Abstract: The arc-sin transformation has long been used as a variance stabilizer for the binomial sample proportion arising out of binary data. The natural back-transformed function is useful for returning an estimate to the original scale of the parameter of interest. However, it is known that such a transformation leads to bias when estimating the original parameter of interest. In this study, we find explicit asymptotic bias-adjusted empirical Bayes (EB) estimators for binomial sample proportions in the context of small area estimation. We obtain an explicit second-order correct approximation of the mean squared errors (MSEs) of such estimators, as well as second-order correct estimators of these MSEs. Moreover, the proposed EB estimators and corresponding MSE estimators outperform their competitors in terms of the bias and variance, as demonstrated in a simulation study. We apply our methodology to real data associated with Coronavirus Disease 2019 (COVID-19) for each prefecture in Japan.

Key words and phrases: Area level model, COVID-19, linear mixed model, mean squared error estimation.