Statistical Inference for Random Recursive Equations

George Mason University, Fairfax VA

Investigators

Abstract

This proposal is concerned with asymptotic and small sample inference related to random recursive equations and associated large deviation problems. Random recursive equations, also referred to as stochastic fixed point equations (SFPE), arise in several areas of contemporary science, including: (i) cell-biology, (ii) analysis of algorithms, (iii) financial time-series modeling, (iv) study of perpetuities, (v) actuarial science, (vi) risk management, (vii) ranking of web-pages, and (viii) processes on complex networks. This proposal is concerned with developing new statistical methods that integrate, refine, and sharpen ideas from large deviation theory, semi-parametric and non-parametric inference, efficient importance sampling. It addresses the following basic questions: (1) How to obtain confidence and prediction intervals for the tail probability in risk models that integrate complex financial and insurance processes? (2) How to efficiently estimate the page-ranks of web-pages and understand the factors that influence them? (3) How to provide statistical comparisons between running times of random recursive algorithms? The answer to question (1) is of significant interest to researchers in actuarial science and risk management. The answer to question (2) will enable the development of policies for better utilization of resources. The answer to (3) will yield quantitative methods for developing and comparing recursive algorithms which are used, for example, in computer science. Random recursive equations allow one to unify a wide class of problems that arise in scientific investigations. In a range of applications, scientists are often interested in understanding the probabilities of occurrence of very rare event, which could nevertheless have catastrophic consequences. These rare events could be rare types of cancer whose prevalence rate is small, or the probability of bankruptcy of a financial institution, or beginning stages of resistance to a drug. A key issue is that, while extensive amounts of data are available to model and analyze the frequently occurring events, the amount of data available to study these rare events is perennially low, making the inferential problem challenging. This proposal is concerned with mathematical, statistical, and computational methods to address these challenging issues and provide concrete answers to some of the problems concerning probabilities of rare events.