Fréchet Kernel Sliced Average Variance Estimation

Along with the development of modern information technology,regression prob-lems with high-dimensional predictors are becoming increasingly common.The cal-culation of such problems is quite challenging for some regression methods.In this case,how to convert high-dimensional regression problems into easy-to-handle low-dimensional problems has become a research hotspot.This transformation aims to re-duce dimensions while minimizing information loss.As a popular dimensionality re-duction method,sufficient dimension reduction technique effectively solves curse of dimensionality by replacing the original predictors with a small number of linear com-binations that do not lose regression information.Therefore,it is of great significance to study the theories and methods in the field of sufficient dimension reduction.At the same time,we encounter more and more non-Euclidean data,especially those not located in the vector space,which makes various new methods that can deal with non-Euclidean data attract much attention.Petersen et al.[1]extended the classical regression to Fréchet regression,which developed the regression relationship between responses which are complex random objects in a metric space and vectors of real-valued predictors based on the idea of Fréchet mean.As in classical regression,there exists the same potential challenge of heavy computation when implementing Fréchet regression.However,existing dimension reduction methods designed for classical re-gression are not directly applicable to Fréchet regression.Therefore,it is necessary to study how to perform sufficient dimension reduction technique for Fréchet regression.In the field of sufficient dimension reduction,there are two important topics:the first is how to estimate the base direction of the central dimension reduction subspace S_Y|X,and the other is how to determine the structural dimension k of S_Y|X.This paper discusses these two aspects based on the Fréchet regression model.Firstly,this paper extends sliced average variance estimation method in classical regression to Fréchet regression,proposes the Fréchet kernel sliced average variance estimation（FKSAVE）method to estimate S_Y|X,and provides theoretical guarantee for this method through the analysis of asymptotic properties.Secondly,FKSAVE is combined with predictor augmentation estimator in Luo et al.[2]to estimate k.Finally,this paper demonstrates the excellent limited sample performance of the proposed method through numerical simulation and empirical analysis.In the numerical simulation part,this paper evaluates the estimation performance of the proposed method on Fréchet regression model of three different response types（probability distribution,spherical and symmetric positive definite matrix）,verifies the effectiveness of the proposed method.In the empirical analysis part,this paper applies the proposed method to analyze the human mortality dataset,further illustrating the application value of the proposed method.