Sharp Inequalities for Sums and Functions of Dependent Variables

Columbia University, New York NY

Investigators

Abstract

0205791 de la Pena In this project the Principal Investigator (PI) introduces three related problem areas of key importance in the study of the probabilistic and statistical properties of sums and functions of independent and dependent random variables. In particular, the PI proposes to develop sharp inequalities for the moments of self-normalized processes, as well as for sums of multilinear forms and U-statistics (unbiased statistics) in independent and dependent variables. In addition, the PI intends to further develop a novel approach to approximating the expected time it takes a process (with a general dependence structure) to hit a given boundary. The interest in the study of self-normalized processes stems from their use as key quantities in the development of non-parametric estimators, as well as for their use as pivotal quantities for the creation of confidence intervals and tests of hypothesis. For example, the t-statistic is a self-normalized and unit-less estimator commonly used in the testing of hypotheses about the mean of a distribution with unknown variance. The interest on sharp results for self-normalized estimators is based in part in the need for approximating p-values and the power of tests in situations when the assumptions on the variables need to be relaxed (e.g. independence, normality and/or identical distribution). The study of results related to sums of multilinear forms and U-statistics is related to their use as building blocks in the development of certain stochastic integrals, as well as for their use as the typical unbiased estimators in statistics. Moreover, sums of multilinear forms in independent random variables are frequently used in approximating non-linear estimators of moving averages, which are of fundamental importance in econometric studies. The proposed work concerning the development of new and improved tools (under relaxed assumptions) for the study of statistical estimators is important for the assessment of hypotheses with direct implications in medical and social sciences as well as engineering through its connection to the comparison of competing treatments and technologies under a wider set of scenarios than is currently possible. The study on how long it takes for a random process to hit a boundary, on the basis of historical evidence, has potential important implications in the physical sciences and economics including in the study of how long it will take for 1) a tornado to hit a city, 2) a person to develop cancer, 3) an earthquake to occur or 4) the stock market to crash