Let X1, , Xn be a sample from a location family with common density f (x ), where the location parameter

Let X1, ··· , Xn be a sample from a location family with common density f (x − θ), where the location parameter θ ∈ R and f (·) is known. Consider testing the null hypothesis that θ = θ0 versus an alternative θ = θ1 for some θ1 > θ0.

Suppose there exists a most powerful levelαtest of the form: reject the null hypothesis iff T = T (X1, ··· , Xn) > C, where C is a constant and T (X1,..., Xn) is location equivariant, i.e., T (X1 + c,..., Xn +

c) = T (X1,..., Xn) + c for all constants c.

Is the test also most powerful level α for testing the null hypothesis θ ≤ θ0 against the alternative θ = θ1. Prove or give a counterexample.