Likelihood ratio inference in nonparametric monotone function estimation problems

[摘要]题目：非参数单调函数估计问题中的似然比推断本研究项目主要关注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单调函数的极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“非规律性”。在过去的几十年里，基于似然的推理在这些问题上取得了一些进展，但似然比的行为在很大程度上是未知的。在这个项目中，P.I.试图为这些“非规则”单调函数模型发展一种似然比推断理论。这是由于基于似然比的推理在正则参数、半参数和非参数问题中的广泛适用性。卡方分布的出现作为对数似然比的极限，允许基于已知的卡方分布为感兴趣的参数构建测试程序和置信区域，并避免了估计干扰参数的需要。因此，很自然地要问，似然比范式的优势是否延续到形状受限（更具体地说，单调）函数估计的领域。目前的研究计划对各种感兴趣的模型和应用进行了调查。更具体地说，拟议研究计划的主要组成部分是：（ii）从非参数和半参数角度研究个体上具有测量协变量的单调函数模型，这在应用中是典型的情况（iii）开发使用基于似然的方法构建感兴趣的单调函数的点向置信集和置信带的方法，并将这些程序与现有方法进行比较。议程上还有相关的研究问题，如竞争似然比统计的研究以及相关极限分布的计算和分析表征。形状受限函数的研究涉及到各种各样的问题。特别是单调性，它是一种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，来估计单调函数，该项目有望对非参数统计的理论和实践产生广泛的影响。它将导致在医学、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，并将引发相关领域类似统计推断方法的发展。这个项目的想法和成果也将通过列入高级课程的课程，在培训和发展未来的统计学家方面取得丰硕成果。

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.