題組內容

II. 1. (20%) Let $X_1, X_2, \ldots, X_n$ be a random sample from Bernoulli(p). Consider two estimators, $\hat{p}_U = \bar{X}$ and $\hat{p}_B = \frac{n}{n+\sqrt{n}}\bar{X} + \frac{\sqrt{n}}{n+\sqrt{n}}\frac{1}{2}$.

a. (5%) Find the limiting distribution of \( \sqrt{n} (\hat{p}_U - p) \) and \( \sqrt{n} (\hat{p}_B - p) \), respectively.