Remarks on definitions of the anisotropic norm of a random vector

Alternative definitions of the concentration ellipsoid of a random vector are surveyed, and an extension of the concentration ellipsoid of Darmois is suggested as being the most convenient and natural definition. The advantage of the proposed definition in providing substantially simplified proofs of results in (linear) estimation theory is discussed, and is illustrated by new and short proofs of two key results. A not-so-well-known, but elementary, extremal representation of a nonnegative definite quadratic form, together with the corresponding Cauchy-Schwarẓ-type inequality, is seen to play a crucial role in these proofs.

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Lifetime estimation from doubly censored data and dirichlet process prior knowledge on the observable random vector

Communication in Statistics- Theory and Methods ◽

10.1080/03610929708831998 ◽

1997 ◽

Vol 26 (6) ◽

pp. 1541-1558 ◽

Cited By ~ 1

Author(s):

Arturo J. Fernández ◽

José I. Bravo ◽

[d]éigo De Fuentes

Keyword(s):

Prior Knowledge ◽

Censored Data ◽

Random Vector ◽

Dirichlet Process ◽

Lifetime Estimation ◽

Dirichlet Process Prior ◽

Doubly Censored Data ◽

Doubly Censored

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Utilization of random vector functional link integrated with manta ray foraging optimization for effluent prediction of wastewater treatment plant

Journal of Environmental Management ◽

10.1016/j.jenvman.2021.113520 ◽

2021 ◽

Vol 298 ◽

pp. 113520

Author(s):

Khaled Elmaadawy ◽

Mohamed Abd Elaziz ◽

Ammar H. Elsheikh ◽

Ahmed Moawad ◽

Bingchuan Liu ◽

...

Keyword(s):

Wastewater Treatment ◽

Random Vector ◽

Wastewater Treatment Plant ◽

Treatment Plant ◽

Functional Link ◽

Manta Ray

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Multivariate extremes, aggregation and dependence in elliptical distributions

Advances in Applied Probability ◽

10.1017/s0001867800011770 ◽

2002 ◽

Vol 34 (03) ◽

pp. 587-608 ◽

Cited By ~ 30

Author(s):

Henrik Hult ◽

Filip Lindskog

Keyword(s):

Random Vector ◽

Positive Constant ◽

Tail Dependence ◽

Constant Factor ◽

Dispersion Matrix ◽

Elliptical Distributions ◽

Spectral Measures ◽

Lower Tail ◽

Explicit Formulae ◽

Dependence Properties

In this paper, we clarify dependence properties of elliptical distributions by deriving general but explicit formulae for the coefficients of upper and lower tail dependence and spectral measures with respect to different norms. We show that an elliptically distributed random vector is regularly varying if and only if the bivariate marginal distributions have tail dependence. Furthermore, the tail dependence coefficients are fully determined by the tail index of the random vector (or equivalently of its components) and the linear correlation coefficient. Whereas Kendall's tau is invariant in the class of elliptical distributions with continuous marginals and a fixed dispersion matrix, we show that this is not true for Spearman's rho. We also show that sums of elliptically distributed random vectors with the same dispersion matrix (up to a positive constant factor) remain elliptical if they are dependent only through their radial parts.

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