Statistics and Actuarial Science
http://hdl.handle.net/10012/9934
2020-09-21T15:39:48ZDependence: From classical copula modeling to neural networks
http://hdl.handle.net/10012/16157
Dependence: From classical copula modeling to neural networks
Prasad, Avinash Srikanta
The development of tools to measure and to model dependence in high-dimensional data is of great interest in a wide range of applications including finance, risk management, bioinformatics and environmental sciences. The copula framework, which allows us to extricate the underlying dependence structure of any multivariate distribution from its univariate marginals, has garnered growing popularity over the past few decades. Within the broader context of this framework, we develop several novel statistical methods and tools for analyzing, interpreting and modeling dependence.
In the first half of this thesis, we advance classical copula modeling by introducing new dependence measures and parametric dependence models. To that end, we propose a framework for quantifying dependence between random vectors. Using the notion of a collapsing function, we summarize random vectors by single random variables, referred to as collapsed random variables. In the context of this collapsing function framework, we develop various tools to characterize the dependence between random vectors including new measures of association computed from the collapsed random variables, asymptotic results required to construct confidence intervals for these measures, collapsed copulas to analytically summarize the dependence for certain collapsing functions and a graphical assessment of independence between groups of random variables. We explore several suitable collapsing functions in theoretical and empirical settings. To showcase tools derived from our framework, we present data applications in bioinformatics and finance.
Furthermore, we contribute to the growing literature on parametric copula modeling by generalizing the class of Archimax copulas (AXCs) to hierarchical Archimax copulas (HAXCs). AXCs are typically used to model the dependence at non-extreme levels while accounting for any asymptotic dependence between extremes. HAXCs then enhance the flexibility of AXCs by their ability to model partial asymmetries. We explore two ways of inducing hierarchies. Furthermore, we present various examples of HAXCs along with their stochastic representations, which are used to establish corresponding sampling algorithms.
While the burgeoning research on the construction of parametric copulas has yielded some powerful tools for modeling dependence, the flexibility of these models is already limited in moderately high dimensions and they can often fail to adequately characterize complex dependence structures that arise in real datasets. In the second half of this thesis, we explore utilizing generative neural networks instead of parametric dependence models. In particular, we investigate the use of a type of generative neural network known as the generative moment matching network (GMMN) for two critical dependence modeling tasks. First, we demonstrate how GMMNs can be utilized to generate quasi-random samples from a large variety of multivariate distributions. These GMMN quasi-random samples can then be used to obtain low-variance estimates of quantities of interest. Compared to classical parametric copula methods for multivariate quasi-random sampling, GMMNs provide a more flexible and universal approach. Moreover, we theoretically and numerically corroborate the variance reduction capabilities of GMMN randomized quasi-Monte Carlo estimators. Second, we propose a GMMN--GARCH approach for modeling dependent multivariate time series, where ARMA--GARCH models are utilized to capture the temporal dependence within each univariate marginal time series and GMMNs are used to model the underlying cross-sectional dependence. If the number of marginal time series is large, we embed an intermediate dimension reduction step within our framework. The primary objective of our proposed approach is to produce empirical predictive distributions (EPDs), also known as probabilistic forecasts. In turn, these EPDs are also used to forecast certain risk measures, such as value-at-risk. Furthermore, in the context of modeling yield curves and foreign exchange rate returns, we show that the flexibility of our GMMN--GARCH models leads to better EPDs and risk-measure forecasts, compared to classical copula--GARCH models.
2020-08-25T00:00:00ZRisk Analysis: Measures of concordance, their compatibility and capital allocation
http://hdl.handle.net/10012/16150
Risk Analysis: Measures of concordance, their compatibility and capital allocation
Koike, Takaaki
This thesis addresses various topics in the field of probability theory and statistics with applications in quantitative risk management.
The first topic concerns matrix compatibility and attainability problems for measures of concordance. We characterize a class of bivariate measures of concordance arising as Pearson's correlations of random variables transformed by a so-called concordance-inducing function. This class of transformed rank correlations includes Spearman's rho, Blomqvist's beta and van der Waerden's coefficient as special cases by taking uniform, Bernoulli and normal distributions as concordance-inducing functions, respectively. For multivariate random vectors, the correlation-based measures are extended as square matrices with entries given by the bivariate measures. We study compatibility and attainability problems for such measures, which ask whether a given square matrix can be realized as a matrix of pairwise measures of concordance for some random vector, and how such a random vector can be constructed. Dimension reduction of compatibility and attainability for block matrices is also studied.
The second topic of this thesis is estimating and comparing transformed rank correlation coefficients. We propose a novel framework for comparing transformed rank correlations in terms of the asymptotic variance of their canonical estimators. A general criterion derived from this framework is that concordance-inducing functions with smaller variances of squared random variables are more preferable. In particular, we show that Blomqvist's beta attains the optimal asymptotic variance and Spearman's rho outperforms van der Waerden's coefficient. We also find that the optimal bounds of the asymptotic variance are attained by Kendall's tau.
The third topic of this thesis is to efficiently estimate risk allocations, which is known to be challenging due to their rare-event nature. We first focus on the problem of estimating Value-at-Risk (VaR) contributions derived by the Euler principle, and propose a novel framework of their estimation by using Markov chain Monte Carlo (MCMC) methods. We prove consistency and asymptotic normality of the proposed estimators under certain assumptions on the underlying marginal and copula densities. The framework of estimating VaR contributions with MCMC methods is then extended to the estimation of wider class of the systemic risk measures and risk allocations whose jth component can be written as a risk measure of the jth conditional marginal loss distribution given the so-called crisis event. Improved sample efficiency of our MCMC estimators is expected since they consist of samples from the conditional loss distribution given the rare event of interest whereas existing estimators are constructed by first simulating the unconditional loss distribution and then extracting the samples satisfying the rare event condition. In a series of numerical experiments, we demonstrate that biases and mean squared errors for our MCMC estimators are reduced in comparison to existing estimators.
The last topic of this thesis is to investigate the conditional distribution of a loss random vector given that the aggregate loss equals an exogenously provided capital. This conditional distribution serves as a building block for calculating risk allocations such as such as VaR contributions. A superlevel set of this conditional distribution can be interpreted as a set of severe and plausible stress scenarios the given capital is supposed to cover. We show that various distributional properties of this conditional distribution are inherited from those of the underlying joint loss distribution. Among these properties,
we find that modality of the conditional distribution is an important feature in risk profile related to the number of risky scenarios likely to occur in a stressed situation. Under unimodality, we study a novel risk allocation method called maximum likelihood allocation (MLA), defined as the mode of the conditional distribution given the total capital. Under multimodality, a single vector of allocations can be less sound. To overcome this issue, we investigate the so-called multimodalty adjustment to increasing the soundness of risk allocations. Properties of the conditional distribution, MLA and multimodality adjustment are demonstrated in numerical experiments. In particular, we observe that negative dependence among losses typically leads to multimodality, and thus to multiple risky scenarios and higher multimodality adjustment.
2020-08-20T00:00:00ZStatistical Methods for Event History Data under Response Dependent Sampling and Incomplete Observation
http://hdl.handle.net/10012/16062
Statistical Methods for Event History Data under Response Dependent Sampling and Incomplete Observation
Shi, Yidan
This thesis discusses statistical problems in event history data analysis including survival analysis and multistate models. Research questions in this thesis are motivated by the Nun Study, which contains longevity data and longitudinal follow-up of cognition functions in 678 religious sisters. Our research interests lie in modeling the survival pattern and the disease process for dementia. These data are subject to a process-dependent sampling scheme, and the homogeneous Markov assumption is violated when using a multistate model to fit the panel data for cognition. In this thesis, we formulated three statistical questions according to the aforementioned issues and propose approaches to deal with these problems.
Survival analysis is often subject to left-truncation when the data are collected within certain study windows. Naive methods ignoring the sampling conditions yield invalid estimates. Much work has been done to deal with the bias caused by left-truncation. However, discussion on the loss-in-efficiency is limited. In Chapter 2, we proposed a method in which auxiliary information is borrowed to improve the efficiency in estimation. The auxiliary information includes summary-level statistics from a previous study on the same cohort and census data for a comparable population. The likelihood and score functions are developed. A Monte Carlo approximation is proposed to deal with the difficulty in obtaining tractable forms of the score and information functions. The method is illustrated by both simulation and real data application to the Nun Study.
Continuous-time Markov models are widely used for analyzing longitudinal data on the disease progression over time due to the great convenience for computing the probability transition matrices and the likelihood functions. However, in practice, the Markov assumption does not always hold. Most of the existing methods relax the Markov assumption while losing the advantage of that assumption in the calculation of transition probabilities. In Chapter 3, we consider the case where the violation of the Markov property is due to multiple underlying types of disease. We propose a mixture hidden Markov model where the underlying process is characterized by a mixture of multiple time-homogeneous Markov chains, one for each disease type, while the observation process contains states corresponding to the common symptomatic stages of these diseases. The method can be applied to modeling the disease process of Alzheimer's disease and other types of dementia. In the Nun Study, autopsies were conducted on some of the deceased participants so that one can know whether these individuals have Alzheimer's pathology in their brains. Our method can incorporate these partially observed pathology data as disease type indicators to improve the efficiency in estimation. The predictions for the overall prevalence and type-specific prevalence for dementia are calculated based on the proposed method. The performance of the proposed methods is also evaluated via simulation studies.
Many prospective cohort studies of chronic diseases select individuals whose observed process history satisfies particular conditions. For instance, studies aiming to estimate the incidence rate of dementia or the effect of genetic factors on the disease would recruit individuals in the condition of being alive and disease-free. In contrast, some other studies may aim to collect information on disease progression or mortality from the time of the disease onset. Under such settings, individuals are recruited if they are in a subset of the states at the study entry, and the methods of estimation need to account for such state-dependent selection conditions. For multistate analysis, one option is to construct the likelihood based on the prospective data given the history up to and including the time at accrual. This approach yields consistent estimates under state-dependent sampling condition with a price of loss in efficiency. Alternatively, the likelihood contribution from the retrospective and current status data at the time of accrual can be incorporated, but with difficulty in obtaining such information. For example, subjects' initial states are often unknown, imposing a challenge for the computation of the contribution from the current status data at the time of recruitment. However, auxiliary information on the initial states may be available, such as the age-specific population prevalence data related to the disease. In Chapter 4, we proposed a weighted-likelihood method to incorporate auxiliary prevalence data and account for the state-dependent selection condition. The method is demonstrated by simulation and applied to the Nun Study of aging and Alzheimer's disease. A Bayesian sensitivity test is conducted to evaluate the impact of misspecification of the auxiliary prevalence.
2020-07-17T00:00:00ZEffect of a Mobile Phone Intervention on Quitting Smoking in a Young Adult Population of Smokers: Randomized Controlled Trial
http://hdl.handle.net/10012/15912
Effect of a Mobile Phone Intervention on Quitting Smoking in a Young Adult Population of Smokers: Randomized Controlled Trial
Baskerville, Neill Bruce; Struik, Laura; Guindon, Godefroy Emmanuel; Norman, Cameron; Whittaker, Robyn; Burns, Catherine; Hammond, David; Dash, Darly; Brown, K. Stephen
Background: Digital mobile technology presents a promising medium for reaching young adults with smoking cessation interventions because they are the heaviest users of this technology.
Objective: The aim of this study was to determine the efficacy of an evidence-informed smartphone app for smoking cessation, Crush the Crave (CTC), on reducing smoking prevalence among young adult smokers in comparison with an evidence-informed self-help guide, On the Road to Quitting (OnRQ).
Methods: A parallel, double-blind, randomized controlled trial with 2 arms was conducted in Canada to evaluate CTC. In total, 1599 young adult smokers (aged 19 to 29 years) intending to quit smoking in the next 30 days were recruited online and randomized to receive CTC or the control condition OnRQ for a period of 6 months. The primary outcome measure was self-reported continuous abstinence at the 6-month follow-up.
Results: Overall follow-up rates were 57.41% (918/1599) and 60.48% (967/1599) at 3 and 6 months, respectively. Moreover, 45.34% (725/1599) of participants completed baseline, 3-, and 6-month follow-up. Intention-to-treat analysis (last observation carried forward) showed that continuous abstinence (N=1599) at 6 months was not significantly different at 7.8% (64/820) for CTC versus 9.2% (72/779) for OnRQ (odds ratio; OR 0.83, 95% CI 0.59-1.18). Similarly, 30-day point prevalence abstinence at 6 months was not significantly different at 14.4% (118/820) and 16.9% (132/779) for CTC and OnRQ, respectively (OR 0.82, 95% CI 0.63-1.08). However, these rates of abstinence were favorable compared with unassisted 30-day quit rates of 11.5% among young adults. Secondary measures of quit attempts and the number of cigarettes smoked per day at 6-month follow-up did not reveal any significant differences between groups. For those who completed the 6-month follow-up, 85.1% (359/422) of young adult smokers downloaded CTC as compared with 81.8% (346/423) of OnRQ, χ21(N=845)=1.6, P=.23. Furthermore, OnRQ participants reported significantly higher levels of overall satisfaction (mean 3.3 [SD 1.1] vs mean 2.6 [SD 1.3]; t644=6.87, P<.001), perceived helpfulness (mean 5.8 [SD 2.4] vs mean 4.3 [SD 2.6], t657=8.0, P<.001), and frequency of use (mean 3.6 [SD 1.2] vs mean 3.2 [SD 1.1], t683=5.7, P<.001) compared with CTC participants.
Conclusions: CTC was feasible for delivering cessation support but was not superior to a self-help guide in helping motivated young adults to quit smoking. CTC will benefit from further formative research to address satisfaction and usage. As smartphone apps may not serve as useful alternatives to printed self-help guides, there is a need to conduct further research to understand how digital mobile technology smoking cessation interventions for smoking cessation can be improved.
Trial Registration: ClinicalTrials.gov NCT01983150; http://clinicaltrials.gov/ct2/show/NCT01983150 (Archived by WebCite at http://www.webcitation.org/6VGyc0W0i)
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2018-10-23T00:00:00Z