Consider the problem of obtaining a (two-sided) confidence band for an unknown continuous cumulative distribution function F.

(i) Show that this problem is invariant both under strictly increasing and strictly decreasing continuous transformations X i = f (Xi), i = 1,..., n, and determine a maximal invariant with respect to this group.

(ii) Show that the problem is not invariant under the transformation X

i =

⎧

⎨

⎩

Xi if |Xi| ≥ 1, Xi − 1 if 0 < Xi < 1, Xi + 1 if − 1 < Xi < 0.

[(ii): For this transformation g, the set g∗ S(x) is no longer a band.]