Suppose that observations X₁, X₂,..., X_n 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

Ha: X_i11 tends to be larger than X_i for =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  When H0 is true, any sequence of ranks has probability 1yn!. Use this to determine the P-value in the case n = 4, d = 2