S Normal approximation for Neyman–Pearson tests. Let .E; E IQ0;Q1/ be a statistical standard model with simple null hypothesis and alternative and strictly positive densities 0, 1. Consider the negative log-likelihood ratio h D log. 0= 1/ and assume that its variance v0 D V0.h/ exists. In the associated infinite product model, let Rn be the likelihood ratio after n observations. Show that the Neyman–Pearson test of a given level 0 < ˛ < 1 has a rejection region of the form

®

logRn > nH.Q0IQ1/ C p

nv0 ˆ

1.1˛Cn/

¯

with n ! 0 as n!1. Hint: Determine the asymptotic size of these tests when n D  ¤ 0 does not depend on n.