Mathematical Sciences: Inference and Curve-Fitting in Generalizations of Isotonic Regression Models

在许多科学假说中，研究人员可以指定一种方向性效应：一种治疗被认为可以改善健康，一种新的教学方法被认为可以改善学习，增加剂量被认为具有增加效应，直到某个(可能未知的)毒性点，然后是减少效应。许多统计方法忽略了这种方向性信息，这导致了统计能力和效率的损失。我们在这项研究中的主要目标是开发非常通用的方法来计算估计和测试，以及评估估计的标准误差和测试的错误率。本研究的主要工作有三个方面：第一，对一般算法的分析，如Dykstra对von Neumann交替投影法的推广。第二，开发和实现平滑和拟合算法，以使用在参数阶数限制的实验中出现的数据。第三，对于具有方向性效应的研究假设，构建具有随之而来的错误率的统计估计和检验方法。通常在科学研究中，研究人员期望有方向性效应，但不能证明某一特定的数学函数是一种关系。例如，如果增加施肥量(至少在一定范围内)，一种作物的产量可以合理地预期增加。在这里，存在方向性关系的事实是显而易见的，但这种关系的确切性质远非显而易见。我们的目标是开发利用定向效应的方法，以便更有效地利用实验数据。我们将开发计算机算法，以服从方向性效应的方式来拟合数据，并使用统计和数学理论来研究我们的拟合方法的特性。

In many scientific hypotheses the researcher can specify a directional effect: a treatment is believed to improve health, a new teaching method is believed to improve learning, an increase in dosage is believed to have an increasing effect up to a certain (possibly unknown) toxicity point and then a decreasing effect. Many statistical methods ignore this directional information, which results in a loss of statistical power and efficiency. Our major goal in this research is to develop very general approaches to the computation of estimates and tests along with assessments of the standard errors of the estimates and the error rates of the tests. The research has three main thrusts: First, the analysis of general algorithms such as Dykstra's generalization of von Neumann's alternating projections method. Second, the development and implementation of smoothing and fitting algorithms to use on data that arise in experiments with order restrictions on the parameters. And third, the construction of statistical estimation and testing methods with attendant error rates for research hypotheses that entail directional effects. Often in scientific investigations the researcher expects a directional effect, but cannot justify a specific mathematical function as a relationship. For example, the yield of a crop can reasonably be expected to increase if the amount of fertilizer applied is increased (at least within a certain range). The fact that there is a directional relationship is obvious here, but the exact nature of the relationship is far from obvious. Our goal is to develop methods that make use of directional effects so that more efficient use may be made of experimental data. We will develop computer algorithms to fit data in ways that obey directional effects, and use statistical and mathematical theory to study the properties of our fitting methods.