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Tests and estimates of shape based on spatial signs and ranks. (English) Zbl 1359.62150

Summary: Nonparametric procedures for testing and estimation of the shape matrix in the case of multivariate elliptic distribution are considered. Testing for sphericity is an important special case. The tests and estimates are based on the spatial sign and rank covariance matrices. The estimates based on the spatial sign covariance matrix and symmetrized spatial sign covariance matrix are D. E. Tyler’s [Ann. Stat. 15, 234–251 (1987; Zbl 0628.62053)] shape matrix and and L. Dümbgen’s [Ann. Inst. Stat. Math. 50, No. 3, 471–491 (1998; Zbl 0912.62061)] shape matrix, respectively. The test based on the spatial sign covariance matrix is the sign test statistic in the class of nonparametric tests proposed by M. Hallin and D. Paindaveine [Ann. Stat. 34, No. 6, 2707–2756 (2006; Zbl 1114.62066)]. New tests and estimates based on the spatial rank covariance matrix are proposed. The shape estimates introduced in the paper play an important role in the inner standardisation of the spatial sign and rank tests for multivariate location. Limiting distributions of the tests and estimates are reviewed and derived, and asymptotic efficiencies as well as finite-sample efficiencies of the proposed tests are compared with those of the classical modified S. John’s [Biometrika 58, 123–127 (1971; Zbl 0218.62055); ibid. 59, 169–173 (1972; Zbl 0231.62072)] test and the van der Waerden test [Hallin and Paindaveine, loc. cit.]. The symmetrised spatial sign- and rank-based estimates and tests seem to have a very high efficiency in the multivariate normal case, and they are much better than the classical estimate (shape matrix based on the regular covariance matrix) and test (John’s test) for distributions with heavy tails.

MSC:

62G10 Nonparametric hypothesis testing
62M30 Inference from spatial processes
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