Likelihood ratio inference in nonparametric monotone function estimation problems

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项目主要涉及基于似然方法的统计推断，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著特征是具有非高斯极限分布的单调函数的最大似然估计器的逐点收敛速度较慢(n的立方根)；这一特性被称为“非正则性”。虽然在过去的几十年里，对这些问题的基于似然的推理已经取得了一些进展，但似然比的行为大体上是未知的。在这个项目中，P.I.试图为这些“非正则”单调函数模型发展一种似然比推断理论。这是因为基于似然比的推理在常规参数、半参数和非参数问题中具有广泛的适用性。作为对数似然比极限的卡方分布的出现，允许在已知卡方分布的基础上为感兴趣的参数构造检验程序和置信域，并避免了估计有害参数的需要。因此，人们自然会问，似然比范式的优势是否会延续到形状受限(更具体地说，单调)函数估计领域。目前的研究计划针对各种感兴趣的模型和应用程序对此进行研究。更具体地说，建议的研究计划的主要组成部分是：(I)调查极限的普适性，D(Ii)从非参数和半参数两个角度研究单调函数模型中关于个体的测量协变量的典型情况(这在应用中是典型的情况)(Iii)开发使用基于似然的方法为感兴趣的单调函数构造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。议程上还有相关的研究问题，如竞争似然比统计量的研究以及相关极限分布的计算和分析表征。形状约束函数的研究出现在各种各样的问题中。特别是，单调性是一种非常自然的形状约束，它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。通过使用像似然和似然比这样有吸引力的统计学概念来估计单调函数，这个项目有望对非参数统计的理论和实践产生广泛的影响。这将导致在医学、公共卫生、可靠性和许多其他应用领域使用基于似然比的方法分析数据的方法得到显著改进，并将引发相关领域统计推断的类似方法的发展。通过将这一项目纳入高级课程的课程，该项目的想法和成果也将在培训和发展未来的统计学家方面卓有成效。

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.