On multivariate skewness and kurtosis

Let $X$ be a $d$-dimensional standardized random variable $({\bf E}(X) = 0, \operatorname{cov} (X) = 1)$. Then for a multivariate analogue of skewness $s = {\bf E}(\| X \|^2 X)$ and kurtosis $k = {\bf E}XX^T XX^T - (d + 2)I$ we show that $\| s \|^2 \leqq {\text{tr}}\, k + 2d$. For infinitely divisib...

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Veröffentlicht in:	Theory of probability and its applications 1994-09, Vol.38 (3), p.547-551
Hauptverfasser:	MORI, T. F, ROHATGI, V. K, SZEKELY, G. J
Format:	Artikel
Sprache:	eng
Schlagworte:	Distribution theory Exact sciences and technology Inequality Kurtosis Mathematics Probability and statistics Probability theory and stochastic processes Random variables Sciences and techniques of general use Skewness
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Beschreibung
Zusammenfassung:	Let $X$ be a $d$-dimensional standardized random variable $({\bf E}(X) = 0, \operatorname{cov} (X) = 1)$. Then for a multivariate analogue of skewness $s = {\bf E}(\\| X \\|^2 X)$ and kurtosis $k = {\bf E}XX^T XX^T - (d + 2)I$ we show that $\\| s \\|^2 \leqq {\text{tr}}\, k + 2d$. For infinitely divisible distributions $\\| s \\|^2 \leqq {\text{tr }}k$.
ISSN:	0040-585X 1095-7219
DOI:	10.1137/1138055