Chapter 6.1, Problem 18E

Let X₁, X₂,..., X_n be a random sample from a pdf f(x) that is symmetric about μ, so that $\tilde{X}$ is an unbiased estimator of μ. If n is large, it can be shown that V( $\tilde{X}$) ≈ 1/(4/n[f(μ)]²).

1. a. Compare V( $\tilde{X}$) to V( $\bar{X}$) when the underlying distribution is normal.
2. b. When the underlying pdf is Cauchy (see Example 6.7), V( $\bar{X}$) = ∞, so $\bar{X}$ is a terrible estimator. What is V( $\tilde{X}$) in this case when n is large?