Shape-Restricted Inference

[摘要]题目：形状限制推理考虑给定观测值的函数估计问题。该函数可以是回归函数、密度函数或概率密度函数，例如在生物测定模型中。形状受限的方法允许从业者只对函数类施加定性的限制，例如增加的、凹的或s型的。可以使用最大似然思想获得估计；推理中有许多问题需要解决。本提案的目标包括使用形状限制为回归函数建立置信限，开发具有形状限制协变量的ANCOVA类型模型的测试，具有形状限制趋势函数的时间序列分析，光滑形状限制函数估计，以及使用形状限制误差密度开发稳健回归估计器。用于回归、密度估计和生物测定问题的传统统计方法包括：1)估计函数，2)估计拟合质量，可能使用置信限，以及3)测试关于函数的假设。研究者希望发展这些非参数的方法，也就是说，不强加函数的参数形式。统计中的形状限制方法用最少的关于函数形式的假设来解决这些估计和推理问题。例如，可以假设增长曲线是递增的凹形曲线，或者概率曲线是s型曲线——这是比通常的逻辑模型更普遍的假设。密度函数可能被假设为对称的和单峰的，在这种情况下，像正态性这样更强的假设可能是不合理的。使用较少的假设对函数进行拟合将使数据更加逼真。也许更重要的是，这些形状受限的拟合可以用来测试参数模型的有效性，或者从几个候选参数模型中进行选择。

AbstractDMS-0204572PI: Mary MeyerTitle: Shape-Restricted InferenceConsider the problem of estimating a function given observations with some random component. The function may be a regression function, a density, or a probability density function such as in bioassay models. Shape-restricted methods allow the practitioner to impose only qualitative restrictions on the class of functions, such as increasing, concave, or sigmoidal. Estimates may be obtained using maximum-likelihood ideas; there are many problems in inference to be solved. Goals for this proposal include developing confidence bounds for regression functions using shape restrictions, developing tests for an ANCOVA type of model with a shape-restricted covariate, time-series analysis with shape-restricted trend function, smooth shape-restricted function estimation, and developing a robust regression estimator using a shape-restricted error density.Traditional statistical methods for regression, density estimation and bioassay problems include: 1) estimating a function, 2) estimating the quality of fit, perhaps using confidence bounds, and 3) testing hypotheses about the function. The investigator wishes to develop these methods nonparametrically, that is, without imposing a parametric form for the function. Shape-restricted methods in statistics approach these estimation and inference problems with a minimum of assumptions about the functional form. For example, a growth curve may be assumed to be increasing and concave, or a probability curve might be sigmoidal- a more general assumption than the usual logistic model. A density function might be assumed to be symmetric and unimodal, in a situation where stronger assumptions like normality might not be justified. A fit to a function using fewer assumptions will have more fidelity to the data. Perhaps more importantly, these shape-restricted fits may be used to test the validity of the parametric models, or to select from several candidate parametric models.