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

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 versus H_a: X _{i + 1}, tends to be larger than X_i for I = 1, …,n(an increasing trend) Suppose the X_i’s are ranked from 1 to n. Then when H_a 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 R_i be the rank of Xi and consider the test statistic Then small values of D give support to H_a (e.g., the smallest value is 0 for so H₀ should be rejected in favor of When H₀ is true, any sequence of ranks has probability 1/n!. Use this to find 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.]