Homework #2 for Stat 610

Due Thursday February 22, in class.

1. From Wasserman,

   Chapter 7:

   Problems:  5, 6, 8

2. Data X_1,...,X_n are considered to be iid Bernoulli trials
   with success probablity theta.
   a. Confirm that the plug-in estimator of theta is
      theta.hat = (1/n) sum_{i=1}^n X_i

   b. Confirm that 

      sqrt{n} ( theta.hat - theta ) -->d Normal[ 0, sigma^2 ]

      as n --> infty where sigma^2 = theta(1-theta)

   c. An approximate 95% confidence interval for theta is

      theta.hat +/- 2* sqrt{ (theta.hat)( 1-theta.hat)/n }

      Suppose that n=4. Work out a formula for the
      coverage probability of this interval as a function of
      theta. Plot this function along a grid of theta values
      in (0,1).  Repeat this calculation for n=10 and also
      for n = 30.

3. From Wasserman

   Chapter 8. Problem 6

4. The following 20 observations are known to be drawn from a 
   population with mean = sqrt(theta) and variance
   equal to 1, although the value of theta>0 is not known:

   2.32 3.50 1.61 0.91 2.87 2.27 3.02 1.53 0.06 2.49
   3.19 3.64 3.24 2.89 1.61 2.96 0.80 0.44 1.09 1.14

   a. Recall from quiz #1 that an approximate 95% CI for
      theta is 

      theta.hat +/- 4 sqrt( theta.hat/n )

      Compute this interval.

   b. By contrast, convert the asymptotic interval for theta^2 into
      one for theta, and compare this to (a).

   c. Use bootstrap sampling to compute two other intervals,
      one using the bootstrap estimate of standard error, and
      one using the percentile method.

5. Consider a realized random sample x_1,...,x_n from which
   a bootstrap sample X_1^*, ..., X_n^* is drawn.
   Show that

    Xbar^*_n = (1/n) sum_{i=1}^n X_i^*
             = (1/n) sum_{i=1}^n M_i x_i

   for count variables M_1,...,M_n that have a
   Multinomial(n, (1/n,...,1/n)) distribution [see Wasserman, page 39].
   Show that
     
    E[ M_i ] = 1
    var[ M_i ] = 1 - 1/n
    cov[ M_i, M_j ] = -1/n, for j .neq. i  [note problem 1 above, Wasserman#5]

   Use this to compute E[ Xbar^*_n ] and var[ Xbar^*_n ] given the data.