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Functional data have received significant attention as they frequently appear in modern applications, such as functional magnetic resonance imaging (fMRI) and natural language processing. The infinite-dimensional nature of functional data makes it necessary to use dimension reduction techniques. Most existing techniques, however, rely on the covariance operator, which can be affected by heavy-tailed data and unusual observations. Therefore, in this paper, we consider a robust sliced inverse regression for multivariate elliptical functional data. For that reason, we introduce a new statistical linear operator, called the conditional spatial sign Kendall’s tau covariance operator, which can be seen as an extension of the multivariate Kendall’s tau to both the conditional and functional settings. The new operator is robust to heavy-tailed data and outliers, and hence can provide a robust estimate of the sufficient predictors. We also derive the convergence rates of the proposed estimators for both completely and partially observed data. Finally, we demonstrate the finite sample performance of our estimator using simulation examples and a real dataset based on fMRI.
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Robust inference for change points in high dimension
This paper proposes a new test for a change point in the mean of high-dimensional data based on the spatial sign and self-normalization. The test is easy to implement with no tuning parameters, robust to heavy-tailedness and theoretically justified with both fixed-and sequential asymptotics under both null and alternatives, where n is the sample size. We demonstrate that the fixed-n asymptotics provide a better approximation to the finite sample distribution and thus should be preferred in both testing and testing-based estimation. To estimate the number and locations when multiple change-points are present, we propose to combine the p-value under the fixed-n asymptotics with the seeded binary segmentation (SBS) algorithm. Through numerical experiments, we show that the spatial sign based procedures are robust with respect to the heavy-tailedness and strong coordinate-wise dependence, whereas their non-robust counterparts proposed in Wang et al. (2022) [28] appear to under-perform. A real data example is also provided to illustrate the robustness and broad applicability of the proposed test and its corresponding estimation algorithm.
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- Award ID(s):
- 2014018
- PAR ID:
- 10528257
- Editor(s):
- Rosen, D
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Journal of Multivariate Analysis
- Edition / Version:
- 1
- Volume:
- 193
- Issue:
- 1
- ISSN:
- 0047-259X
- Page Range / eLocation ID:
- 105-114
- Subject(s) / Keyword(s):
- Change points High dimensional data Segmentation Self-normalization Spatial sign
- Format(s):
- Medium: X Size: 2MB Other: pdf
- Size(s):
- 2MB
- Sponsoring Org:
- National Science Foundation
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