Data depth for measurable noisy random functions

JOURNAL OF MULTIVARIATE ANALYSIS

卷:170

Article

Nagy, Stanislav¹  Ferraty, Frederic² 

[1] Charles Univ Prague, Fac Math & Phys, Dept Probabil & Math Stat, Sokolovska 83, Prague 18675 8, Czech Republic

[2] Univ Paul Sabatier, Inst Math Toulouse, 118 Route Narbonne, F-31062 Toulouse 9, France

关键词: Asymptotics;    Data depth;    Functional data;    Measurement error;    Rate of convergence;    Smoothing;   

DOI  :  10.1016/j.jmva.2018.11.003

来源: Elsevier

【 摘 要 】

In the literature on data depth applicable to random functions, it is usually assumed that the trajectories of all the random curves are continuous, known at each point of the domain, and observed exactly. These assumptions turn out to be unrealistic in practice, as the functions are often observed only on a finite grid of time points, and in the presence of measurement errors. In this work, we provide the necessary theoretical background enabling the extension of the statistical methodology based on data depth to measurable (not necessarily continuous) random functions observed within the latter framework. It is shown that even if the random functions are discontinuous, observed discretely, and contaminated with additive noise, many common depth functionals maintain the fine consistency properties valid in the ideal case of completely observed noiseless functions. For the integrated depth for functions, we provide uniform rates of convergence over the space of integrable functions. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free