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Statistical Missing Data and Computation Problems: Theories and Applications in Astrophysics, Finance and Economics.

机译：统计缺失数据和计算问题：天体物理学，金融和经济学中的理论和应用。

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摘要

The missing data problems are everywhere in statistics. The groundbreaking EM algorithm offers a simple algorithm with sound theoretical properties for dealing with missing data problems where there is a parametric likelihood. The EM algorithm guarantees the monotonicity of the observed likelihood when it iterates and its output generally is a (local) MLE. However, when we have missing data problems where there is no parametric likelihood, life becomes much more complicated. Many approaches have been proposed in the literature but they tend to be based on case by case consideration without unified principles. We aim to fill this gap in the estimation with missing data by establishing a general framework of "self-consistent" estimators, which not only have an easy-to-follow algorithm for their computation but also have nice theoretical properties.;We began with an overview of the concept and examples of the "self-consistent" estimator, which is first proposed in Efron (1967) and gained considerable interest in the literature on nonparametric estimation with missing data, especially for the survival or distribution functions. We also reviewed the ES algorithm of Elashoff and Ryan (2004), which is a generalization of EM to the cases with estimation equations. It therefore forms an intermediary step between the EM algorithm with parametric likelihood and our algorithm for cases with no parametric likelihood or estimating equations. Subsequently, we presented two general approaches in establishing the algorithmic convergence and theoretical properties of the "self-consistent" estimator. The first approach is based on contraction mapping theories, and we applied it to wavelet denoising with soft thresholding. The second approach is based on fixed point theories. We gave wavelet denoising with hard thresholding and lasso regression as the examples for the second approach.;Besides theoretical and methodological developments, we also applied the missing data methods in several interdisciplinary settings. We first presented the application in astrophysics, where we provide algorithms adapted from the EM algorithm for analyzing a set of astrophysics data for the purpose of identifying the lightcurves for the stars and then event detection based on the light curves. The second application is on a general equilibrium model of the U.S. economy in order to evaluate the impact of the tax policy on U.S. economic growth. Latent variables of production and consumption biases are introduced into the model and Kalman filters are adopted to deal with the missing latent variables. The last application is on financial credit risk modeling. Latent gamma variables are introduced to credit risk model by Duffie and Singleton (1999) to capture the contagious effects between the defaults. We also presented efficient algorithms and large deviation theories for the credit risk models.
机译：缺失的数据问题在统计中无处不在。突破性的EM算法提供了一种简单的算法，具有合理的理论属性，可以处理存在参数可能性的数据丢失问题。 EM算法保证迭代时观察到的似然性的单调性，并且其输出通常是（局部）MLE。但是，当我们缺少没有参数可能性的数据问题时，生活将变得更加复杂。文献中已经提出了许多方法，但是它们往往基于个案考虑而没有统一的原则。我们的目的是通过建立一个“自洽”估计量的通用框架来填补缺失数据中的估计缺口，该框架不仅具有易于遵循的算法，而且具有良好的理论特性。 Efron（1967）首次提出了“自洽”估计器的概念和示例的概述，并引起了关于缺少数据的非参数估计的文献的极大兴趣，尤其是对于生存或分布函数。我们还回顾了Elashoff和Ryan（2004）的ES算法，该算法是EM对带有估计方程的情况的推广。因此，它在具有参数似然性的EM算法与我们针对无参数似然性或估计方程的情况的算法之间形成了中间步骤。随后，我们提出了两种建立“自洽”估计器的算法收敛和理论性质的一般方法。第一种方法基于收缩映射理论，我们将其应用于具有软阈值的小波去噪。第二种方法基于定点理论。我们以硬阈值和套索回归的小波去噪为例，作为第二种方法。除了理论和方法上的发展外，我们还在几种跨学科的环境中应用了缺失数据方法。我们首先介绍了在天体物理学中的应用，其中我们提供了从EM算法改编而来的算法，用于分析一组天体物理学数据，目的是识别恒星的光曲线，然后根据光曲线进行事件检测。第二个应用是在美国经济的一般均衡模型上，以评估税收政策对美国经济增长的影响。将生产和消费偏差的潜在变量引入模型，并采用卡尔曼滤波器处理缺失的潜在变量。最后一个应用程序是金融信用风险建模。 Duffie和Singleton（1999）将潜在的伽玛变量引入信用风险模型以捕获违约之间的传染性影响。我们还介绍了信用风险模型的高效算法和大偏差理论。

著录项

  • 作者

    Li, Zhan.;

  • 作者单位

    Harvard University.;

  • 授予单位 Harvard University.;
  • 学科 Statistics.
  • 学位 Ph.D.
  • 年度 2011
  • 页码 154 p.
  • 总页数 154
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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