Shape-Restricted Inference

摘要DMS-0204572PI：Mary MeyerTitle：形状受限推理考虑给出带有随机分量的观测值估计函数的问题。该函数可以是回归函数、密度或概率密度函数，例如在生物测定模型中。形状限制方法允许实践者仅对函数类施加定性限制，如递增、凹或S型。估计可以使用最大似然思想来获得；在推理中有许多问题需要解决。该方案的目标包括：使用形状约束建立回归函数的置信限，对具有形状约束协变量的ANCOVA型模型进行测试，使用形状约束趋势函数进行时间序列分析，平滑形状约束函数估计，以及使用形状约束误差密度开发稳健的回归估计器。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。研究人员希望以非参数的方式开发这些方法，也就是说，不将参数形式强加给函数。统计学中的形状限制方法以关于函数形式的最小假设来处理这些估计和推理问题。例如，增长曲线可以被假设为递增和凹陷，或者概率曲线可能是S形的--这是一个比通常的Logistic模型更一般的假设。在正态等更强的假设可能不成立的情况下，密度函数可能被假定为对称的单峰函数。用更少的假设对函数进行拟合将使数据更逼真。也许更重要的是，这些形状受限的拟合可用于测试参数模型的有效性，或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.