Testing Model Utility for Single Index Models Under High Dimension

Qian Lin (), Zhigen Zhao () and Jun S. Liu ()
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Qian Lin: Tsinghua University, Center for Statistical Science and Department of Industrial Engineering
Zhigen Zhao: Temple University, Department of Statistical Science
Jun S. Liu: Harvard University, Department of Statistics

A chapter in Festschrift in Honor of R. Dennis Cook, 2021, pp 65-86 from Springer

Abstract: Abstract For the single index model y = f(β τ x, 𝜖) with Gaussian design, where f is unknown and β is a sparse p-dimensional unit vector with at most s nonzero entries, we are interested in testing the null hypothesis that β, when viewed as a whole vector, is zero against the alternative that some entries of β is nonzero, with n i.i.d observations. Assuming that v a r ( 𝔼 [ x ∣ y ] ) $$var(\mathbb {E}[\boldsymbol {x} \mid y])$$ is non-vanishing, we define the generalized signal-to-noise ratio (gSNR) λ of the model as the unique non-zero eigenvalue of v a r ( 𝔼 [ x ∣ y ] ) $$var(\mathbb {E}[\boldsymbol {x} \mid y])$$ . We show that if s 2 log 2 ( p ) ∧ p $$s^{2}\log ^2(p)\wedge p$$ is of a smaller order of n, denoted as s 2 log 2 ( p ) ∧ p ≺ n $$s^{2}\log ^2(p)\wedge p\prec n$$ , one can detect the existence of signals if and only if gSNR ≻ p 1 ∕ 2 n ∧ s log ( p ) n $$\succ \frac {p^{1/2}}{n}\wedge \frac {s\log (p)}{n}$$ . Furthermore, if the noise is additive (i.e., y = f(β τ x) + 𝜖), one can detect the existence of the signal if and only if gSNR ≻ p 1 ∕ 2 n ∧ s log ( p ) n ∧ 1 n $$\succ \frac {p^{1/2}}{n}\wedge \frac {s\log (p)}{n} \wedge \frac {1}{\sqrt {n}}$$ . It is rather surprising that the detection boundary for the single index model with additive noise matches that for linear regression models. These results pave the road for thorough theoretical analysis of single/multiple index models in high dimensions.

Keywords: Sliced inverse regression; Optimal detection; Higher criticism; Minimax rate (search for similar items in EconPapers)
Date: 2021
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-030-69009-0_4

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DOI: 10.1007/978-3-030-69009-0_4

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