Likelihood ratio inference in nonparametric monotone function estimation problems

非参数单调函数估计问题中的似然比推断

• 批准号: 0306235
• 负责人: Moulinath Banerjee
• 金额: $ 10.5万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-06-01 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0306235&HistoricalAwards=false
• 关键词: Likelihood; ratio; inference; nonparametric; monotone

AbstractPI: M. Banerjee, DMS-0306235Title: Likelihood ratio inference in nonparametric monotone function estimation problemsThe research program primarily concerns statistical inference using likelihood based methods and especially, likelihood ratios in nonparametric monotone function estimation problems. A distinguishing feature of the monotone function models is a slower (cube root of n) pointwise rate of convergence of maximum likelihood estimators of the monotone function of interest, with a non-Gaussian limit distribution; this property is referred to as ``non-regularity''. While some progress in likelihood based inference for these problems has been achievedover the past few decades, the behavior of likelihood ratios is by andlarge unknown. In this project, the P.I. seeks to develop a theory oflikelihood ratio inference for these ``non-regular'' monotone functionmodels. This is motivated by the wide applicability of likelihood ratio based inference in regular parametric, semiparametric and nonparametric problems. The emergence of a chi-squared distribution as the limit of log-likelihood ratios allows the construction of test procedures and confidence regions for the parameters of interest, based on the known chi-squared distributions and circumvents the need to estimate nuisanceparameters. It is thus natural to ask whether the advantages of the likelihood ratio paradigm carry over to the domain of shape-restricted (and more particularly, monotone) function estimation. The current research program investigates this for various models and applications of interest. More specifically, the main components of the proposed research program are: (i) Investigation of the universality of the limit, D (ii) Studying monotone function models with measured covariates on the individuals, which is typically the case in applications, from both nonparametric and semiparametric angles(iii) Developing methods of constructing pointwise confidence sets and confidence bands for monotone functions of interest using likelihood based methods and comparison of these procedures to currently existing methods. Also on the agenda are related research issues, like the study of competing likelihood ratio statistics and the computational and analytical characterization of the associated limit distributions.The study of shape--restricted functions arises in a wide variety of problems. In particular, monotonicity, which is a very natural shape-constraint appears in many different areas of application, such as reliability, renewal theory, survival analysis, epidemiology, biomedical studies and astronomy. Through its use of attractive statistical concepts like likelihood and likelihood ratios, for estimating monotone functions, this project is expected to have a broad impact on the theory and practice of nonparametric statistics. It will lead to significantly improved methods for analyzing data using likelihood ratio based methods in medicine, public health, reliability and numerous other application areas and will trigger the development of analogous methods of statistical inference in related fields. The ideas and results of this project will also be fruitful in the training and development of future statisticians through inclusion in the curriculum of advanced courses.

摘要PI：M. Banerjee，DMS-0306235标题：非参数单调函数估计问题中的似然比推断研究计划主要涉及使用基于似然的方法进行统计推断，特别是非参数单调函数估计问题中的似然比。单调函数模型的一个显着特征是感兴趣的单调函数的最大似然估计的逐点收敛速度较慢（n的立方根），具有非高斯极限分布;这种特性被称为“非正则性”。虽然在过去的几十年里，这些问题的基于似然的推理取得了一些进展，但似然比的行为基本上是未知的。在这个项目中，P.I.试图为这些“非正则”单调函数模型建立一个似然比推断理论。这是由于基于似然比的推理在常规参数、半参数和非参数问题中的广泛适用性。卡方分布作为对数似然比的极限的出现允许基于已知的卡方分布来构造感兴趣的参数的检验程序和置信区域，并且规避了估计nuisanceparameters的需要。因此，很自然地会问，似然比范式的优点是否延续到形状限制（更具体地说，单调）函数估计的领域。目前的研究计划调查了各种模型和感兴趣的应用。更具体地说，拟议研究方案的主要组成部分是：（i）极限的普适性研究，D（ii）研究个体上具有测量协变量的单调函数模型，这是应用中的典型情况，从非参数和半参数角度（iii）开发使用基于似然的方法为感兴趣的单调函数构建逐点置信集和置信带的方法，并将这些方法与现有方法进行比较。议程上还包括相关的研究问题，如竞争似然比统计的研究和相关极限分布的计算和分析表征。形状限制函数的研究出现在各种各样的问题中。特别是，单调性，这是一个非常自然的形状约束出现在许多不同的应用领域，如可靠性，更新理论，生存分析，流行病学，生物医学研究和天文学。通过使用有吸引力的统计概念，如似然和似然比，估计单调函数，该项目预计将对非参数统计的理论和实践产生广泛的影响。它将导致显着改进的方法，用于分析数据，在医学，公共卫生，可靠性和许多其他应用领域的方法，并将触发在相关领域的统计推断的类似方法的发展。该项目的想法和成果也将通过纳入高级课程的课程表，在培训和发展未来的统计人员方面取得丰硕成果。