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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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##### References:
 [1] Amari, S. (2002). Independent component analysis and method of estimating functions. IEICE Trans. Fundamentals Electronics, Communications and Computer Sciences E85-A 540-547. [2] Bickel, P. J. (1982). On adaptive estimation. Ann. Statist. 10 647-671. · Zbl 0489.62033 [3] Bickel, P. J., Klaassen, C. A. J., Ritov, Y. and Wellner, J. A. (1993). Efficient and Adaptive Statistical Inference for Semiparametric Models . Johns Hopkins Univ. Press, Baltimore. · Zbl 0786.62001 [4] Cardoso, J. F. (1989). Source separation using higher moments. In Proceedings of IEEE International Conference on Acoustics , Speech and Signal Processing , Glasgow 2109-2112. [5] Cassart, D., Hallin, M. and Paindaveine, D. (2010). On the estimation of cross-information quantities in R-estimation. In Nonparametrics and Robustness in Modern Statistical Inference and Time Series Analysis : A Festschrift in Honor of Professor Jana Jurečková (J. Antoch, M. Hušková and P. K. Sen, eds.) 35-45. IMS, Beachwood, OH. [6] Chen, A. and Bickel, P. J. (2006). Efficient independent component analysis. Ann. Statist. 34 2825-2855. · Zbl 1114.62033 [7] Hallin, M., Oja, H. and Paindaveine, D. (2006). Semiparametrically efficient rank-based inference for shape. II. Optimal R -estimation of shape. Ann. Statist. 34 2757-2789. · Zbl 1115.62059 [8] Hallin, M. and Paindaveine, D. (2006). Semiparametrically efficient rank-based inference for shape. I. Optimal rank-based tests for sphericity. Ann. Statist. 34 2707-2756. · Zbl 1114.62066 [9] Hallin, M. and Paindaveine, D. (2008). Semiparametrically efficient one-step R-estimation. Unpublished manuscript. Univ. Libre de Bruxelles. [10] Hallin, M., Vermandele, C. and Werker, B. (2006). Serial and nonserial sign-and-rank statistics: Asymptotic representation and asymptotic normality. Ann. Statist. 34 254-289. · Zbl 1091.62033 [11] Hallin, M. and Werker, B. J. M. (2003). Semi-parametric efficiency, distribution-freeness and invariance. Bernoulli 9 137-165. · Zbl 1020.62042 [12] Hyvärinen, A. (1999). Fast and robust fixed-point algorithms for independent component analysis. IEEE Trans. Neural Networks 10 626-634. [13] Hyvärinen, A. and Oja, E. (1997). A fast fixed-point algorithm for independent component analysis. Neural Comput. 9 1483-1492. [14] Ilmonen, P., Nevalainen, J. and Oja, H. (2010). Characteristics of multivariate distributions and the invariant coordinate system. Statist. Probab. Lett. 80 1844-1853. · Zbl 1202.62068 [15] Ilmonen, P., Nordhausen, K., Oja, H. and Ollila, E. (2011). Independent component (IC) functionals and a new performance index. Unpublished manuscript. Univ. Tampere. [16] Ilmonen, P. and Paindaveine, D. (2011). Supplement to “Semiparametrically efficient inference based on signed ranks in symmetric independent component models.” . · Zbl 1231.62043 [17] Kreiss, J.-P. (1987). On adaptive estimation in stationary ARMA processes. Ann. Statist. 15 112-133. · Zbl 0616.62042 [18] Le Cam, L. (1986). Asymptotic Methods in Statistical Decision Theory . Springer, New York. · Zbl 0605.62002 [19] Oja, H., Paindaveine, D. and Taskinen, S. (2011). Parametric and nonparametric tests for multivariate independence in IC models. Unpublished manuscript. Univ. Libre de Bruxelles. [20] Oja, H., Sirkiä, S. and Eriksson, J. (2006). Scatter matrices and independent component analysis. Austrian J. Statist. 35 175-189. [21] Ollila, E. (2010). The deflation-based FastICA estimator: Statistical analysis revisited. IEEE Trans. Signal Process. 58 1527-1541. · Zbl 1392.94370 [22] Ollila, E. and Kim, H. J. (2011). On testing hypotheses of mixing vectors in the ICA model using FastICA. In Proceedings of IEEE International Symposium on Biomedical Imaging ( ISBI ’), Chicago , IL 11 325-328. [23] Puri, M. L. and Sen, P. K. (1985). Nonparametric Methods in General Linear Models . Wiley, New York. · Zbl 0569.62024 [24] Rao, C. R. and Mitra, S. K. (1971). Generalized Inverse of Matrices and Its Applications . Wiley, New York. · Zbl 0236.15004 [25] Theis, F. J. (2004). A new concept for separability problems in blind source separation. Neural Comput. 16 1827-1850. · Zbl 1090.68567
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