Asymptotic expansions for distribution functions and densities of sums of independent random vectors

1981 ◽  
Vol 21 (2) ◽  
pp. 153-162
Author(s):  
N. Lazakovičius
Keyword(s):  
Asymptotic Expansions ◽  
Distribution Functions ◽  
Random Vectors

scholarly journals The asymptotic expansions of the distribution function and of density function for the sums of independent identically distributed random vectors

1968 ◽  
Vol 8 (3) ◽  
pp. 405-422
Author(s):  
A. Bikelis
Keyword(s):  
Distribution Function ◽  
Density Function ◽  
Asymptotic Expansions ◽  
Random Vectors ◽  
Independent Identically Distributed

The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: А. Бикялис. Асимптотические разложения для плотностей и распределений сумм независимых одинаково распределенных случайных векторов A. Bikelis. Nepriklausomų vienodai pasiskirsčiusių atsitiktinių vektorių sumų tankių ir pasiskirstymo funkcijų asimptotiniai išdėstymai

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 The difference between the product and the convolution product of distribution functions in Rn

2011 ◽  
Vol 89 (103) ◽  
pp. 19-36 ◽  
Author(s):  
E. Omey ◽  
R. Vesilo
Keyword(s):  
Distribution Function ◽  
Distribution Functions ◽  
Convolution Product ◽  
Random Vectors ◽  
The Difference

Assume that X? and Y? are independent, nonnegative d-dimensional random vectors with distribution function (d.f.) F(x?) and G(x?), respectively. We are interested in estimates for the difference between the product and the convolution product of F and G, i.e., D(x?) = F(x?)G(x?) ? F ? G(x?). Related to D(x?) is the difference R(x?) between the tail of the convolution and the sum of the tails: R(x?) = (1 ? F ? G(x?))?(1 ? F(x?) + 1 ? G(x?)). We obtain asymptotic inequalities and asymptotic equalities for D(x?) and R(x?). The results are multivariate analogues of univariate results obtained by several authors before.

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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