Let t and s be two statistics such that s t for a

Question

Let T and S be two statistics such that S = ψ(T ) for a measurable ψ. Show that

(a) if T is complete, then S is complete;

(b) if T is complete and sufficient and ψ is one-to-one, then S is complete and sufficient;

(c) the results in (a) and (b) still hold if the completeness is replaced by the bounded completeness.