Abstract:
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Nearly a century ago Jerzy Neyman & Egon Pearson showed that test-statistics with monotone likelihood ratio (MLR) functions yield uniformly most powerful tests (UMPT). The theory seems complete, but an egregious inconsistency lingers between 1-sided and 2-sided tests, devaluing confidence in inferences, and discomfiting novice to veteran statisticians. I suggest that issues arise from marginal probability measures of the risk of Type I Error: commission of error requires both that the null be the true state of nature and that we reject the null, i.e., empirical evidence must favor the alternative. As conditional probabilities generally imply data-dependent procedures, however, their use requires further justification: I prove that test-statistics with MLRs yield not only UMPTs, but also uniformly bound conditional probabilities of Type I Error with no data dependence. Based on this result, I propose tweaking testing theory by assessing error risk based on probabilities conditioned on both causes of Type I Error: these yield more satisfying interpretation, and synchronize results for 1-sided and 2-sided tests based on symmetric sampling distributions.
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