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We derive the maximum likelihood estimate (MLE) of a population proportion when it differs from the same of a second population by a known value. This constrained MLE (CMLE) has a closed form in limited scenarios, which are completely characterized. These include the cases when the CMLE takes a boundary value in the parameter space. The existence of solution is established in the other cases and numerical methods are adopted in R and Excel to obtain the estimates solving a nonlinear equation. The standard error of the CMLE is estimated via bootstrap which also yields a confidence interval estimate; this is compared with a second method based on asymptotic distribution. The CMLE is of particular importance in the two sample testing of hypothesis of proportions based on independent samples, when these parameters differ by a non-zero value under the null hypothesis. Numerical computation establishes that the test statistic using the standard error based on this CMLE leads to a more reliable decision than the existing alternatives when the sample sizes are moderate to large.

Maximum likelihood estimation of two-sample population proportions under constraint on their difference