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

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 level α test 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.