Inference for Restricted Parameters

摘要：研究已知密度函数或回归函数是光滑的，并且满足单调性或凸性等形状限制的问题。形状限制回归和光滑样条是一种很有前途的方法，形状限制与局部多项式方法的结合是另一种方法。在大样本中，重点是渐近分布。对于大量的平滑，形状限制对渐近分布的贡献可以忽略不计；只要稍微平滑，形状限制就占主导地位，导致非正态渐近分布。研究了少平滑多过渡的性质。在许多情况下，特别是对于光滑样条曲线，估计量是微分方程的解。在这种情况下，可以使用格林函数构造一个渐近等效核估计量，从中可以找到渐近分布。在中等样本中，即使存在大量平滑，形状限制也会影响估计量的分布。这种效应的大小是分析研究，使用收缩估计技术，并通过模拟。在只有少数参数（如方差成分）的模型中，也会出现已知参数不等式的问题，而在此类问题中，经典置信区间可能是空的或退化的。目前的研究兴趣集中在寻找具有良好频域性质的贝叶斯可信区间。形状限制密度估计和回归的工作是由一个项目激发的，该项目旨在绘制附近矮球状星系（如小熊座）中暗物质的分布。暗物质是看不见的物质。它的存在是从引力效应推断出来的，但它的组成是一个悬而未决的问题。人们预计，知道暗物质的位置可能会对它是什么有所启发。对少参数模型的研究是由物理学和天文学中信号和背景事件的解纠缠问题所驱动的。置信区间问题是寻找基本粒子（特别是希格斯粒子）的一个重要部分，在最近的几次高能物理学家会议上一直是一个特色话题。在一个关于物理科学统计的跨学科研讨会上，研究者正在使用这两个例子和其他例子。

ABSTRACTPI: Michael Woodroofe proposal: 0405584 INFERENCE FOR RESTRICTED PARAMETERS Problems in which a density or regression function is known to be smooth and to satisfy shape restrictions, like monotonicity or convexity, are investigated. Shape restricted regression and smoothing splines offer one promising approach to the problem, and the combination of shape restrictions with local polynomial methods another. In large samples the emphasis is on asymptotic distributions. With a lot of smoothing the contribution of the shape restrictions to the asymptotic distribution is negligible; with only a little smoothing, the shape restrictions dominate, leading to non-normal asymptotic distributions. The nature of the transition between a little smoothing a lot is investigated. In many cases, especially with smoothing splines, an estimator is the solution to a differential equation. In such cases it is possible to use the Green's function to construct an asymptotically equivalent kernel estimator from which the asymptotic distribution may be found. In moderate samples, the shape restrictions do affect the distribution of estimators, even in the presence of substantial smoothing. The size of this effect is investigated both analytically, using techniques of shrinkage estimation, and by simulation. Problems with known inequalities for parameters also arise in models with only a few parameters, like variance components, and classical confidence intervals can be empty, or degenerate, in such problems. In current work interest centers on finding Bayesian credible intervals with good frequentist properties. The work on shape restricted density estimation and regression is motivated by a project to map the distribution of dark matter in nearby dwarf spheroidal galaxies, like Ursa Minor. Dark matter is matter that cannot be seen. Its existence is inferred from gravitational effects, but what it is comprised of is an open question. It is expected that knowing where the dark matter is may shed some light on what it is. The work on models with few parameters is motivated by the problem of disentangling signal and background events in both physics and astronomy. The confidence interval problem is an important part of the search for an elementary particle, the Higgs, in particular, and has been a featured topic at several recent meetings of high-energy physicists. The investigator is using these two examples and others in an interdisciplinary seminar on Statistics in the physical sciences.