Consider the model II analogue of the two-way layout of Section 7.5, according to which Xijk = μ + Ai + Bj + Ci j + Eijk (7.68)

(i = 1,..., a; j = 1,..., b; k = 1,..., n), where the Ai , Bj , Ci j , and Eijk are independently normally distributed with mean zero and with variances σ2 A, σ2 B, σ2 C and σ2 respectively. Determine tests which are UMP among all tests that are invariant (under a suitable group) and unbiased of the hypotheses that the following ratios do not exceed a given constant (which may be zero):

(i) σ2 C/σ2;

(ii) σ2 A/(nσ2 C + σ2);

(iii) σ2 B/(nσ2 C + σ2).

Note that the test of (i) requires n > 1, but those of (ii) and (iii) do not.

[Let S2 A = nb (Xi·· − X···)2, S2 B = na (X· j· − X···)2, S2 C = n

(Xi j· −

Xi·· − X· j· + X···)2, S2 = (Xijk − Xi j·)2, and make a transformation to new variables Zijk (independent, normal, and with mean zero except when i = j = k =

1) such that S2 A = a i=2 Z2 i11, S2 B = 

b j=2 Z2 1 j1, S2 C = a i=2



b j=2 Z2 i j1, S2 = a i=1



b j=1

n k=2 Z2 ijk .]