Test Martingales for bounded random variables
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Test martingales have been proposed as a more intuitive approach to hypothesis testing than the classical approach based on the p-value of a sample. If one is interested in a test procedure that in some sense allows you to do minimal work when lucky and to do further work if less lucky the test martingale technique simply allows to accomodate the new results, much like Bayesian decision techniques do. Unlike Bayesian techniques the test martingale technique allows for a classical frequentist interpretation of the test result. An elaboration is presented of the use of test martingales to make inference about the expected value E(T) of a random variable T. Assuming that T is bounded from above, we construct processes that are positive supermartingales under a hypothesis of the type H_0: E(T)>μ. These lead to tests of such a hypothesis with a given significance level. We consider the power of such tests under the assumption that T is also bounded from below, and construct test martingales that lead to tests with power 1. We also show how to construct confidence upper (and lower) bounds. In financial auditing random sampling is proposed as one of the possible techniques to gather enough assurance to be able to state that there are no 'material' misstatements in a financial report. The original goal of our work is to provide a mathematical context that could represent such process of gathering assurance, if random sampling were the only technique used.
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