Supplement to “A Framework For Estimation of Convex Functions”

In this supplement we prove the additional technical lemmas stated in Section 7.1 which are used in the proofs of the main results. Lemma 4 The function H−1 defined in Section 2.1 is concave and nondecreasing. It is strictly increasing for all x where H−1(x) < 12 . Moreover for C ≥ 1 it satisfies H−1(Ct) ≤ C 2 3H−1(t). (60) The function K defined in Section 2.1 is also increasing and satisfies for C ≥ 1 C 2 3K(t) ≤ K(Ct) ≤ CK(t). (61) Proof of Lemma 4: First note that H is a nondecreasing convex function. Moreover there is a unique point x0 such that it is strictly increasing on some open interval (x0, 1 2) where fs(x0) = 0. The inverse function H −1(x) is thus strictly increasing on the interval (0, H( 2)). In this interval H −1(x) < 12 . For x > H( 1 2), H −1(x) = 12 . It follows that H −1 is nondecreasing. The concavity of H−1 is guaranteed because it is the inverse of an increasing convex function. Now let C ≥ 1. Then since fs is convex and fs(0) = 0 it follows that whenever C2/3y ≤ 12 , Cfs(y) ≤ fs(Cy) and hence also CH(y) = C √ yfs(y) ≤ C √ yfs(C y) = H(Cy). ∗The research of Tony Cai was supported in part by NSF Grant DMS-0604954 and NSF FRG Grant DMS-0854973.