Section 6.7 (p.362)

2.

4.

6.

8.

10.

11.

12. (a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

13. There exists a map f such that f(T) = T ^‘ and f ^–1(T ^‘) = T. If T is a sufficient statistic, then by Factorization Criterion we have

Therefore, T ^‘ is also a sufficient statistic. By the same token we can show that if T ^‘ is sufficient then T is also sufficient.

14. T = S log X_i = log (P X_i). That is, S log X_i is a one-to-one function of P X_i. From Exercise 6 and 14 we know that T = S log X_i is a sufficient statistic.