6.10 The positive-part Lindley estimator of Problem 6.9 has an interesting interpretation in the one-way analysis of variance, in particular with respect to the usual test performed, that of H0 : θ1 = θ2 = ··· = θs. This hypothesis is tested with the statistic F = (y¯i − ¯

y¯)

2/(s − 1)

(yij − ¯yi)2/s(n − 1), which, under H0, has an F-distribution with s − 1 and s(n − 1) degrees of freedom.

(a) Show that the positive-part Lindley estimator can be written as

δi = ¯

y¯ +



1 − c s − 3 s − 1 1

F

+

(y¯i − ¯

y¯).

(b) The null hypothesis is rejected if F is large. Show that this corresponds to using the MLE under H0 if F is small, and a Stein estimator if F is large.

(c) The null hypothesis is rejected at level α if F >Fs−1,s(n−1),α. For s = 8 and n = 6:

(i) What is the level of the test that corresponds to choosing c = 1, the optimal risk choice?

(ii) What values of c correspond to choosing α = .05 or α = .01, typical α levels. Are the resulting estimators minimax?