The conventional proportion estimator (...) = ~iX/n~i (number of events divided by sample sizes) for estimatingthe binomial parameter ~ip~i encounters a number of problems when data are sparse. It is suggested that ~ip~ibe estimated using the class (...)_c, where (...)_c = (~iX~i + ~ic) / (~in~i + 2~ic~i).Choosing optimal point c is investigated in a center study from various perspectives. For a multi-center studyof size ~ik~i, the optimal weights (...)_cj that minimize the MSE of (Formula - Statistics) are derived,subject to (Formula - Statistics) = 1, where (...)_cj = (~iX^j~i + ~ic~i) /(~in^j~i + 2~ic~i). The performance in terms of the smallest MSE of (...)_cw iscompared with the well-known summary proportion estimators. The results show that the optimal value of ~ic~i for minimizing the MSE of Formula is dependent on ~ip~iand equals ~ic~i = 2~ip~i(1- ~ip~i)/(1- 2~ip~i)². To eliminate ~ip~i, Bayes risk with respect to theuniform prior and Euclidean loss are used, leading to the minimum point ~ic~i = 1. This result is true and can beextended for application in a multi-center study of size ~ik~i. The best performance in terms of the smallest Bayesrisk (...)_cw is provided at the point ~ic~i = 1 for moderate to large sample sizes(~in_j~i ≥

16), followed by (...)_cw defined by ~ic~i = 0.5 (at least for smallersample sizes ~in_j~i ≤

8). For estimating the risk difference ~iθ

~i = ~ip~i₁ - ~ip~i₂, the optimal point(~ic~i₁, ~ic~i₂) with the smallest MSE of the adjusted risk difference estimator(Formula - Statistics) found in a particular case of ~ic~i = ~ic~i₁ = ~ic~i ₂ and~in~i = ~in~i₁ = ~in~i₂ as ~ic~i = (Formula - Statistics), ~iθ

~i ≠

0.Another idea from a Bayesian perspective found the optimal point (~ic~i₁, ~ic~i₂)for minimizing the average MSE of (...)_c is (~ic~i₁, ~ic~i₂) = (1, 1).In a multi-center study of size ~ik~i, the optimal weights (...)_cj that minimize the MSE of(Formula - Statistics) are derived subject to (Formula - Statistics) = 1, where (Formula - Statistics). Under acommon risk difference ~iθ

~i over all centers, the performance in the sense of smallest MSE of(...)_cw is that, if ~iθ

~i = 0, ~iθ

~i = 0.1, ~iθ

~i = 0.2, and ~iθ

~i = 0.3, theproposed summary estimator (...)_cw adjusted by ~ic~i = ~ic~i₁ = ~ic~i₂ = 2is the best choice, with the smallest MSE. For ~iθ

~i = 0.4, the proposed estimator (...)_cwadjusted by ~ic~i = ~ic~i₁ = ~ic~i₂ = 1 performs best. However, since the true valueof ~iθ

~i is usually not available

in practice, we suggest choosing the estimator (...)_cw adjustedby ~ic~i = ~ic~i₁ = ~ic~i₂ = 1, which can minimize the Bayes risk with respectto uniform prior and Euclidean loss function. Alternatively, using the ratio estimation method, the optimal point ~ic~i that reduces the bias of(Formula - Statistics) wher (Formula - Statistics) and (Formula - Statistics) are the sample means over all ~ik~icenters, i.e. (Formula - Statistics) where (Formula - Statistics), and (Formula - Statistics) We suggest selecting(Formula - Statistics) when center size ~ik~i or sample size ~in_j are moderate to large.