Using the estimating equations                           X i≤n xi{f ′ (Yi − x ⊤ i βb n)/f(Yi − x ⊤ i βb n)} = 0 for the MLE of β, we can show, under some the regularity conditions, that...

Using the estimating equations

                          X i≤n xi{f ′ (Yi − x ⊤ i βb n)/f(Yi − x ⊤ i βb n)} = 0

for the MLE of β, we can show, under some the regularity conditions, that

                         limn→∞ DnIE[(βb n − β)(βb n − β) ⊤] = I(f)Ip.

Show that the Pitman efficacy for the likelihood ratio test based on the density f (= F ′ ) is proportional to I(f). For the likelihood ratio test statistic we may use the approximation by quadratic form in the MLE and Problems 2.2.2 and 2.2.4.