Estimation of Change-Points in Spline Models
Yang, Guangyu
2022
Abstract
In this dissertation thesis, we present novel, rigorously studied and computationally efficient methods for change-points estimation in different spline models, including linear spline models, generalized linear spline models and constrained spline models. In Chapter II, we estimate change-points in linear spline models. In this chapter, we study influence functions of regular and asymptotically linear estimators using semiparametric theory. Based on the theoretical development, we propose a novel and simple method to circumvent the nondifferentiability, the key challenge in linear spline models, using the modified derivative idea. Consistency and asymptotic normality are rigorously derived for the proposed estimator. A two-step semismooth Newton-Raphson algorithm is further developed for the proposed method. Simulation studies have shown that the proposed method performs well in terms of both statistical and computational properties and improves over existing methods. For example, the existing smoothing-based method sometimes only has a 60% convergence rate and is sensitive to the initial value of the algorithm. And estimates from the highly cited R package "segmented" sometimes exhibit large outliers and may even have a bimodal distribution with around 99% of the coverage probability. In comparison, our proposed method is more stable in terms of almost 100% convergence rates, more robust to choices of different initial values, and has better coverage probabilities. In Chapter III, we extend the estimation of change-points from linear spline models to generalized linear spline models. In this chapter, to overcome the nondifferentiability, we follow the idea of modified derivative from which we propose a novel method to estimate change-points as well as other unknown parameters in generalized linear spline models. Furthermore, we improved the two-step semismooth Newton-Raphson algorithm so that this algorithm is applicable for the proposed method in generalized linear spline models. The statistical properties (consistency, asymptotic normality, and asymptotic efficiency) of the proposed estimator are rigorously studied. Based on simulation studies, the statistical and computational properties for the proposed method performs well. In Chapter IV and Chapter V, we aim to estimate the threshold in constrained spline models, which assume no effect between the factor of interest and the outcome under or above the unknown threshold according to clinical knowledge. In Chapter IV, using a constrained linear spline model, we estimate the threshold of nadir oxygen delivery level, below which there is an increased risk of postoperative acute kidney injury, during a cardiac surgery. Our proposed method is built upon Chapter III. Through simulation studies, we have shown that the proposed method is more robust and efficient than existing methods. In Chapter V, we extend the constrained linear spline model to the constrained penalized spline model, which is able to account for a flexible pattern after the threshold instead of assuming a linear pattern as in the constrained linear spline model. Using the study of Pregnancy Research on Inflammation, Nutrition, & City Environment: Systematic Analyses, we explore the threshold of exposure to air pollution above which there is an adverse effect in terms of low birth weight for pregnant women.Deep Blue DOI
Subjects
Asymptotic Efficiency Broken-Stick Model Differentiable in Quadratic Mean Threshold Estimation
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Thesis
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