## Abstract

The central subspace of a pair of random variables (y, x) ∈ Rp+^{1} is the minimal subspace S such that y x|P_{S}x. In this paper, we consider the minimax rate of estimating the central space under the multiple index model y = f (β^{τ}_{1} x, β^{τ}_{2} x,..., β^{τ}_{d}x, ε) with at most s active predictors, where x ∼ N(0, Σ) for some class of Σ. We first introduce a large class of models depending on the smallest nonzero eigenvalue λ of var(E[x|y]), over which we show that an aggregated estimator based on the SIR procedure converges at rate d ∧ ((sd + s log(ep/s))/(nλ)). We then show that this rate is optimal in two scenarios, the single index models and the multiple index models with fixed central dimension d and fixed λ. By assuming a technical conjecture, we can show that this rate is also optimal for multiple index models with bounded dimension of the central space.

Original language | English (US) |
---|---|

Pages (from-to) | 1-20 |

Number of pages | 20 |

Journal | Annals of Statistics |

Volume | 49 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2021 |

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty