Section 6.8 (p.369)

 

1.        

 

2.        

 

3.        

 

4.        

 

5.         Write down the joint pdf and use the following equations:

 

6.        

 

7.         Using the notations given in Exercise 6, we can show that:

            (a) ,  b(x) = 1,  ,  d1(x) = x2,

            ,  d2(x) = x.

 

            (b) ,  b(x) = 1,  c1(q) = a-1,  d1(x) = log x,  c2(q) = -b,  d2(x) = x.

 

            (c) ,  b(x) = 1,  c1(q) = a-1,  d1(x) = log x,  c2(q) = b-1,  d2(x) = log(1-x).

 

8.         The MLE is (Example 1 of Sec. 6.5) and it is also a sufficient statistic (Exercise 1 of Sec. 6.7). Therefore  is minimal sufficient.

 

9.         The MLE is max{X1,…,Xn} (Example 4 of Sec. 6.5) and it is also a sufficient statistic (Exercise 4 of Sec. 6.7). Therefore max{X1,…,Xn}  is minimal sufficient.

 

11.       , otherwise it is zero.

The above likelihood function has a maximum when q is chose as small as possible. Therefore the MLE is max(x1,…,xn). The median of this distribution is , so the MLE of the median is .

            Furthermore, max(x1,…,xn) is a sufficient statistic. This implies that  is also sufficient. Since it is both MLE and sufficient, it’s minimal sufficient.