Nonparametric Survival Analysis under Shape Restrictions
The main problem studied in this thesis is to analyse and model time-to- event data, particularly when the survival times of subjects under study are not exactly observed. One of the primary tasks in the analysis of survival data is to study the distribution of the event times of interest. In order to avoid strict assumptions associated with a parametric model, we resort to nonparametric methods for estimating a function. Although other nonparametric approaches, such as Kaplan-Meier, kernel-based, and roughness penalty methods, are popular tools for solving function estimation problems, they suffer from some non-trivial issues like the loss of some important information about the true underlying function, difficulties with bandwidth or tuning parameter selection. In contrast, one can avoid these issues at the cost of enforcing some qualitative shape constraints on the function to be estimated. We con ne our survival analysis studies to estimating a hazard function since it may make a lot of practical sense to impose certain shape constraints on it. Specifically, we study the problem of nonparametric estimation of a hazard function subject to convex shape restrictions, which naturally entails monotonicity constraints. In this thesis, three main objectives are addressed. Firstly, the problem of nonparametric maximum-likelihood estimation of a hazard function under convex shape restrictions is investigated. We introduce a new nonparametric approach to estimating a convex hazard function in the case of exact observations, the case of interval-censored observations, and the mixed case of exact and interval-censored observations. A new idea to handle the problem of choosing the minimum of a convex hazard function estimate is proposed. Based on this, a new fast algorithm for nonparametric hazard function estimation under convexity shape constraints is developed. Theoretical justification for the convergence of the new algorithm is provided. Secondly, nonparametric estimation of a hazard function under smoothness and convex shape assumptions is studied. Particularly, our nonparametric maximum-likelihood approach is generalized for smooth estimation of a function by applying a higher-order smoothness assumption of an estimator. We also evaluate the performance of the estimators using simulation studies and real-world data. Numerical studies suggest that the shape-constrained estimators generally outperform their unconstrained competitors. Moreover, the empirical results indicate that the smooth shape-restricted estimator has more capability to model human mortality data compared to the piecewise linear continuous estimator, specifically in the infant mortality phase. Lastly, our nonparametric estimation of a hazard function approach under convex shape restrictions is extended to the Cox proportional hazards model. A new algorithm is also developed to estimate both convex baseline hazard function and the effects of covariates on survival times. Numerical studies reveal that our new approaches generally dominate the traditional partial likelihood method in the case of right-censored data and the fully semiparametric maximum likelihood estimation method in the case of interval-censored data. Overall, our series of studies show that the shape-restricted approach tends to provide more accurate estimation than its unconstrained competitors, and further investigations in this direction can be highly fruitful.
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