4.20 Extra Questions

1. What is the difference between consistency and unbiasedness?

2. Suppose you have an estimator that is unbiased. Will it necessarily be consistent? If not, provide an example of an unbiased estimator that is not consistent.

3. Suppose you have an estimator that is consistent. Will it necessarily be unbiased? If not, provide an example of a consistent estimator that is not unbiased.

4. The Central Limit Theorem says that, \(\sqrt{n}\left(\frac{1}{n} \sum_{i=1}^n (Y_i - \mathbb{E}[Y])\right) \rightarrow N(0,V)\) as \(n \rightarrow \infty\) where \(V = \mathrm{var}(Y)\).

   1. What happens to \(n \left(\frac{1}{n} \sum_{i=1}^n (Y_i - \mathbb{E}[Y])\right)\) as \(n \rightarrow \infty\)? Explain.

   2. What happens to \(n^{1/3} \left(\frac{1}{n} \sum_{i=1}^n (Y_i - \mathbb{E}[Y])\right)\) as \(n \rightarrow \infty\)? Explain.