On the optimality of sliced inverse regression in high dimensions
نویسندگان
چکیده
The central subspace of a pair random variables $(y,\boldsymbol{x})\in \mathbb{R}^{p+1}$ is the minimal $\mathcal{S}$ such that $y\perp\!\!\!\!\!\perp \boldsymbol{x}|P_{\mathcal{S}}\boldsymbol{x}$. In this paper, we consider minimax rate estimating space under multiple index model $y=f(\boldsymbol{\beta }_{1}^{\tau }\boldsymbol{x},\boldsymbol{\beta }_{2}^{\tau }\boldsymbol{x},\ldots,\boldsymbol{\beta }_{d}^{\tau }\boldsymbol{x},\epsilon )$ with at most $s$ active predictors, where $\boldsymbol{x}\sim N(0,\boldsymbol{\Sigma })$ for some class $\boldsymbol{\Sigma }$. We first introduce large models depending on smallest nonzero eigenvalue $\lambda $ $\operatorname{var}(\mathbb{E}[\boldsymbol{x}|y])$, over which show an aggregated estimator based SIR procedure converges $d\wedge ((sd+s\log (ep/s))/(n\lambda ))$. then optimal in two scenarios, single and fixed dimension $d$ $. By assuming technical conjecture, can also bounded space.
منابع مشابه
Localized Sliced Inverse Regression
We developed localized sliced inverse regression for supervised dimension reduction. It has the advantages of preventing degeneracy, increasing estimation accuracy, and automatic subclass discovery in classification problems. A semisupervised version is proposed for the use of unlabeled data. The utility is illustrated on simulated as well as real data sets.
متن کاملStudent Sliced Inverse Regression
Sliced Inverse Regression (SIR) has been extensively used to reduce the dimension of the predictor space before performing regression. SIR is originally a model free method but it has been shown to actually correspond to the maximum likelihood of an inverse regression model with Gaussian errors. This intrinsic Gaussianity of standard SIR may explain its high sensitivity to outliers as observed ...
متن کاملOn Sliced Inverse Regression With High-Dimensional Covariates
Sliced inverse regression is a promising method for the estimation of the central dimension-reduction subspace (CDR space) in semiparametric regression models. It is particularly useful in tackling cases with high-dimensional covariates. In this article we study the asymptotic behavior of the estimate of the CDR space with high-dimensional covariates, that is, when the dimension of the covariat...
متن کاملAsymptotics of Sliced Inverse Regression
Sliced Inverse Regression is a method for reducing the dimension of the explanatory variables x in non-parametric regression problems. Li (1991) discussed a version of this method which begins with a partition of the range of y into slices so that the conditional covariance matrix of x given y can be estimated by the sample covariance matrix within each slice. After that the mean of the conditi...
متن کاملA note on shrinkage sliced inverse regression
We employ Lasso shrinkage within the context of sufficient dimension reduction to obtain a shrinkage sliced inverse regression estimator, which provides easier interpretations and better prediction accuracy without assuming a parametric model. The shrinkage sliced inverse regression approach can be employed for both single-index and multiple-index models. Simulation studies suggest that the new...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Annals of Statistics
سال: 2021
ISSN: ['0090-5364', '2168-8966']
DOI: https://doi.org/10.1214/19-aos1813