Mathematical Sciences: Three Topics in Mathematical Statistics

DMS 9626118 布朗 这项研究涉及三个统计主题。i）存在涉及非参数函数估计的若干有用公式，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的观察设置的渐近等价性。 研究人员建议构建一个渐进等效的公式，以获得有效的等效结果 多维响应变量。ii）在某些复杂参数情况下构造统计检验的一种标准方法涉及对辅助统计量进行调节，并在所需的统计显著性水平下在（更简单的）条件问题内构造检验。 不幸的是，这一过程带来了一些特殊的统计决策理论的后果。 这些都进行了更充分的调查，并提出了替代方案，这个标准的方法，几乎是简单的实施，但避免这些不良后果。iii）生物等效性的标准测试本质上是一维的，因为它们一次只考虑一个性能方面。 本研究部分基于研究者最近研究中包含的原则构建了多维生物等效性程序， 合作者 统计过程通常分为单参数，多参数或非参数，这取决于研究是否分别着眼于一个性能特征，几个相关的，或它们的非结构化连续体，如随时间变化的响应函数的形状不明。（一）现代生物技术事业的一个重要组成部分是测试 新开发的现有技术的复制品是否具有与原始技术相同的效果。 例如，一种（更便宜的）仿制药是否与它打算取代的原始处方药完全一样。 已经制定了“生物等效性”的标准统计检验，以根据收集的关于这两种技术的数据来检验这一假设。 这些标准测试本质上是一维的，并且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，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学的另一个极端。 这种技术几乎不涉及先验结构和先验假设。 因此，它们在涉及分析大量复杂数据的各种现代科学事业中，包括对环境和全球变化的研究中，具有极其重要的价值。 影响该领域的一个技术问题是存在几个明显的问题 不同的数学公式来解决这些问题。 这项研究涉及为这些问题建立一个统一的理论，这将使这些方法更有效地使用和更有效地研究。iii）本研究的第三个方面涉及渐近辅助统计量的概念。 这是一种技术装置，部分是为了简化复杂多参数模型的分析而发明的。 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导致的某些特点和缺陷。 这项研究提出了替代技术建议，这是几乎一样简单的使用，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.