3.14 This problem is associated with Section 3.7.

(i) Verify that the LRT statistic is given by (3.21) and (3.22).

(ii) For fixed θ, consider the random variable Yi defined therein. Show that for any real number k, E(Y k i ) = exp k(k − 1)

2

θ2

.

It follows that E(Zi ) = 0 and var(Zi ) = eθ2 − 1, where Zi = Yi − 1.

(iii) Show that pˆ < 1 with probability tending to 1.

(iv) Show by the inequality (3.30) that w.p.→ 1, and 2L∗ ≤ M2 −1 implies that max1≤j≤m Un(θj ) ≤ M.

(v) Show that for any δ > 0 and any l ≥ 1, one can choose θ1, . . . , θl > 0 such that all pairwise correlations (3.28) are less than δ.

(vi) Furthermore, let U1, . . . , Ul be jointly normal, each have N(0, 1) distribution, and the correlations between Uj and Uk be given by (3.28). Show that as δ → 0, P(max1≤j≤l Uj ≤ x) → Φ(x)l for every x, where δ is the maximum absolute value of the correlations between the Uj ’s.