Refined Approximation of Tail Probabilities, Constrained Expectations, Data Analysis in Multidimensional and Metric Spaces, Plus Optimal Stable Growth in Finance

0205054Klass在与不同的合著者合作中，PI计划开展以下五项工作。1)完成一篇关于独立或任意随机变量和的分位数位置的(接近最优)逼近的论文。2)完成一篇金融学论文，该论文引入了一个新的约束条件，以增强最优增长标准，从而提供当地财富稳定。可以展示一种自然的次优策略。它进一步展示了如何创建一个长期投资组合策略，该策略以投资组合增长的(相同)渐近速度渐进地提取资金，而不会减少长期利率(！)。3)推广了前人关于自正规鞅的工作。4)尝试构造和/或识别在各种情况下出现的极值分布，这些极值分布使某个受无穷多个线性不等式约束的随机数的期望值最大化。5)发展了David L.Allen教授在1992年向他传达的一个想法，即如何在仅给定所获得的数据的情况下，构造关于R中密度的自然后验分布，并通过Radon变换将结果推广到R^d。这种方法允许人们建立参数估计的置信度区间，仅用有限样本进行假设检验和数据分类。拟合度测试也被考虑在内。主要研究人员计划在五个领域开展工作：任意独立随机变量和的分位数近似，旨在防止超过预先设定的先前最大累积财富的预定比例的局部资本损失的最优投资策略(加上一种允许在不牺牲长期增长率的情况下以长期增长率消费的方法)，扩展先前关于自归一化鞅的工作，解决或找到某些无限维线性规划问题的近似最优解，演示如何充分利用在涉及假设检验或数据分类(分类)的统计环境中获得的数据。投资工作应对个人、企业和社区(政府)投资具有根本利益和重要性。在给定关于A和B的有限的先前数据的情况下，数据分析工作可能提出确定特定数据来自A或B的最佳可能方法。这种工作在字符识别、手写破译等方面可能是有用的。其他主题是由PI先前的工作所激发的。

0205054Klass In work with various co-authors, the PI plans to work on the following five tasks. 1) Complete a paper on (nearly optimal) approximation of quantile location for sums of independent and otherwise arbitrary random variables. 2) Complete a paper in finance which introduces a new constraint to augment the optimal growth criterion and thereby provide local wealth stability. A natural sub-optimal strategy can be exhibited. It is further shown how one can create a long-term portfolio strategy which asymptotically withdraws money at the (same) asymptotic rate at which the portfolio is growing, without diminishing that long-term rate(!). 3) Extend previous work on self-normalized martingales. 4) Attempt to construct and/or identify extremal distributions which arise in various situations maximizing the expected value of some random quantity subject to infinitely many linear inequality constraints. 5) Develop an idea, conveyed to him in 1992 by Prof. David L. Allen, to show how to construct a natural posterior distribution on densities in R, given only the data acquired, and extending the result to R^d by means of the Radon Transform. This approach permits one to establish confidence intervals for parameter estimation, to perform hypothesis testing and data classification with only finite samples. Goodness of fit tests are also contemplated. The principle investigator plans work in five areas: quantile approximation for sums of arbitrary independent random variables, optimal investment strategy designed to guard against local capital losses of more than a pre-set proportion of the previous maximal accumulated wealth (plus a method of permitting consumption at the long run growth rate without sacrificing that long-run rate), extend previous work on self-normalized martingales, solve or find approximately optimal solutions to certain infinite dimensional linear programming problems, demonstrate how to make full use of the data acquired in statistical settings which involve hypothesis testing or classification (categorization) of data. The investment work should be of fundamental interest and importance to individual, corporate and community (governmental) investment. The data analysis efforts may suggest a best possible approach to determining whether a particular datum came from A or B given limited prior data on A and B. Such work could be useful in character recognition, hand-writing decipherment, etc. The other topics were motivated by prior work of the PI.