Moments of skew-normal random vectors and their quadratic forms

2001 ◽  
Vol 51 (4) ◽  
pp. 319-325 ◽  
Author(s):  
Marc G. Genton ◽  
Li He ◽  
Xiangwei Liu
Keyword(s):  
Quadratic Forms ◽  
Random Vectors ◽  
Skew Normal

scholarly journals Improved Approximation of the Sum of Random Vectors by the Skew Normal Distribution

2014 ◽  
Vol 51 (2) ◽  
pp. 466-482 ◽  
Author(s):  
Marcus C. Christiansen ◽  
Nicola Loperfido
Keyword(s):  
Normal Distribution ◽  
Normal Approximation ◽  
Distribution Functions ◽  
Skew Normal Distribution ◽  
Random Vectors ◽  
Multivariate Distributions ◽  
Uniform Distance ◽  
Special Case ◽  
Independent Identically Distributed ◽  
Skew Normal

We study the properties of the multivariate skew normal distribution as an approximation to the distribution of the sum of n independent, identically distributed random vectors. More precisely, we establish conditions ensuring that the uniform distance between the two distribution functions converges to 0 at a rate of n-2/3. The advantage over the corresponding normal approximation is particularly relevant when the summands are skewed and n is small, as illustrated for the special case of exponentially distributed random variables. Applications to some well-known multivariate distributions are also discussed.

scholarly journals On quadratic forms in multivariate generalized hyperbolic random vectors

Biometrika ◽  
2020 ◽  
Author(s):  
Simon A Broda ◽  
Juan Arismendi Zambrano
Keyword(s):  
Least Squares ◽  
Quadratic Forms ◽  
Inversion Formula ◽  
Expected Shortfall ◽  
Distribution Functions ◽  
Least Squares Estimator ◽  
Tail Probabilities ◽  
Random Vectors ◽  
Two Stage ◽  
Approximate Expressions

Summary This article presents exact and approximate expressions for tail probabilities and partial moments of quadratic forms in multivariate generalized hyperbolic random vectors. The derivations involve a generalization of the classic inversion formula for distribution functions (Gil-Pelaez, 1951). Two numerical applications are considered: the distribution of the two-stage least squares estimator and the expected shortfall of a quadratic portfolio.

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