Let X1,...,Xn be independent and normally distributed. Suppose Xi has mean i and variance 2 (which is

Let X1,...,Xn be independent and normally distributed. Suppose Xi has mean µi and variance σ2 (which is the same for all i). Consider testing the null hypothesis that µi = 0 for all i. Using invariance considerations, find a UMP invariant test with respect to a suitable group of transformations in each of the following cases:

(i). σ2 is known and equal to one.

(ii). σ2 is unknown.

Section 6.4