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

Let X1,...,Xn be i.i.d. according to a model {Pθ, θ ∈ Ω}, 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 nH2(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.