On the Asymptotic Behavior of the Leading Eigenvector of Tyler’s Shape Estimator Under Weak Identifiability

Davy Paindaveine () and Thomas Verdebout ()
Additional contact information
Davy Paindaveine: ECARES and Mathematics Department
Thomas Verdebout: ECARES and Mathematics Department

A chapter in Robust and Multivariate Statistical Methods, 2023, pp 45-64 from Springer

Abstract: Abstract We consider point estimation in an elliptical Principal Component Analysis framework. More precisely, we focus on the problem of estimating the leading eigenvector θ1 of the corresponding shape matrix. We consider this problem under asymptotic scenarios that allow the difference rn := λn1 − λn2 between both largest eigenvalues of the underlying shape matrix to converge to zero as the sample size n diverges to infinity. Such scenarios make the problem of estimating θ1 challenging since this leading eigenvector is then not identifiable in the limit. In this framework, we study the asymptotic behavior of θ ̂ 1 $$\hat {{\boldsymbol {\theta }}}_1$$ , the leading eigenvector of Tyler’s M-estimator of shape. We show that consistency and asymptotic normality survive scenarios where n r n $$\sqrt {n}r_n$$ diverges to infinity as n does, although the faster the sequence (rn) converges to zero, the poorer the corresponding consistency rate is. We also prove that consistency is lost if r n = O ( 1 ∕ n ) $$r_n=O(1/\sqrt {n})$$ , but that θ ̂ 1 $$\hat {{\boldsymbol {\theta }}}_1$$ still bears some information on θ1 when n r n $$\sqrt {n}r_n$$ converges to a positive constant. When n r n $$\sqrt {n}r_n$$ diverges to infinity, we provide asymptotic confidence zones for θ1 based on θ ̂ 1 $$\hat {{\boldsymbol {\theta }}}_1$$ . Our non-standard asymptotic results are supported by Monte Carlo exercises.

Keywords: Principal component analysis; Point estimation; Confidence zone estimation; Robustness; Weak identifiability (search for similar items in EconPapers)
Date: 2023
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-22687-8_3

Ordering information: This item can be ordered from
http://www.springer.com/9783031226878

DOI: 10.1007/978-3-031-22687-8_3

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().