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Robust group and simultaneous inferences for high-dimensional single index model
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Recasts the high-dimensional single index model into a pseudo-linear model with the empirical CDF of the response as transform, giving robust, asymptotically honest group tests (max-type statistic with Gumbel limit and multiplier bootstrap) and FDR-controlling multiple testing.
Problem
Statistical inference (group testing and simultaneous testing of coefficients) for high-dimensional single index models when errors are heavy-tailed or contain outliers, where existing methods require sub-Gaussian errors and well-separated signals.
Approach
Under the linearity condition, the SIM is equivalent to a linear model in the transformed response F(Y) (distribution function of Y); orthogonalized (decorrelated) score statistics built on F_n(Y) yield a max-type group test that needs no moment condition on the error and is honest without well-separated signals.
Key results
- Under linear and nonlinear models with Cauchy errors and outliers, the proposed group test achieves power 1.000 for the nonzero group G2 and G4 while controlling type I error near 0.05 (e.g., 0.012 to 0.058).
- Versus ST, under linear model with outliers, the power for G4 is 1.000 for the proposed method against 0.050 for ST; under nonlinear model with Cauchy errors, power is 1 vs 0.040.
- The FDR procedure controls FDR near the 0.2 nominal level (e.g., 0.188-0.213) with power 1.000 in simulations, while BH-Vn and BH-Wn lose power or fail to control FDR under Cauchy errors.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use response-distribution transformation instead of the ST procedure | Group inference for high-dimensional SIM · Nonlinear SIM, original data, ϵ ∼ N(0, 1), G2 | empirical rejection rate (power): 0.785 -> 1.000 | Helped |
| Use response-distribution transformation instead of the ST procedure | Group inference for high-dimensional SIM · Linear SIM with outliers, G4 | empirical rejection rate (power): 0.050 -> 1 | Helped |
| Use response-distribution transformation instead of the ST procedure | Group inference for high-dimensional SIM · Nonlinear SIM with Cauchy error, G4 | empirical rejection rate (power): 0.040 -> 1 | Helped |
| Use the proposed FDR procedure instead of BH-Vn | Simultaneous coefficient testing · Linear model, ϵ ∼ t(1), n = 1000, sparsity 4 | power: 0.017 -> 1.000 | Helped |
| Use the proposed FDR procedure instead of BH-Wn | Simultaneous coefficient testing · Linear model, ϵ ∼ t(1), n = 1000, sparsity 4 | power: 0.028 -> 1.000 | Helped |
| Use response-distribution transformation rather than h(Y) = Y (ablation) | Group inference for high-dimensional SIM · Outlier-contaminated simulations | empirical rejection rate (power) | Helped |
Surprising findings
- Using the sigmoid-transformed response fails to control type I error in the nonlinear model, while the empirical CDF transform does.
- The debiased-lasso BH procedure (BH-Vn) yields FDR above the nominal level (0.458 in one setting) and nearly zero power under Cauchy errors.
Limitations
- Requires the linearity condition on predictors and kappa_h nonzero, excluding even link functions such as sparse phase retrieval.
- Theory requires sparsity s = o(sqrt(n)/(log p log p0)) and sub-Gaussian predictors; FDR control needs p = O(n^c).
- Does not provide quantitative bounds on robustness.
Experimental setup
Datasets: riboflavin (Bacillus Subtilis) dataset, n=71, p=4088 (from R package hdi) simulated Gaussian and multivariate t(1) data
Benchmarks: simulated group inference settings (n=200,500; p=800) simulated FDR settings (p=1000) riboflavin real data
Metrics: empirical type I error empirical power empirical FDR p-values
Baselines: ST (three-step studentized test, Tang et al.) BH-Vn (debiased lasso p-values with BH) BH-Wn (decorrelated score p-values with BH) sigmoid-transformed response raw-response statistic
Builds on: debiased LASSO decorrelated score test rank-LASSO SILM three-step testing linearity condition
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