Let X1,..., Xn be i.i.d. according to a model {P , }, where is

Let X1,..., Xn be i.i.d. according to a model {Pθ , θ ∈ }, where θ

is real-valued. Consider testing θ = θ0 versus θ = θn at level-α (α fixed, 0 <α< 1).

Show that it is possible to have n H2(Pθ0 , Pθn ) → c < ∞ and still have a sequence of level-α tests φn = φn(X1,..., Xn) such that Eθn (φn) → 1. Hint: Take Pθ uniform on [0, θ] and θn = θ0 − h/n for h > 0.