## Due Date

Due Fri, Nov 01, 2013.

## Problems From Efron and Tibshirani

Problems 4.1, 4.2, 5.2, and 5.4.

The file

contains two variables `x`, which is a random sample from a normal distribution, and `y`, which is a random sample from a Cauchy distribution.

1. For the variable `x` compute

1. the sample mean
2. the sample median
3. the sample 25% trimmed mean, for which R statement is `mean(x, trim=0.25)`

And for each of these three point estimates calculate a bootstrap standard error (use bootstrap sample size 1000 at least).

2. Repeat part (a) for the variable `y`

3. Using the fact that `x` is actually a random sample from a normal distribution, we actually know the asymptotic relative efficiency (ARE) of estimators (i) and (ii) in part (a). (See the efficiency page for details). What is it, and how close is the square of the ratio of bootstrap standard errors?

4. Using the fact that `y` is actually a random sample from a Cauchy distribution, also denoted t(1) because it is the Student t distribution for one degree of freedom, we actually know the asymptotic relative efficiency (ARE) of estimators (i) and (ii) in part (b). (See the efficiency page for details). What is it, and how close is the square of the ratio of bootstrap standard errors?