The Frechet mean or barycenter generalizes the idea of averaging in spaces where pairwise addition is not well-defined. In general metric spaces, the Frechet sample mean is not a consistent estimator of the theoretical Frechet mean. For graph-valued random variables, for instance, the Frechet sample mean may fail to converge to a unique value. Hence, it becomes necessary to consider the convergence of sequences of sets of graphs. We show that a specific type of almost sure convergence for the Frechet sample mean previously introduced by Ziezold (1977) is, in fact, equivalent to the Kuratowski outer limit of a sequence of Frechet sample means. Equipped with this outer limit, we provide a new proof of the strong consistency of the Frechet sample mean for graph-valued random variables in separable (pseudo-)metric space. Our proof strategy exploits the fact that the metric of interest is bounded, since we are considering graphs over a finite number of vertices. In this setting, we describe two strong laws of large numbers for both the restricted and unrestricted Frechet sample means of all orders, thereby generalizing a previous result, due to Sverdrup-Thygeson (1981).
Non-Euclidean data that are indexed with a scalar predictor such as time are increasingly encountered in data applications, while statistical methodology and theory for such random objects are not well developed yet. To address the need for new methodology in this area, we develop a total variation regularization technique for nonparametric Fréchet regression, which refers to a regression setting where a response residing in a metric space is paired with a scalar predictor and the target is a conditional Fréchet mean. Specifically, we seek to approximate an unknown metric-space valued function…
Budapest University of Technology and Economics（布达佩斯技术与经济大学）；Université Paris-Saclay, CNRS, Univ Evry, Laboratoire de Mathématiques et Modélisation d’Evry（巴黎萨克雷大学，法国国家科学研究中心，埃夫里大学，埃夫里数学与建模实验室）；Univ. Lille, Inria, CNRS, Centrale Lille, UMR 9189, CRIStAL（里尔大学，法国国家信息与自动化研究所，法国国家科学研究中心，里尔中央理工学院，CRIStAL实验室）；LPSM of Sorbonne University（索邦大学LPSM）
With the rapid development of data collection techniques, complex data objects that are not in the Euclidean space are frequently encountered in new statistical applications. Fréchet regression model (Peterson & Müller 2019) provides a promising framework for regression analysis with metric space-valued responses. In this paper, we introduce a flexible sufficient dimension reduction (SDR) method for Fréchet regression to achieve two purposes: to mitigate the curse of dimensionality caused by high-dimensional predictors and to provide a visual inspection tool for Fréchet regression. Our approac…
We consider a nonparametric version of the integer-valued GARCH(1,1) model for time series of counts. The link function in the recursion for the variances is not specified by finite-dimensional parameters, but we impose nonparametric smoothness conditions. We propose a least squares estimator for this function and show that it is consistent with a rate that we conjecture to be nearly optimal.
A nonparametric regression setting is considered with a real-valued covariate and responses from a metric space. One may approach this setting via Fréchet regression, where the value of the regression function at each point is estimated via a Fréchet mean calculated from an estimated objective function. A second approach is geodesic regression, which builds upon fitting geodesics to observations by a least squares method. These approaches are applied to transform two of the most important nonparametric regression estimators in statistics to the metric setting -- the local linear regression est…
In this paper, we introduce the concept of isotropic Hilbert-valued spherical random field, thus extending the notion of isotropic spherical random field to an infinite-dimensional setting. We then establish a spectral representation theorem and a functional Schoenberg's theorem. Following some key results established for the real-valued case, we prove consistency and quantitative central limit theorem for the sample power spectrum operators in the high-frequency regime.
