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：M。Banerjee，DMS-0306235TITLE：非参数单调函数函数估计问题的可能性比推断研究计划主要涉及使用基于可能性的方法，尤其是非参数单调函数功能估计问题的统计推断。单调函数模型的一个区别特征是，单调的单调函数的最大似然估计器的收敛速率较慢（n的立方根），而非高斯极限分布。该属性称为``非注册性''。在过去的几十年中，基于这些问题的推断已经取得了一些基于这些问题的推论，但似然比的行为是由Andlarge未知。在这个项目中，P.I.试图为这些``非规范''单调函数模型发展一种理论。这是由基于可能性比的广泛适用性在常规参数，半参数和非参数问题中的推论所激发的。卡方分布的出现作为对数似然比的极限，可以根据已知的卡方分布来构建感兴趣参数的测试程序和置信区域，并规避估算nuisanceParameters的必要性。因此，很自然地询问似然比范式的优势是否会延伸到形状限制（尤其是单调）功能估计的领域。当前的研究计划为各种感兴趣的模型和应用调查了这一点。更具体地说，拟议的研究计划的主要组成部分是：（i）研究极限的普遍性，d（ii）研究单个函数模型具有对个体的协变量的单调函数模型，这通常是适用的情况，来自非参数，和半参数角（III）开发了使用基于似然的方法以及这些过程与当前现有方法的这些过程的单调函数来构建点置信点和置信频段的方法。议程上也是相关的研究问题，例如研究竞争的可能性比率统计数据以及相关极限分布的计算和分析表征。对形状限制功能的研究发生在各种问题中。特别是，单调性是一种非常自然的形状约束，出现在许多不同的应用领域，例如可靠性，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（例如可能性和似然比，估计单调功能），该项目有望对非参数统计的理论和实践产生广泛的影响。这将导致使用基于似然比的医学，公共卫生，可靠性和许多其他应用领域的方法分析数据的方法可显着改进，并将触发相关领域中统计推断的类似方法的发展。该项目的想法和结果也将在培训和发展未来的统计学家通过纳入高级课程的课程。