## Convergence rates for the generalized Fréchet mean via the quadruple inequality.(English)Zbl 1432.62080

Summary: For sets $$\mathcal{Q}$$ and $$\mathcal{Y}$$, the generalized Fréchet mean $$m\in \mathcal{Q}$$ of a random variable $$Y$$, which has values in $$\mathcal{Y}$$, is any minimizer of $$q\mapsto \mathbb{E}[\mathfrak{c}(q,Y)]$$, where $$\mathfrak{c}\colon \mathcal{Q}\times \mathcal{Y}\to \mathbb{R}$$ is a cost function. There are little restrictions to $$\mathcal{Q}$$ and $$\mathcal{Y}$$. In particular, $$\mathcal{Q}$$ can be a non-Euclidean metric space. We provide convergence rates for the empirical generalized Fréchet mean. Conditions for rates in probability and rates in expectation are given. In contrast to previous results on Fréchet means, we do not require a finite diameter of the $$\mathcal{Q}$$ or $$\mathcal{Y}$$. Instead, we assume an inequality, which we call quadruple inequality. It generalizes an otherwise common Lipschitz condition on the cost function. This quadruple inequality is known to hold in Hadamard spaces. We show that it also holds in a suitable way for certain powers of a Hadamard-metric.

### MSC:

 62G05 Nonparametric estimation 62G20 Asymptotic properties of nonparametric inference 62R20 Statistics on metric spaces
Full Text:

### References:

 [1] Ahidar-Coutrix, A., Gouic, T. L. and Paris, Q. (2018). Convergence rates for empirical barycenters in metric spaces: curvature, convexity and extendible geodesics., arXiv: 1806.02740 [2] Bacak, M. (2018). Old and new challenges in Hadamard spaces., arXiv: 1807.01355 [3] Banerjee, A., Guo, X. and Wang, H. (2005). On the optimality of conditional expectation as a Bregman predictor., IEEE Trans. Inform. Theory 51 2664-2669. · Zbl 1284.94025 [4] Banholzer, D., Fliege, J. and Werner, R. (2017). On almost sure rates of convergence for sample average approximations., http://www.optimization-online.org/DB_HTML/2017/01/5834.html. [5] Barden, D., Le, H. and Owen, M. (2018). Limiting behaviour of Fréchet means in the space of phylogenetic trees., Ann. Inst. Statist. Math. 70 99-129. · Zbl 1394.62153 [6] Bačák, M. (2014). Computing medians and means in Hadamard spaces., SIAM J. Optim. 24 1542-1566. · Zbl 1306.49046 [7] Bačák, M. (2014)., Convex analysis and optimization in Hadamard spaces. De Gruyter Series in Nonlinear Analysis and Applications 22. De Gruyter, Berlin. [8] Bednorz, W. and Latała, R. (2014). On the boundedness of Bernoulli processes., Ann. of Math. (2) 180 1167-1203. · Zbl 1309.60053 [9] Berg, I. D. and Nikolaev, I. G. (2008). Quasilinearization and curvature of Aleksandrov spaces., Geom. Dedicata 133 195-218. · Zbl 1144.53045 [10] Bhattacharya, R. and Patrangenaru, V. (2003). Large sample theory of intrinsic and extrinsic sample means on manifolds. I., Ann. Statist. 31 1-29. · Zbl 1020.62026 [11] Billera, L. J., Holmes, S. P. and Vogtmann, K. (2001). Geometry of the space of phylogenetic trees., Adv. in Appl. Math. 27 733-767. · Zbl 0995.92035 [12] Bousquet, O. (2002). A Bennett concentration inequality and its application to suprema of empirical processes., C. R. Math. Acad. Sci. Paris 334 495-500. · Zbl 1001.60021 [13] Choi, H. I., Choi, S. W. and Moon, H. P. (1997). Mathematical theory of medial axis transform., Pacific J. Math. 181 57-88. · Zbl 0885.53004 [14] del Barrio, E., Deheuvels, P. and van de Geer, S. (2007)., Lectures on empirical processes. EMS Series of Lectures in Mathematics. European Mathematical Society (EMS), Zürich Theory and statistical applications, With a preface by Juan A. Cuesta Albertos and Carlos Matrán. · Zbl 1132.62001 [15] Deza, M. M. and Deza, E. (2016)., Encyclopedia of distances, Fourth ed. Springer, Berlin. · Zbl 1351.51001 [16] Dubey, P. and Müller, H.-G. (2017). Fréchet analysis of variance for random objects., arXiv: 1710.02761 [17] Eltzner, B. and Huckemann, S. F. (2018). A Smeary Central Limit Theorem for Manifolds with Application to High Dimensional Spheres., arXiv: 1801.06581 · Zbl 1405.62070 [18] Federer, H. (1959). Curvature measures., Trans. Amer. Math. Soc. 93 418-491. · Zbl 0089.38402 [19] Fréchet, M. (1948). Les éléments aléatoires de nature quelconque dans un espace distancié., Ann. Inst. H. Poincaré 10 215-310. · Zbl 0035.20802 [20] Huckemann, S. F. (2011). Intrinsic inference on the mean geodesic of planar shapes and tree discrimination by leaf growth., Ann. Statist. 39 1098-1124. · Zbl 1216.62084 [21] Nye, T. M. W. (2011). Principal components analysis in the space of phylogenetic trees., Ann. Statist. 39 2716-2739. · Zbl 1231.62110 [22] Pennec, X., Fillard, P. and Ayache, N. (2006). A Riemannian framework for tensor computing., International Journal of Computer Vision 66 41-66. doi: 10.1007/s11263-005-3222-z. · Zbl 1287.53031 [23] Petersen, A. and Müller, H.-G. (2018). Fréchet regression for random objects with Euclidean predictors., Annals of Statistics, to be pusblished. arXiv: 1608.03012 [24] Pollard, D. (1990)., Empirical processes: theory and applications. NSF-CBMS Regional Conference Series in Probability and Statistics 2. Institute of Mathematical Statistics, Hayward, CA; American Statistical Association, Alexandria, VA. [25] Sturm, K.-T. (2002). Nonlinear martingale theory for processes with values in metric spaces of nonpositive curvature., Ann. Probab. 30 1195-1222. · Zbl 1017.60050 [26] Sturm, K.-T. (2003). Probability measures on metric spaces of nonpositive curvature. In, Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002). Contemp. Math. 338 357-390. Amer. Math. Soc., Providence, RI. [27] Talagrand, M. (2014)., Upper and lower bounds for stochastic processes. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] 60. Springer, Heidelberg Modern methods and classical problems. [28] van de Geer, S. A. (2000)., Applications of empirical process theory. Cambridge Series in Statistical and Probabilistic Mathematics 6. Cambridge University Press, Cambridge. · Zbl 0953.62049 [29] van der Vaart, A. W. and Wellner, J. A. (1996)., Weak convergence and empirical processes. Springer Series in Statistics. Springer-Verlag, New York With applications to statistics. · Zbl 0862.60002 [30] Ziezold, H. (1977). On expected figures and a strong law of large numbers for random elements in quasi-metric spaces., 591-602. · Zbl 0413.60024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.