Inference for Restricted Parameters

摘要：Michael Woodroofe提案：0405584 限制参数的推断问题，其中密度或回归函数是已知的是光滑的，并满足形状的限制，如单调性或凸性，进行了研究。 形状约束回归和光滑样条提供了一个很有前途的方法来解决这个问题，形状约束与局部多项式方法的组合是另一个。在大样本中，重点是渐近分布。 通过大量的平滑，形状限制对渐近分布的贡献可以忽略不计;只有很少的平滑，形状限制占主导地位，导致非正态渐近分布。 的性质之间的过渡一点点平滑了很多的调查。 在许多情况下，特别是光滑样条，估计量是微分方程的解。 在这种情况下，可以使用绿色的功能，以构建一个渐近等价的核估计，从该渐近分布可以发现。在中等的样本，形状的限制确实影响估计的分布，即使在存在大量的平滑。 这种影响的大小进行了研究分析，使用技术的收缩估计，并通过模拟。 已知参数不等式的问题也出现在只有少数参数的模型中，如方差分量，经典的置信区间可以是空的，或退化，在这样的问题。 在目前的工作中，人们的兴趣集中在寻找具有良好频率论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的动机，以映射暗物质在附近的矮球状星系，如小熊星座的分布。 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，但它是由什么组成的是一个悬而未决的问题。 人们期望知道暗物质在哪里可能会揭示它是什么。 在物理学和天文学中，解开信号和背景事件的问题激发了对具有较少参数的模型的工作。 置信区间问题是寻找基本粒子（特别是希格斯粒子）的重要组成部分，也是最近几次高能物理学家会议的主题。 调查员正在使用这两个例子和其他关于物理科学统计的跨学科研讨会。

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.