8. Suppose that observations X1, X2, . . . , Xn are made on a process at times 1, 2, . . . , n. On the basis of this data, we wish to test H0: the Xi’s constitute an independent and identically distributed sequence versus Ha: Xi1 tends to be larger than Xi for i  1, . . . , n (an increasing trend)

Suppose the Xi s are ranked from 1 to n. Then when Ha is true, larger ranks tend to occur later in the sequence, whereas if H0 is true, large and small ranks tend to be mixed together. Let Ri be the rank of Xi and consider the test statistic D  gn (Ri  i)2. Then small values of D give support to Ha (e.g., the smallest value is 0 for R1  1, R2  2, . . . , Rn  n), so H0 should be rejected in favor of Ha if d

c. When H0 is true, any sequence of ranks has probability 1/n!.
Use this to nd c for which the test has a level as close to .10 as possible in the case n  4. [Hint: List the 4! rank sequences, compute d for each one, and then obtain the null distribution of D. See the Lehmann book (in the chapter bibliography), p. 290, for more information.]