In this exercise you are encouraged to study the large-sample behavior of the bootstrap through some simulation studies. Two cases will be considered, as follows. In each case, consider n = 50, 100, and 200. In both cases, a sample X1, . . . , Xn are drawn independently from a distribution, F. Then 2000 bootstrap samples are drawn, given the values of X1, . . . , Xn. A parameter,

θ, is of interest. Let ˆ θ be the estimator of θ based on X1, . . . , Xn and ˆ θ
∗ be the bootstrap version of ˆ θ.
(i) F = Uniform[0, 1], θ = the median of F (which is 1/2), ˆ θ = the sample median of X1, . . . , Xn, and ˆ θ
∗ = the sample median of X ∗
1, . . . , X ∗
n, the bootstrap sample. Make a histogram based on the 2000 ˆ θ
∗’s.
(ii) F = Uniform[0, θ], ˆ θ = X(n) = max1≤i≤n Xi , and ˆ θ
∗ = X ∗
(n)
= max1≤i≤n X ∗
i . Make a histogram based on the 2000 ˆ θ
∗’s.
(iii) Make a histogram of the true distribution of ˆ θ for case (i). This can be done by drawing 2000 sample of X1, . . . , Xn and computing ˆ θ for each sample. Compare this histogram with that of (i). What do you conclude?
(iv) Make a histogram of the true distribution of ˆ θ for case (ii). This can be done by drawing 2000 sample of X1, . . . , Xn, and compute ˆ θ for each sample. Compare this histogram with that of (ii). What do you conclude?