To “ Global Testing under Sparse Alternatives : Anova , Multiple Comparisons and the Higher Criticism ”

We prove the results stated in the main paper. We start by providing a brief summary of the notations used in the paper. Set [p] = {1, . . . , p} and for a subset J ⊂ [p], let |J | be its cardinality. Bold upper (resp. lower) case letters denote matrices (resp. vectors), and the same letter not bold represents its coefficients, e.g. aj denotes the jth entry of a. For an n × p matrix A with column vectors a1, . . . ,ap, and a subset J ⊂ [p], AJ denotes the n-by-|J | matrix with column vectors aj, j ∈ J . Likewise, aJ denotes the vector (aj , j ∈ J ). The Euclidean norm of a vector is ‖a‖ and the supnorm ‖a‖∞. For a matrix A = (aij), ‖A‖∞ = supi,j |aij |, and this needs to be distinguished from ‖A‖∞,∞, which is the operator norm induced by the sup norm, ‖A‖∞,∞ = sup‖x‖∞≤1 ‖Ax‖∞. The Frobenius (Euclidean) norm of A is ‖A‖F . Φ (resp. φ) denotes the cumulative distribution (resp. density) function of a standard normal random variable, and Φ̄ its survival function. For brevity, we say that β is S-sparse if β has exactly S nonzero coefficients. Finally, we say that a random variable X ∼ FX is stochastically smaller than Y ∼ FY , denoted X ≤sto Y , if FX(t) ≥ FY (t) for all scalar t.