(i) Suppose X is a random variable taking values in a sample space S with probability law P. Let 0 and 1 be disjoint families

(i) Suppose X is a random variable taking values in a sample space S with probability law P. Let ω0 and ω1 be disjoint families of probability laws.

Assume that, for every Q ∈ ω1 and any > 0, there exists a subset A of S (which may depend on ) such that Q(A) ≥ 1 − and such that, if X has distribution Q, then the conditional distribution of X given X ∈ A is a distribution in ω0; call it P .

Show Q − P1 → 0 as → 0.

(ii) Based on data X with probability law P, consider the problem of testing the null hypothesis P ∈ ω0 versus P ∈ ω1. Suppose that, for every Q ∈ ω1, there exists a sequence {Pk } with Pk ∈ ω0 such that Q − Pk1 → 0 as k → ∞. Show that if a test φ is level-α, then EQ[φ(X)] ≤ α for all Q ∈ ω1.

(iii) Suppose X1,..., Xn are i.i.d. on the real line. Let ω0 be distributions with a finite mean and ω1 those without a finite mean. Apply (i) and (ii) to show that no level-α test of ω0 versus ω1 has power > α against any Q ∈ ω1.

[Such nonexistence results data back to Bahadur (1955); see Lemma 13.4.4. This example in (iii) and others are treated in Romano (2004), which also contains many references on such problems.]