Section 7.8 (p.430)

6. . From Example 1, . We can show that the equality holds in the information inequality (24) for every p, therefore is efficient for p.

7. . From Example 2, . It’s easy to show that the equality holds in the information inequality (24), therefore is efficient.

Section 7.9 (p.433)

18. According to the discussion from p.427-429, the MLE of b will has approximately a normal distribution with mean b and variance 1/[nI(b)].

Therefore, . Thus the MLE of b is approximately normal with mean b and variance b²/n.

19. (a) Let a(b) = 1/b. Delta method (p.429-430) shows that has approximately normal distribution with mean a(b) and variance [a¢(b)]²/[nb(b)] where b(q) = 1/b². Therefore, the asymptotic distribution of is normal with mean 1/b and variance 1/(nb²).

(b) It’s easy to show that the MLE (Quiz 1), therefore . Since the exponential distribution has mean 1/b and variance 1/b². The CLT shows that has normal distribution with mean 1/b and variance 1/(nb²).