Abstract: 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.
Authors’ Names: Shubhabrata Das
Journal Name: Communications in Statistics – Theory and Methods
URL: https://www.tandfonline.com/doi/full/10.1080/03610926.2021.1961152
Journal Article: 'Maximum Likelihood Estimation of Two-sample Population Proportions Under Constraint on Their Difference' - Prof. Shubhabrata Das
Abstract: 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.
Authors’ Names: Shubhabrata Das
Journal Name: Communications in Statistics – Theory and Methods
URL: https://www.tandfonline.com/doi/full/10.1080/03610926.2021.1961152
