Let X1,...,Xn be i.i.d. with density p0 or p1, and consider testing the null hypothesis H that p0 is true. The MP level-α test rejects when

Πn i=1r(Xi) ≥ Cn, where r(Xi) = pi(Xi)/p0(Xi), or equivalently when 1

√n

-log r(Xi) − E0[log r(Xi)].

≥ kn. (11.89)

(i) Show that, under H, the left side of (11.89) converges in distribution to N(0, σ2) with σ2 = Var0[log r(Xi)], provided σ < ∞.

(ii) From (i) it follows that kn → σz1−α, where zα is the α quantile of N(0, 1).

(iii) The power of the test (11.89) against p1 tends to 1 as n → ∞. Hint: Use Problem 3.39(iv).