The Annals of Statistics

Semiparametrically efficient inference based on signed ranks in symmetric independent component models

Pauliina Ilmonen and Davy Paindaveine

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Abstract

We consider semiparametric location-scatter models for which the p-variate observation is obtained as X = ΛZ + μ, where μ is a p-vector, Λ is a full-rank p × p matrix and the (unobserved) random p-vector Z has marginals that are centered and mutually independent but are otherwise unspecified. As in blind source separation and independent component analysis (ICA), the parameter of interest throughout the paper is Λ. On the basis of n i.i.d. copies of X, we develop, under a symmetry assumption on Z, signed-rank one-sample testing and estimation procedures for Λ. We exploit the uniform local and asymptotic normality (ULAN) of the model to define signed-rank procedures that are semiparametrically efficient under correctly specified densities. Yet, as is usual in rank-based inference, the proposed procedures remain valid (correct asymptotic size under the null, for hypothesis testing, and root-n consistency, for point estimation) under a very broad range of densities. We derive the asymptotic properties of the proposed procedures and investigate their finite-sample behavior through simulations.

Article information

Source
Ann. Statist., Volume 39, Number 5 (2011), 2448-2476.

Dates
First available in Project Euclid: 30 November 2011

Permanent link to this document
https://projecteuclid.org/euclid.aos/1322663464

Digital Object Identifier
doi:10.1214/11-AOS906

Mathematical Reviews number (MathSciNet)
MR2906874

Zentralblatt MATH identifier
1231.62043

Subjects
Primary: 62G05: Estimation 62G10: Hypothesis testing
Secondary: 62G20: Asymptotic properties 62H99: None of the above, but in this section

Keywords
Independent component analysis local asymptotic normality rank-based inference semiparametric efficiency signed ranks

Citation

Ilmonen, Pauliina; Paindaveine, Davy. Semiparametrically efficient inference based on signed ranks in symmetric independent component models. Ann. Statist. 39 (2011), no. 5, 2448--2476. doi:10.1214/11-AOS906. https://projecteuclid.org/euclid.aos/1322663464


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Supplemental materials

  • Supplementary material: Further results on tests and a proof of Theorem 4.3. This supplement provides a simple explicit expression for the proposed test statistics, derives local asymptotic powers of the corresponding tests, and presents simulation results for hypothesis testing. It also gives a proof of Theorem 4.3.