Let F0 be the uniform (0,1) c.d.f. and consider testing F = F0 by the Kolmogorov Smirnov test.

(i) Construct a sequence of alternatives Fn to F0 satisfying n1/2dK(Fn, F0) → δ

with 0 <δ< ∞ such that the limiting power against Fn is α, even though there exist tests whose limiting power against Fn exceeds α.

(ii) Construct a sequence of alternatives Fn to F0 satisfying n1/2dK(Fn, F0) → δ

with 0 <δ< ∞ such that the limiting power against Fn is one.

[Hint: Fix 1 > γn > 0 with n1/2γn → δ > 0 and let Fn(t) be defined by Fn(t) = 

0 if t<γn t if γn ≤ t ≤ 1. (14.81)

Note that dK(Fn, F0) = γn by construction. Let U1,...,Un be i.i.d. according to the uniform distribution on (0, 1), and let Gˆn(t) denote the empirical c.d.f. of the Ui. Set Xi =



Ui if Ui ≥ γn

γn if Ui < γn, (14.82)

so that X1,...,Xn are i.i.d. with c.d.f. Fn. Let Fˆn(t) denote the empirical c.d.f.

of the Xi. Argue that sup t

|Fˆn(t) − t| ≤ max sup t

|Gˆn(t) − t|, γn



and PFn {Tn > sn,1−α} ≤ P{n1/2 sup t

|Gˆn(t) − t| > sn,1−α}

if n1/2γn < sn,1−α. If δ