Statistical tests of the mean are quite common. Sometimes the analyst cannot validate the assumptions underlying the test, such as normality, symmetry, independence of measurements, etc. This causes unknown deviation of the actual sampling distribution from the distribution assumed by the test, and thus unknown size and power of the test. This distributional uncertainty makes it difficult to reliably choose the decision threshold (critical value) and sample size. We present a method for evaluating the robustness of a test to an unknown degree of distributional uncertainty, based on info-gap decision theory. Analysis of robustness is useful in evaluating effective size and power, and for selecting the decision threshold and sample-size. We study binary simple-hypothesis tests of the mean and consider both type I and type II errors. We show quantitatively that robustness to distributional uncertainty improves, at fixed nominal level of significance, as the effective level of significance deteriorates. Likewise, robustness improves as the effective power of the test deteriorates. Furthermore, we show how to choose the decision threshold and sample size in light of distributional uncertainty. We illustrate our results by application to the t-test and to a test of false nulls in epidemiology.
Keywords. binary hypothesis tests, distributional uncertainty, info-gaps, robustness, tests of the mean, t test, chronic wasting disease, false nulls.
The paper is availabe in the following formats:
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Prof. Yakov Ben-Haim
Yitzhak Moda'i Chair in Technology and Economics
Faculty of Mechanical Engineering
Technion - Israel Institute of Technology
Haifa 32000 Israel
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