Binomial Parameter Estimation: Asymptotic analysis of Normality and Confidence Regions Using Moments Estimators

Confidence regions for binomial distributions are widely used in practical research. Extensive application has interested many statisticians in improving new methods for constructing these regions in asymptotic and exact ways. This research primarily aims to construct confidence ellipses for parameters p and m of the binomial distribution Bin(m,p) using a sample of fixed size n in the method of moments estimators. Although the conventional method of moments estimators for p and m are well-established and straightforward to derive, their denominators can reach zero with positive probabilities. This study introduces modified estimators to effectively address this issue. By using the asymptotic normality of the joint distribution of both the conventional and modified estimators, we examine the quadratic forms of these estimators that follow a chi-square distribution, enabling the construction of simultaneous confidence regions (ellipses) for the parameters p and m. The method of moments estimators for binomial parameters p and m are found to be approximately normal, which allows for confidence regions to be constructed based on the chi-square distribution. Presenting modified estimators to handle zero-denominator issues ensures robust and reliable inferences. The theoretical side of the confidence region results show the effectiveness of binomial parameters p and m using modified estimators, solving the limitations of conventional approaches. This method provides a powerful framework for statistical inference, relying on asymptotic normality and a chi-square distribution. This method provides a powerful framework for statistical inference because it depends on asymptotic normality and the chi-square distribution.