Test Martingales for bounded random variables

Abstract

Given a random sample from a random variable $T$ which is bounded from above, $T\le\tau$ a.s., we define processes that are positive supermartingales if $E(T)\ge\mu$. Such processes are called test martingales. Tests of the supermartingale hypothesis implicitly test the hypothesis $H_0:E(T)\ge\mu$. We construct test martingales that lead to tests with power 1. We also construct confidence upper bounds. We extend the techniques to testing $H_0:E(T)=\mu$ and constructing confidence intervals. In financial auditing random sampling is proposed as one of the possible techniques to gather enough assurance to be able to state that there is no 'material' misstatement in a financial report. The goal of our work is to provide a mathematical context that could represent such process of gathering assurance by means of repeated random sampling.