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- Koizumi, K., et al.
(författare)
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Measures of multivariate skewness and kurtosis in high-dimensional framework
- 2014
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Ingår i: SUT Journal of Mathematics. - : Tokyo University of Science. - 0916-5746. ; 50:2, s. 483-511
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Tidskriftsartikel (refereegranskat)abstract
- Skewness and kurtosis characteristics of a multivariate p-dimensional distribution introduced by Mardia (1970) have been used in various testing procedures and demonstrated attractive asymptotic properties in large sample settings. However these characteristics are not designed for high-dimensional problems where the dimensionality, p can largely exceeds the sample size, N. Such type of high-dimensional data are commonly encountered in modern statistical applications. This the suggests that new measures of skewness and kurtosis that can accommodate high-dimensional settings must be derived and carefully studied. In this paper, we show that, by exploiting the dependence structure, new expressions for skewness and kurtosis are introduced as an extension of the corresponding Mardia’s measures, which uses the potential advantages that the block-diagonal covariance structure has to offer in high dimensions. Asymptotic properties of newly derived measures are investigated and the cumulant based characterizations are presented along with of applications to a mixture of multivariate normal distributions and multivariate Laplace distribution, for which the explicit expressions of skewness and kurto-sis are obtained. Test statistics based on the new measures of skewness and kurtosis are proposed for testing a distribution shape, and their limit distributions are established in the asymptotic framework where N → ∞ and p is fixed but large, including p > N. For the dependence structure learning, the gLasso based technique is explored followed by AIC step which we propose for optimization of the gLasso candidate model. Performance accuracy of the test procedures based on our estimators of skewness and kurtosis are evaluated using Monte Carlo simulations and the validity of the suggested approach is shown for a number of cases when p > N.
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| 2. |
- Koizumi, K., et al.
(författare)
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Modified jarque-bera type tests for multivariate normality in a high-dimensional framework
- 2014
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Ingår i: Journal of Statistical Theory and Practice. - : Springer Science and Business Media LLC. - 1559-8608 .- 1559-8616. ; 8:2, s. 382-399
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Tidskriftsartikel (refereegranskat)abstract
- In this article, we introduce two types of new omnibus procedures for testing multivariate normality based on the sample measures of multivariate skewness and kurtosis. These characteristics, initially introduced by, for example, Mardia (1970) and Srivastava (1984), were then extended by Koizumi, Okamoto, and Seo (2009), who proposed the multivariate Jarque-Bera type test () based on the Srivastava (1984) principal components measure scores of skewness and kurtosis. We suggest an improved MJB test () that is based on the Wilson-Hilferty transform, and a modified MJB test () that is based on the F-approximation to. Asymptotic properties of both tests are examined, assuming that both dimensionality and sample size go to infinity at the same rate. Our simulation study shows that the suggested test outperforms both and for a number of high-dimensional scenarios. The test is then used for testing multivariate normality of the real data digitalized character image.
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