{"title":"FUNCTIONAL SUFFICIENT DIMENSION REDUCTION THROUGH AVERAGE FRÉCHET DERIVATIVES.","authors":"Kuang-Yao Lee, Lexin Li","doi":"10.1214/21-aos2131","DOIUrl":null,"url":null,"abstract":"<p><p>Sufficient dimension reduction (SDR) embodies a family of methods that aim for reduction of dimensionality without loss of information in a regression setting. In this article, we propose a new method for nonparametric function-on-function SDR, where both the response and the predictor are a function. We first develop the notions of functional central mean subspace and functional central subspace, which form the population targets of our functional SDR. We then introduce an average Fréchet derivative estimator, which extends the gradient of the regression function to the operator level and enables us to develop estimators for our functional dimension reduction spaces. We show the resulting functional SDR estimators are unbiased and exhaustive, and more importantly, without imposing any distributional assumptions such as the linearity or the constant variance conditions that are commonly imposed by all existing functional SDR methods. We establish the uniform convergence of the estimators for the functional dimension reduction spaces, while allowing both the number of Karhunen-Loève expansions and the intrinsic dimension to diverge with the sample size. We demonstrate the efficacy of the proposed methods through both simulations and two real data examples.</p>","PeriodicalId":8032,"journal":{"name":"Annals of Statistics","volume":null,"pages":null},"PeriodicalIF":3.2000,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10085580/pdf/nihms-1746366.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Statistics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1214/21-aos2131","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2022/4/7 0:00:00","PubModel":"Epub","JCR":"Q1","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0
Abstract
Sufficient dimension reduction (SDR) embodies a family of methods that aim for reduction of dimensionality without loss of information in a regression setting. In this article, we propose a new method for nonparametric function-on-function SDR, where both the response and the predictor are a function. We first develop the notions of functional central mean subspace and functional central subspace, which form the population targets of our functional SDR. We then introduce an average Fréchet derivative estimator, which extends the gradient of the regression function to the operator level and enables us to develop estimators for our functional dimension reduction spaces. We show the resulting functional SDR estimators are unbiased and exhaustive, and more importantly, without imposing any distributional assumptions such as the linearity or the constant variance conditions that are commonly imposed by all existing functional SDR methods. We establish the uniform convergence of the estimators for the functional dimension reduction spaces, while allowing both the number of Karhunen-Loève expansions and the intrinsic dimension to diverge with the sample size. We demonstrate the efficacy of the proposed methods through both simulations and two real data examples.
期刊介绍:
The Annals of Statistics aim to publish research papers of highest quality reflecting the many facets of contemporary statistics. Primary emphasis is placed on importance and originality, not on formalism. The journal aims to cover all areas of statistics, especially mathematical statistics and applied & interdisciplinary statistics. Of course many of the best papers will touch on more than one of these general areas, because the discipline of statistics has deep roots in mathematics, and in substantive scientific fields.