Mathematical Sciences: Three Topics in Mathematical Statistics

这项研究涉及三个统计主题。i)有几个有用的公式涉及非参数函数估计，包括非参数回归，非参数密度估计和标准非参数信号处理模型。最近，研究者和其他人已经证明了这些模型在涉及一维响应变量观察的情况下的渐近等价性。为了得到多维响应变量的有效等价结果，研究者提出了一个渐近等价的公式。ii)在某些复杂参数情况下构建统计检验的一种标准方法包括在辅助统计量上进行条件反射，并在（更简单的）条件问题中以期望的统计显著性水平构建检验。不幸的是，这个过程带来了一些特殊的统计决策理论后果。对这些问题进行了更全面的调查，并提出了替代该标准方法的方法，这些方法几乎同样容易实现，但避免了这些不良后果。iii)生物等效性的标准试验本质上是一维的，因为它们一次只考察一个性能方面。本研究构建了多维生物等效性程序，部分基于研究者和合作者最近研究中包含的原则。统计过程通常被分类为单参数、多参数或非参数，这取决于研究是分别关注一个性能特征、几个相关特征，还是它们的非结构化连续体，例如未指定形状的随时间变化的响应函数。1)现代生物技术企业的一个重要组成部分是检验现有技术的新开发的复制品是否具有与原技术相同的效果。例如，一种（更便宜的）仿制药是否与它想要取代的原始处方药完全相同？根据收集到的关于这两种技术的数据，已经制定了“生物等效性”的标准统计检验来检验这一假设。这些标准测试本质上是一维的，一次只能检查一个性能方面。本研究涉及生物等效性的一般统计检验的构建，可以同时检查几个相关的性能方面。ii)非参数函数估计是统计方法的另一个极端。这种技术涉及很少的先验结构和很少的先验假设。因此，它们在涉及分析包括环境和全球变化研究在内的大量复杂数据的各种现代科学事业中非常有价值。影响这一领域的一个技术问题是这些问题的几种明显不同的数学公式的存在。这项研究涉及为这些问题建立一个统一的理论，这将使研究更有效，并更有效地使用这些方法。iii)本研究的第三个方面涉及渐近辅助统计的概念。这是一种技术装置，部分是为了简化复杂的多参数模型的分析。现代数据收集和计算大大增加了分析这些模型的重要性，但对它们的熟悉程度揭示了使用这种技术设备所产生的某些特性和缺陷。这项研究提出了另一种技术建议，它几乎同样简单易用，但避免了这些特点。

DMS 9626118 Brown This research involves three statistical topics. i) There are several useful formulations involving nonparametric function estimation including nonparmetric regression, nonparametric density estimation and the standard nonparametric signal processing model. Recently the investigator and others have proved the asymptotic equivalence of these models in settings involving observation of one dimensional response variables. The investigators propose to construct a formulation for asymptotic equivalence to enable obtaining effective equivalence results for multidimensional response variables. ii) One standard method for constructing statistical tests in certain complex parametric situations involves conditioning on an ancillary statistic and constructing a test within the (simpler) conditional problem at the desired level of statistical significance. Unfortunately this procedure entails some peculiar statistical decision theoretic consequences. These are investigated more fully and alternatives are proposed to this standard methodology which are nearly as simple to implement but avoid these undesirable consequences. iii) Standard tests for bioequivalence are inherently one - dimensional in that they look at only one performance aspect at a time. This research constructs multidimensional bioequivalence procedures based in part on principles contained in recent research by the investigator and collaborators. Statistical procedures are often classified as single parameter, multiparameter, or nonparametric depending whether the research looks respectively at one performance characteristic, several related ones, or an unstructured continuum of them, such as a response function over time of unspecified shape. i) An important component of the modem biotechnology enterprise involves testing whether newly developed replicas of existing technologies have the same effect as the original. For example, does a (cheaper) generic drug work exactly t he same as the original prescription drug it is intended to replace. Standard statistical tests of "bioequivalence" have been formulated to test this hypothesis based on data gathered about the two technologies. These standard tests are inherently one-dimensional, and can examine only one performance facet at a time. This research involves the construction of general statistical tests of bioequivalence which can simultaneously examine several related performance aspects. ii) Nonparametric function estimation lies at another extreme in statistical methodology. Such techniques involve little prior structure and few apriori assumptions. They are thus extremely valuable in a variety of modern scientific enterprises involving analysis of masses of complex data including research on the environment and on global change. One technical problem impacting this area has been the existence of several apparently different mathematical formulations for such problems. This research involves the creation of a unified theory for such problems which will enable more efficient research and more effective use of these methods. iii) A third aspect of this research involves the notion of asymptotically ancillary statistics. These are a technical device invented in part to simplify the analysis of complex multiparameter models. Modern data collection and computing have vastly increased the importance of analyzing such models, but more familiarity with them reveals certain peculiarities and deficiencies which result from using this technical device. This research proposes alternate technical advice which is nearly as simple to use but avoids these peculiarities.