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

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

0205054 Klass 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.