Hypothesis tests and confidence interval estimates for the overlap of two normal distributions with equal variances

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Abstract
A special case of the overlapping coefficient, the common area under two probability density curves, that has received intermittent attention in the scientific and statistical literature concerns the overlap of two normal distributions with equal variances. Here we consider the problem of constructing tests of hypotheses and interval estimation for the true overlap in this special situation. Direct and conditional tests for the true value of the overlap are discussed. A method of constructing an exact confidence interval estimator for the true overlap is presented. Several alternative methods of obtaining confidence intervals for the true overlap are compared in a Monte Carlo investigation. In an example, we use the normal theory results discussed and an invariance property of the overlapping coefficient to estimate the overlap between two log‐normal distributions from sample data.