With the increasing availability of non-Euclidean data objects, statisticians are faced with the task of developing appropriate statistical methods for their analysis. For regression models in which the predictors lie in $\mathbb{R}^p$ and the response variables are situated in a metric space, conditional Fr\'echet means can be used to define the Fr\'echet regression function. Global and local Fr\'echet methods have recently been developed for modeling and estimating this regression function as extensions of multiple and local linear regression, respectively. This paper expands on these methodologies by proposing the Fr\'echet Single Index model, in which the Fr\'echet regression function is assumed to depend only on a scalar projection of the multivariate predictor. Estimation is performed by combining local Fr\'echet along with M-estimation to estimate both the coefficient vector and the underlying regression function, and these estimators are shown to be consistent. The method is illustrated by simulations for response objects on the surface of the unit sphere and through an analysis of human mortality data in which lifetable data are represented by distributions of age-of-death, viewed as elements of the Wasserstein space of distributions.

翻译：随着非欧几里得数据对象的日益普及，统计学家面临着开发适当统计方法以进行其分析的挑战。对于预测变量位于$\mathbb{R}^p$而响应变量位于度量空间中的回归模型，条件Fréchet均值可用于定义Fréchet回归函数。近期已开发出全局和局部Fréchet方法，分别作为多元线性回归和局部线性回归的扩展，用于建模和估计该回归函数。本文通过提出Fréchet单指标模型来拓展这些方法论，其中假设Fréchet回归函数仅依赖于多变量预测变量的标量投影。通过结合局部Fréchet方法与M估计进行估计，以同时估计系数向量和潜在回归函数，并证明这些估计量具有一致性。该方法通过单位球面上响应对象的模拟以及人类死亡率数据分析进行了说明，其中寿命表数据由死亡年龄分布表示，被视为Wasserstein分布空间中的元素。