomial-based method to model and estimate the biomarker distributions using the maximum empirical likelihood framework. From the estimated distributions, we derive the ROC curve and its summary indices. We establish the asymptotic consistency of our estimators and validate their performance through extensive simulations and compare them with existing methods. A real-data example is used to demonstrate the practical applicability of our approach. Chapter 3 considers the ROC curve analysis for medical data with non-ignorable missingness in the disease status. In the framework of the logistic regression models for both the disease status and the verification status, we first establish the identifiability of model parameters, and then propose a likelihood method to estimate the model parameters, the ROC curve, and the area under the ROC curve (AUC) for the biomarker. The asymptotic distributions of these estimators are established. Via extensive simulation studies, we compare our method with competing methods in the point estimation and assess the accuracy of confidence interval estimation under various scenarios. To illustrate the application of the proposed method in practical data, we apply our method to the Alzheimer's disease dataset from the National Alzheimer's Coordinating Center. Chapter 4 explores the combination of multiple biomarkers when disease status is determined by an imperfect reference standard, which can lead to misclassification. Previous methods for combining multiple biomarkers typically assume that all disease statuses are determined by a gold standard test, limiting their ability to accurately estimate the ROC curve and AUC in the presence of misclassification. We propose modeling the distributions of biomarkers from truly healthy and diseased individuals using a semiparametric density ratio model. Additionally, we adopt two assumptions from the literature: (1) the biomarkers are conditionally independent of the classification of the imperfect reference standard given the true disease status, and (2) the classification accuracy of the imperfect reference standard is known. Using this framework, we establish the identifiability of model parameters and propose a maximum empirical likelihood method to estimate the ROC curve and AUC for the optimal combination of biomarkers. An Expectation-Maximization algorithm is developed for numerical calculation. Additionally, we propose a bootstrap method to construct the confidence interval for the AUC and the confidence band for the ROC curve. Extensive simulations are conducted to evaluate the robustness of our method with respect to label misclassification. Finally, we demonstrate the effectiveness of our method in a real-data application. In Chapter 5, we provide a brief summary of Chapters 2-4 and outline several directions for future research.
Causal Inference in the Presence of Heterogeneous Treatment Effects
(University of Waterloo, 2025-07-07) Liang, Wei
Causal inference has been widely accepted as a statistical tool in various areas for demystifying causality from data. Treatment effect heterogeneity is a common issue in causal inference which refers to variation in the causal effect of a treatment across different subgroups or individuals within a population. This thesis explores three topics in causal inference in the presence of heterogeneous treatment effects, aiming to provide some insights for this critical issue. Chapter 2 introduces basic notation, frameworks, models, and parameters in causal inference, serving as preliminary material for the three topics studied in Chapters 3 - 5, with a focus on the Rubin causal model. In Chapter 3, we discuss the first topic: causal inference with survey data. In the presence of heterogeneous treatment effects, a causal conclusion based on sample data may not generalize to a broader population if selection bias exists. We propose estimators for population average treatment effects by incorporating survey weights into the propensity score weighting approach to simultaneously mitigate confounding bias and selection bias. A robust sandwich variance estimator is developed to permit valid statistical inference for the population-level causal parameters under a proposed "two-phase randomization model" framework. The proposed estimators and associated inferential procedure are shown to be robust against model misspecifications. We further extend our results to observational non-probability survey samples and demonstrate how to combine auxiliary population in- formation from multiple external reference probability samples for more reliable estimation. We illustrate our proposed methods through Monte Carlo simulation studies and the analysis of a real-world survey dataset. Chapter 4 explores the second topic: estimation of treatment harm rate (THR), the proportion of individuals in a population who are negatively affected by a treatment. The THR is a measure of treatment risk and reveals the treatment effect heterogeneity within a subpopulation. However, the measure is generally non-identifiable even when the treatments are randomly assigned, and existing works focus primarily on the estimation of the THR under either untestable identification or ambiguous model assumptions. We develop a class of partitioning-based bounds for the THR with data from randomized controlled trials with two distinct features: Our proposed bounds effectively use available auxiliary covariates information and they can be consistently estimated without relying on any untestable or ambiguous model assumptions. Our methods are motivated from a key observation that the sharp bounds of the THR can be attained under a partition of the covariates space with at most four cells. Probabilistic classification algorithms are employed to estimate nuisance parameters to realize the partitioning. The resulting interval estimators of the THR are model-assisted in the sense that they are highly efficient when the underlying models are well fitted, while their validity relies solely on the randomization of the trials. Finite sample performances of our proposed interval estimators along with a conservatively extended confidence interval for the THR are evaluated through Monte Carlo simulation studies. An application of the proposed methods to the ACTG 175 data is presented. A Python package named partbte for the partitioning-based algorithm has been developed and is available on https://github.com/w62liang/partition-te. Chapter 5 investigates the third topic: causal mediation analysis in randomized controlled trials with noncompliance. The average causal mediation effect (ACME) and the natural direct effect (NDE) are two parameters of primary interest in causal mediation analysis. However, the two causal parameters are not identifiable in randomized controlled trials in the presence of mediator-outcome confounding and assignment-treatment noncompliance. In such scenarios, we explore partial identification of parameters and derive nonparametric bounds on the ACME and the NDE when the treatment assignment serves as an instrumental variable. The nonparametric sharp bounds for the local causal parameters defined on the subpopulation of treatment-assignment compliers are also provided. We demonstrate the practical application of the proposed bounds through an empirical analysis of a large-scale randomized online advertising dataset. The thesis concludes in Chapter 6 with a brief summary and discussions of future work. Technical details, including the proofs of key propositions and theorems as well as additional simulation results, are provided at the end of each chapter.
Resilient Machine Learning Approaches for Fast Risk Evaluation and Management in Financial Portfolios and Variable Annuities
(University of Waterloo, 2025-05-22) Li, Xintong
Risk management of financial derivatives and actuarial products is intricate and often requires modeling the underlying stochasticity with Monte Carlo simulations. Monte Carlo simulation is flexible and can easily adapt to changes in model assumptions and market conditions. However, as multiple sources of risk are considered over long time horizons, the simulation model becomes complex and time-consuming to run. Tremendous research effort has been dedicated to designing computationally efficient machine learning-based procedures that mitigate the computational burden of a standard simulation procedure. In machine learning, model flexibility comes at the expense of model resilience, which is crucial for risk management tasks. This study considers estimating tail risks of complex financial and actuarial products with resilient machine learning-based nested simulation procedures. We propose a novel metamodeling approach that integrates deep neural networks within a nested simulation framework for efficient risk estimation. Our approaches offer substantial improvements over the associated standard simulation procedures. This study also illustrates how to build and assess resilient machine learning models for different problem complexities and different data structures, qualities, and quantities. To further enhance adaptability to new variable annuity contracts and changing market conditions, this thesis explores transfer learning techniques. By reusing and fine-tuning pre-trained metamodels, the proposed approach accelerates the adaptation process to different contract features and evolving market dynamics without retraining models from scratch. Transfer learning improves computational efficiency and enhances the robustness and flexibility of neural network metamodels in dynamic hedging of variable annuities. Extensive numerical experiments in this thesis demonstrate that the proposed methods substantially improve computational efficiency, sometimes shortening runtime by orders of magnitude compared to standard nested simulation procedures. The results indicate that deep neural network metamodels with transfer learning can quickly adapt to new market scenarios and contract specifications. This research contributes to the advancement of risk management practices for complex actuarial products and financial derivatives. By leveraging advanced machine learning techniques, this thesis offers a practical and scalable solution for insurers to perform timely and accurate risk assessments. The integration of long short-term memory metamodels and transfer learning into a nested simulation framework represents a major step forward toward more efficient, adaptable, and robust methodologies in actuarial science and quantitative finance.
Statistical Analyses of Lumber Strength Properties and a Likelihood-Free Method using Empirical Likelihood
(University of Waterloo, 2025-04-29) Yang, Yunfeng
Wood materials should meet expected strength and reliability standards for safe and stable construction. The strength of lumber and wood products may degrade over time due to sustained applied stresses, a phenomenon known as the duration-of-load (DOL) effect. The inherent variability of lumber, combined with DOL, makes structural reliability analyses particularly challenging. This thesis develops statistical methodologies to address these challenges, focusing on reliability analysis, wood strength modeling, and likelihood-free inference. Chapter 2 evaluates the reliability of lumber, accounting for the DOL effect under different load profiles based on a multimodel Bayesian framework. Three individual DOL models previously used for reliability assessment are considered: the US model, the Canadian model, and the Gamma process model. Procedures for stochastic generation of residential, snow, and wind loads are also described. We propose Bayesian model-averaging (BMA) as a method for combining the reliability estimates of individual models under a given load profile that coherently accounts for statistical uncertainty in the choice of model and parameter values. The method is applied to the analysis of a Hemlock experimental dataset, where the BMA results are illustrated via estimated reliability indices together with 95% interval bands. Chapter 3 explores proof-loading experiments, another industrial procedure for ensuring lumber reliability and quality, besides the DOL experiment from Chapter 2. In proof-loading, a pre-determined load is applied to remove weak specimens, but this may also weaken the surviving specimens (survivors) — a phenomenon we term the damage effect. To capture and assess this effect, we propose a statistical framework that includes a damage model and a likelihood ratio test, offering advantages over existing methods by directly quantifying the damage effect. When applied to experimental data, the proposed framework successfully detects and measures the damage effect while showing good model fit. The framework also provides correlation estimates between strength properties, potentially reducing monitoring costs in industry. Chapter 4 investigates statistical models with intractable likelihoods, such as the Canadian model discussed in Chapter 2. To address the challenge they pose to parameter inference, various likelihood-free methods have been developed, including a recently proposed synthetic empirical likelihood (SEL) approach. We introduce a new SEL estimator based on the reparametrization trick, which greatly reduces the computational burden. The asymptotic property of our SEL estimator is derived for the situation where the number of parameters equals the number of summary statistics, leading to a method that is not only faster, but also yields more accurate uncertainty quantification than conventional MCMC. The SEL approach is further extended by incorporating exponential tilting, which empirically improves performance when summary statistics outnumber parameters. Simulation studies validate the robustness and efficiency of our approach across various scenarios.
Deep Learning Frameworks for Anomaly Detection in Time Series and Graphs with Limited Labels
(University of Waterloo, 2025-04-29) Chen, Jiazhen
Anomaly detection involves identifying patterns or behaviors that substantially differ from normal instances in a dataset. It has a wide range of applications in diverse fields such as cybersecurity, manufacturing, finance, and e-commerce. However, real-world anomaly detection often grapples with two main challenges: label scarcity, as anomalies are rare and hard to label, and the complexity of data structures, which can involve intricate dependencies that require careful analysis. In this thesis, we develop deep learning frameworks designed to work effectively for label-free or extremely limited labeling scenarios, with a focus on time series anomaly detection (TSAD) and graph anomaly detection (GAD). To overcome the issue of label scarcity, our initial work investigates unsupervised TSAD methods that extract meaningful patterns from abundant unlabeled data. Building on recent advances in contrastive learning from NLP and computer vision, we introduce the Contrastive Neural Transformation (CNT) framework. This approach integrates temporal contrastive learning with neural transformations to capture context-aware and discriminative patterns effectively. Moreover, the dual-loss formulation prevents representation collapse by avoiding reliance on negative samples - a common challenge in anomaly detection, where the majority of instances represent normal behavior. While capturing temporal context is essential, understanding inter-series relationships is equally important in multivariate TSAD. Anomalies may seem normal in isolation but reveal abnormal patterns when compared to other series. To address this, we introduce DyGraphAD, a dynamic graph-driven dual forecasting framework that models both intra- and inter-series dependencies through a combination of graph and time series forecasting tasks. This allows anomalies to be detected via significant forecasting errors in both the channel-wise and time-wise dimensions. To further enhance computational efficiency, we propose an alternative framework, termed Prospective Multi-Graph Cohesion (PMGC). PMGC leverages graph structure learning to model inter-series relationships in a task-specific manner, reducing computational load compared to manual sequential graph construction in DyGraphAD. Furthermore, it introduces a multi-graph cohesion mechanism to adaptively learn both long-term dependencies and diverse short-term relationships. A prospective graphing strategy is also introduced to encourage the model to capture concurrent inter-series relationships, reducing reliance solely on historical data. Beyond TSAD, GAD is also critical due to its prevalence in numerous applications. Graphs provide structural information alongside node and edge attributes, and understanding the interplay between graph structure and attributes is essential for uncovering subtle anomalies not apparent when examining nodes or edges alone. Given that obtaining labeled data is relatively more feasible in graphs than in time series data for experimental purposes, we focus on GAD settings with limited labeling, more reflecting practical real-world scenarios. Specifically, we make the first attempt to address GAD in cross-domain few-shot settings, aiming to detect anomalies in a sparsely labeled target graph by leveraging a related but distinct source graph. To handle domain shifts, our CDFS-GAD framework incorporates a domain-adaptive graph contrastive learning and a domain-specific prompt tuning, aiming to align the features across two domains while preserving unique characteristics tailored to each domain. A domain-adaptive hypersphere classification loss and a self-training phase are introduced to further refine predictions in the target domain exploiting the limited labeling information. In addition to static graphs, many real-world applications involve dynamic graph data, where both the structure and attributes evolve over time. This adds complexity to anomaly detection, as both temporal and structural variations must be accounted for. Moreover, obtaining sufficient labeled data remains challenging, and related-domain labeled data may not be available in certain scenarios. To tackle the two more practical issues, we propose the EL$^2$-DGAD framework, specifically designed for detecting anomalies in dynamic graphs in extremely labeled conditions. This framework enhances model robustness through a transformer-based temporal graph encoder that captures evolving patterns from local and global perspectives. An ego-context hypersphere classification loss is further introduced to adjust the anomaly detection boundary contextually under limited supervision, supplemented by an ego-context contrasting module to improve generalization with unlabeled data. Overall, this thesis tackles anomaly detection for two commonly used data types, addressing unsupervised, semi-supervised, and cross-domain few-shot scenarios to meet the demands of real-world applications. Our extensive experiments show that the proposed frameworks perform well against various benchmark datasets and competitive anomaly detection baselines.
Contributions to Change Point and Functional Data Analysis
(University of Waterloo, 2025-04-29) VanderDoes, Jeremy
The advent and progression of computers has led to consideration of data previous considered too unwieldy. So called high-dimensional, or big, data can be considered large in both the size of observations and the number of observations. In this thesis, we consider such data which may be infinite dimensional and is often collected over some dimension, such as time. Methodology for detection of changes and exploration of this information-rich data is explored. Chapter 1 provides a review of concepts and notation used throughout the thesis. Topics related to time series, functional data, and change point analysis are of particular interest and form the foundation of the thesis. The chapter concludes with an overview of the main contributions contained in the thesis. An empirical characteristic functional-based method for detecting distributional change points in functional time series is presented in chapter 2. Although various methods exist to detect changes in functional time series, they typically require projection or are tuned to specific changes. The characteristic functional-based approach is fully functional and sensitive to general changes in the distribution of functional time series. Integrated- and supremum-type test statistics are proposed. Theoretical considerations for the test statistics are examined, including asymptotic distributions and the measure used to integrate the test statistic over the function space. Simulation, permutation, and approximation approaches to calibrate detection thresholds for the test statistics are investigated. Comparisons to existing methods are conducted via simulation experiments. The proposed methods are applied to continuous electricity prices and high-frequency asset returns. Chapter 3 is devoted to graph-based change point detection. Graph-based approaches provide another method for detecting distributional changes in functional time series. Four test statistics and their theoretical properties are discussed. Extensive simulations provide context for graph-based tuning parameter choices and compare the approaches to other functional change point detection methods. The efficacy of graph-based change point detection is demonstrated on multi-year pedestrian counts, high-resolution stock returns, and continuous electricity prices. Despite increased interest in functional time series, available implementations are largely missing. Practical considerations for applying functional change point detection are covered in chapter 4. We present fChange, a functional time series package in R. The package combines and expands functional time series and change point methods into an easy-to-use format. The package provides functionality to store and process data, summarize and validate assumptions, characterize and perform inference of change points, and provide visualizations. The data are stored as discretely observed observations, promoting usability and accuracy. Applications to continuous electricity prices, cancer mortality, and long-term treasury rates are shown. In chapter 5, we propose novel methodology for analyzing tumor microenvironments (TMEs) in cancer research. TMEs contain vast amounts of information on patient's cancer through their cellular composition and the spatial distribution of tumor cells and immune cell populations. We present an approach to explore variation in TMEs, and determine the extent to which this information can predict outcomes such as patient survival or treatment success. Our approach can identify specific interactions which are useful in such predictions. We use spatial $K$ functions to summarize interactions, and then apply a functional random forest-based model. This approach is shown to be effective in simulation experiments at identifying important spatial interactions while also controlling the false discovery rate. We use the proposed approach to interrogate two real data sets of Multiplexed Ion Beam Images of TMEs in triple negative breast cancer and lung cancer patients. The publicly available companion R package funkycells is discussed. The random coefficient autoregressive model of order 1, RCA(1), is a model well-suited for volatile time series. Detection of changes between stable and explosive regimes of scalar data modeled with the RCA(1) is explored in chapter 6. We derive a (maximally selected) likelihood ratio statistic and show that it has power versus breaks occurring even as close as O(\log \log N) periods from the beginning/end of sample. Moreover, the use of quasi maximum likelihood-based estimates yields better power properties, with the added bonus of being nuisance-free. Our test statistic has the same distribution - of the Darling-Erd\H{o}s type - irrespective of whether the data are stationary or not, and can therefore be applied with no prior knowledge on this. Our simulations show that the test has very good power and, when applying a suitable correction to the asymptotic critical values, the correct size. We illustrate the usefulness and generality of our approach through applications to economic and epidemiological time series. Chapter 7 provides summaries and discussions on each chapter. Directions for future work are considered. These directions, with the provided commentary, extend the scope of the models and may behoove practitioners and researchers alike.
Robust Methods and Model Selection for Causal Inference under Missingness of the Study Exposure
(University of Waterloo, 2025-04-28) Shi, Yuliang
The goal of this thesis is to develop new robust methods for the estimation of the causal effect and propose a model selection algorithm when the exposure variable is not fully observed. We mainly discuss the methods using propensity score (PS) and imputation approaches to address both missingness and confounding issues in the observational dataset. How to deal with missing data in observational studies is a common concern for causal inference. However, if the exposure is missing at random (MAR), few approaches are available, and careful adjustments on both missingness and confounding issues are required to ensure a consistent estimate of the true causal effect on the response. In Chapter 2, a new inverse probability weighting (IPW) estimator based on weighted estimating equations (WEE) is proposed to incorporate weights from both the missingness and PS models, which can reduce the joint effect of extreme weights in finite samples. Additionally, we develop a triply robust (TR) estimator via WEE to protect against the misspecification of the missingness model. The asymptotic properties of WEE estimators are shown using properties of estimating equations. Based on simulation studies, WEE methods outperform others, including imputation-based approaches, in terms of bias and variability. Additionally, properly selecting the PS model is a popular topic and has been widely investigated in observational studies. However, there are very few studies investigating the model selection issue for estimating the causal effect when the exposure is MAR. In Chapter 3, we discuss how to select both imputation and PS models, which can result in the smallest root mean squared error (RMSE) of the estimated causal effect. Then, we provide a new criterion, called the “rank score”, for evaluating the overall performance of both models. The simulation studies show that the full imputation plus the outcome-related PS models leads to the smallest RMSE, and the newly proposed criterion is also able to pick the best models. Compared to the MAR assumption, the missing not at random (MNAR) assumption allows for the association between the missingness and the exposure variable, which is a weaker assumption and more reasonable in some application studies. Even though many researchers have discussed how to deal with the missing outcome or confounders in observational studies, very little discussion focuses on the missing exposure under the MNAR assumption. In Chapter 4, we propose the IPW estimators using joint modelling, called “IPW-Joint'', to estimate the causal effect when the exposure is MNAR, which combines estimated weights from the missingness and PS models. Furthermore, to address the problem of model selection in the high-dimensional setting, we apply outcome-adaptive LASSO with weighted absolute mean difference (OAL-WAMD) as a new algorithm to select the outcome-related covariates when the exposure is MNAR. The simulation studies show that IPW-Joint contains more robust properties and smaller variance than the traditional IPW approach. In addition, OAL-WAMD outperforms the traditional LASSO in terms of higher true positive rates (TPR) and smaller variance in both low and high-dimensional settings. In Chapter 5, we summarize the major findings and discuss the potential extensions. The proposed methodology can be applied to several areas, including new drug development, complex missing data problems and finding real-world evidence in observational data.
Efficiency and Equilibria in Centralized and Decentralized Insurance Markets
(University of Waterloo, 2025-04-28) Zhu, Michael Boyuan
This thesis studies Pareto efficiency and market equilibrium in the context of insurance markets. Given a specific model of insurance markets, it is of great practical interest in identifying those risk allocations that are deemed desirable to each agent in the market. Equally as important are market equilibria, the allocations and prices that result from agents’ decisions given the structure of the market. The research presented in this thesis studies the relationship between these two concepts in both centralized and decentralized markets of insurance. Throughout, we provide various characterization results for Pareto-efficient contracts and market equilibria in a variety of settings. These results are illustrated with numerical examples, including an in-depth application to markets of flood risk insurance.
High-Dimensional Scaling Limits of Online Stochastic Gradient Descent in Single-Index Models
(University of Waterloo, 2025-04-25) Rangriz, Parsa
We analyze the scaling limits of stochastic gradient descent (SGD) with a constant step size in the high-dimensional regime in single-index models. Specifically, we prove limit theorems for the trajectories of finite-dimensional summary statistics of SGD as the dimension tends to infinity. These scaling limits enable the analysis of both ballistic dynamics, described by a system of ordinary differential equations (ODEs), and diffusive dynamics, captured by a system of stochastic differential equations (SDEs). Additionally, we analyze a critical step-size scaling regime where, below this threshold, the effective ballistic dynamics align with the gradient flow of the population loss. In contrast, a new diffusive correction term appears at the threshold due to fluctuations around the fixed points. Furthermore, we discuss nearly sharp thresholds for the number of samples required for consistent estimation, which depend solely on an intrinsic property of the activation function known as the information exponent. Our main contribution is demonstrating that if a single-index model has an information exponent greater than two, the deterministic scaling limit, corresponding to the ballistic phase, or so-called dynamical mean-field theory in statistical physics, fails to achieve consistent estimation in high-dimensional inference problems. This shows the necessity of diffusive correction terms to accurately describe the dynamics of online SGD in single-index models via SDEs such as an Ornstein-Uhlenbeck process.
Statistical developments for network meta-analysis and methane emissions quantification
(University of Waterloo, 2025-04-22) Wigle, Augustine
This thesis provides statistical contributions to solve challenges in Network Meta-Analysis (NMA) and the quantification of methane emissions from the oil and gas industry. NMA is an extension of pairwise meta-analysis which facilitates the simultaneous comparison of multiple treatments using data from randomized controlled trials. Some treatments may involve combinations of components, such as one or more drugs given in different combinations. Component NMA (CNMA) is an extension of NMA which allows the estimation of the relative effects of components. In Chapter 2, we compare the popular Bayesian and frequentist approaches to additive CNMA and show that there is an important difference in the assumptions underlying these commonly used models. We prove that the most popular Bayesian CNMA model is prone to misspecification, while the frequentist approach makes a less restrictive assumption. We develop novel Bayesian CNMA models which avoid the restrictive assumption and are robust, and demonstrate in a simulation study that the proposed Bayesian models have favourable statistical properties compared to the existing Bayesian model. The use of all CNMA approaches is demonstrated on a published network. A commonly reported item in an NMA is a list of treatments ranked from most to least preferred, also known as a treatment hierarchy. In Chapter 3, we present the Precision Of Treatment Hierarchy (POTH), a metric which quantifies the level of certainty in a treatment hierarchy from Bayesian or frequentist NMA. POTH summarises the level of certainty into a single number between 0 and 1, making it simple to interpret regardless of the number of treatments in the network. We propose modifications of POTH which can be used to investigate the role of individual treatments or subsets of treatments in the level of certainty in the hierarchy. We calculate POTH for a database of published NMAs to investigate its distribution and relationships with network characteristics. We also provide an in-depth worked example to demonstrate the methods on a real dataset. In the second part of the thesis, we focus on some problems in the quantification of methane emissions from the oil and gas industry. Measurement-based methane inventories, which involve surveying oil and gas facilities and compiling data to estimate methane emissions, are becoming the gold standard for quantifying emissions. However, there is a current lack of statistical guidance for the design and analysis of such surveys. In Chapter 4, we propose the novel application of multi-stage survey sampling techniques to analyse measurement-based methane survey data, providing estimators of total and stratum-level emissions and an interpretable variance decomposition. We also suggest a potentially more efficient approach involving the Hajek estimator, and outline a simple Monte Carlo approach which can be combined with the multi-stage approach to incorporate measurement error. We investigate the performance of the multi-stage estimators in a simulation study and apply the methods to aerial survey data of oil and gas facilities in British Columbia, Canada, to estimate the methane emissions in the province. In Chapter 5, we introduce a Bayesian model for measurements from a methane quantification technology given a true emission rate. The models are fit using data collected in controlled releases (CR) of methane for six different technology types. We use a weighted bootstrap algorithm to provide the distribution of the true emission rate given a new measurement, which synthesizes the new measurement data with the CR data and external information about the possible true emission rate. We present results for the measurement uncertainty of six quantification technologies. Finally, we demonstrate the use of the weighted bootstrap algorithm with different priors and data.
Applications of Lévy Semistationary Processes to Storable Commodities
(University of Waterloo, 2025-04-16) Lacoste-Bouchet, Simon
Volatility Modulated Lévy-driven Volterra (VMLV) processes have been applied by Barndorff-Nielsen, Benth and Veraart (2013) to construct a new framework for modelling spot prices of non-storable commodities, namely energy. In this thesis, we extend this framework to storable commodities by showing that successful classical models belong to the framework albeit under some parameter restrictions (a result which to our knowledge is new). Additionally, we propose a new model for spot prices of storable commodities which is built on the VMLV processes and their important subclass of so-called Lévy semi-stationary (LSS) processes. The main feature of the framework exploited in the model proposed in this thesis is the memory of the VMLV processes which is used judiciously to account for cumulative changes in inventory over time and the corresponding expected changes in prices and volatility. To the best of our knowledge, this is the first study which uses the LSS processes to investigate pricing in storable (as opposed to non-storable) commodity markets to account for the impact of inventory on pricing. To complement the theoretical development of the new model, we also provide in this thesis a companion set of calibration and empirical analyses to shed light on the new model’s performance compared to previously established models in the literature.
Advances in the Analysis of Irregular Longitudinal Data Using Inverse Intensity Weighting
(University of Waterloo, 2025-04-14) Tompkins, Grace
The analysis of irregular longitudinal data can be complicated by the fact that the timing at which individuals are observed in the data are related to the longitudinal outcome. For example, this can occur when patients are more likely to visit a clinician when their symptoms are worse. In such settings, the observation process is referred to as informative, and any analysis that ignores the observation process can be biased. Inverse intensity weighting (IIW) is a method that has been developed to handle specific cases of informative observation processes. IIW weights observations by the inverse probability of being observed at any given time, and creates a pseudopopulation where the observation process is subsequently ignorable. IIW can also be easily combined with inverse probability of treatment weighting (IPTW) to handle non-ignorable treatment assignment processes. While IIW is relatively intuitive and easy to implement compared to other existing methods, there are few peer-reviewed papers examining IIW and its underlying assumptions. In this thesis, we begin by evaluating a flexible weighting method which combines IIW and IPTW through multiplication to handle informative observation processes and non-randomized treatment assignment processes. We show that the FIPTIW weighting method is sensitive to violations of the noninformative censoring assumption and show that a previously proposed extension fails under such violations. We also show that variables confounding the observation and outcome processes should always be included in the observation intensity model. Finally, we show scenarios where weight trimming should and should not be used, and highlight sensitivities of the FIPTIW method to extreme weights. We also include an application of the methodology to a real data set to examine the impacts of household water sources on malaria diagnoses of children in Uganda. Next, we investigate the impact of missing data on the estimation of IIW weights, and evaluate the performance of existing missing data methods through empirical simulation. We show that there is no "one-size-fits-all" approach to handling missing data in the IIW model, and show that the results are highly dependent on the type of covariates that are missing in the observation times model. We then apply the missing data methods to a real data set to estimate the association between sex assigned at birth and malaria diagnoses in children living in Uganda. Finally, we provide an in-depth evaluation on the assumptions made on IIW across various peer-reviewed papers published in the literature. For each set of assumptions, we construct directed acyclic graphs (DAGs) to visualize the assumptions made on the observation and censoring processes which we use to highlight inconsistencies and potential ambiguity among the assumptions presented in existing works involving IIW. We also discuss when causal estimates of the marginal outcome model can be obtained, and propose a general set of assumptions for IIW.

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