Supplemental Material for EigenPrism: Inference for High-Dimensional Signal-to-Noise Ratios

whose distribution we may hope to be close to Gaussian. T1 can be bootstrapped (potentially with standard biascorrection and acceleration) to obtain bootstrap CIs, nonparametrically dealing with the unknown variance v1. We ran simulations with n = 800, p = 1500, X having i.i.d. Bernoulli(0.05) entries (the columns of X were then standardized to have mean 0 and variance 1), θ = σ = 10, and εi i.i.d. t5 (rescaled to have variance 10). We generated a single β uniformly on the θ-radius sphere and ran 1000 simulations (so that β did not change across simulations). Bias-corrected, accelerated 95% bootstrap CIs achieved 93.8% coverage (this is within statistical uncertainty of the nominal 95%, as a 95% CI for the CI coverage is [0.923, 0.953]).