Let F be the family of distributions having density F
= f on

(0, 1) and let F

0 = f0 be the uniform density. Consider testing the null hypothesis that F = F0 based on the Kolmogorov Smirnov test. Show that, if dk(f,f0) is the sup distance between densities and 0

, f0) ≥ c} ≤ α . (14.83)

Argue that the result applies if dK is replaced by the L2 distance between densities. Hint: Consider densities of the form fθ(t) = 1+ c sin(2πθt). [Compare this result with Theorem 14.2.2. Ingster and Suslina (2003) argue that alternatives based on the sup distance between distribution functions are less natural than metrics between densities. This problem shows it is impossible for the Kolomogorv-Smirnov test to have power bounded away from α against such alternatives. In fact, this is true for any test; see Ingster (1993) and Section 14.6. However, by restricting the family of densities to have further smoothness properties, Ingster and Suslina (2003) have obtained positive results.]