Suppose that observations X1, X2,..., Xn are made on a process at times 1, 2,..., n. On

Question:

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: Xi+1 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

Then small values of D give support to Ha (e.g., the smallest value is 0 for R1 = 10, R2 = 2,..., Rn = n). When H0 is true, any sequence of ranks has probability 1/n!. Use this to determine the P-value in the case n = 4, d = 2.