Supplement to “Adaptive Confidence Bands for Nonparametric Regression Functions”

This supplement contains the proofs of Theorem 2, Propositions 2 and 3, Lemma 1 and Eq.(14). 7 Proof of Theorem 2 Rather than prove Theorem 2 directly it is convenient to first prove an analogue of the Theorem in the context of multivariate Normal random vectors. This is done in section 7.1. The proof of Theorem 2 is then given in section 7.2 7.1 Confidence Bound For Multivariate Normal Vectors In the first proposition let Xi, i = 1, 2, . . . , n be independent Normal random variables, N(cnθi, 1). Let X = (X1, X2, . . . Xn). Let θ = (θ1, θ2, θn). For θ given we shall write PX|θ and EX|θ for computing probabilities and expectations under this model. We shall also assume that each θi is 0 or 1 and let Θn be the collection of such parameter values. Suppose that C(X) = (C1(X), C2(X), Cn(X)) is a confidence set for θ = (θ1, θ2, . . . , θn) where Ci(X) is a confidence interval for θi. Let L(Ci(X)) be the length of Ci(X). Proposition 4. Suppose that C(X) is a confidence set for θ with uniform coverage of at least 1− α over Θn. Suppose cn = √ c log n with c < 1. Then for any a < 1 and � > 0 there is an M such that for n ≥M sup θ∈Θn PX|θ( � L(Ci(X)) ≥ an) ≥ (1− α− �) (89) and hence for any � > 0 sup θ∈Θn EX|θ( � L(Ci(X))) ≥ (1− α− �)n (90) when n is sufficiently large. If the confidence set C(X) also satisfies L(Ci(X)) = L(C1(X)) for all i then for any � > 0 there is an M such that for n ≥M and all θ ∈ Θn EX|θ( � L(Ci(X))) ≥ (1− α− �)n (91) For any α < 2 there is a c > 0 such that if cn = c then for any � > 0 there is an M and a C > 0 such that for n ≥M and all θ ∈ Θn EX|θ( � L(Ci(X))) ≥ (1− α− �)Cn (92)