For testing = 0 versus >0, define two test sequences n and n to be asymptotically equivalent under the null hypothesis if n

For testing θ = θ0 versus θ>θ0, define two test sequences φn and

ψn to be asymptotically equivalent under the null hypothesis if φn − ψn → 0 in probability under θ0. Does this imply that, if θ0 is the true value, the probability the tests reach the same conclusion tends to 1? Show that, under q.m.d., asymptotic equivalence under the null hypothesis also implies that, under an alternative sequence

θn,h = θ0 + hn−1/2, Eθn,h (φn) − Eθn,h (ψn) → 0 .

Furthermore, assume at least one of the two, say φn is nonrandomized. Then, conclude the tests are asymptotically equivalent in the sense that the probability the tests reach the same conclusion tends to 1, both under θ0 and a sequence θn,h.