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Semiparametrically efficient inference based on signed ranks in symmetric independent component models. (English) Zbl 1231.62043
Summary: We consider semiparametric location-scatter models for which a \(p\)-variate observation is obtained as \(X = \Lambda Z + \mu \), where \(\mu \) is a \(p\)-vector, \(\Lambda \) is a full-rank \(p \times p\) 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 \(\Lambda \). 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 \(\Lambda \). 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.

MSC:
62G05 Nonparametric estimation
62G10 Nonparametric hypothesis testing
62H12 Estimation in multivariate analysis
62H15 Hypothesis testing in multivariate analysis
62G20 Asymptotic properties of nonparametric inference
62G08 Nonparametric regression and quantile regression
62J05 Linear regression; mixed models
65C60 Computational problems in statistics (MSC2010)
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