Consider the problem of testing (F) = 0 versus (F) = 0, for F F0, the

Consider the problem of testing µ(F) = 0 versus µ(F) = 0, for F ∈ F0, the class of distributions supported on [0, 1]. Let φn be Anderson’s test.

(i) If

|n1/2

µ(Fn)| ≥ δ > 2sn,1−α , then show that EFn (φn) ≥ 1 − 1 2(2sn,1−α − δ)2 , where sn,1−α is the 1 − α quantile of the null distribution of the KolmogorovSmirnov statistic. Hint: Use (11.88) and Chebyshev’s inequality.

(ii) Deduce that the minimum power of φn over {F : n1/2µ(F)| ≥ δ} is at least 1 − [2(2sn,1−α − δ)

−2] if δ > 2sn,1−α.

(iii) Use (ii) to show that, if Fn ∈ F0 is any sequence of distributions satisfying n1/2|µ(Fn)|→∞, then EFn (φn) → 1.