Consider testing moment inequalities under the setting of Example 14.4.8. Rather than the likelihood ratio procedure discussed there, consider the following moment selection procedure. Let J = {j : √nX¯ n,j > − log(n)}. Then, reject if the likelihood ratio statistic 2 log(Rn) > c|J |,1−α, where |J | is the cardinality of J .

(i) For any fixed θ ∈ 0, show that the probability of a Type 1 error under θ is asymptotically no bigger than α. Does the size of the test tend to α?

(ii) Show that this procedure asymptotically achieves the power in (14.92). Can you think of any criticism of the procedure?

(iii) Consider the test that rejects H0 when Mn = √n max 1≤i≤k

(X¯ n,i) > dk,1−α , where dk,1−α is the distribution of max1≤i≤k Zi when the Zi are i.i.d. standard normal.

Compute the limiting rejection probability under θ = θ0 + hn−1/2 for any θ0 on the boundary of 0 and any h ∈ IRk .

(iv) As in (ii) above, apply a moment selection procedure based on the test statistic Mn, and repeat (iii) for the procedure. [Moment selection methods for testing moment inequalities are discussed in Andrews and Barwick (2012) and Romano, Shaikh and Wolf (2014). Special emphasis is placed on error control that is uniform in the underlying distribution.]