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

Abstract

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.

Singular value decomposition of the third multivariate moment

“…Random vectors whose third standardized moments have only one nonzero singular value appear in hidden truncation models, finite mixture models and multivariate density approximations (Christiansen and Loperfido [2]). …”

MaxSkew and MultiSkew: Two R Packages for Detecting, Measuring and Removing Multivariate Skewness

“…Similarly, the third moment (cumulant) of a random vector is a matrix containing all moments (cumulants) of order three which can be obtained from the random vector itself. Statistical applications of the third moment include, but are not limited to: factor analysis ( [1,2]), density approximation ( [3][4][5]), independent component analysis ( [6]), financial econometrics ( [7,8]), cluster analysis ( [4,[9][10][11][12]), Edgeworth expansions ( [13], page 189), portfolio theory ( [14]), linear models ( [15,16]), likelihood inference ( [17]), projection pursuit ( [18]), time series ( [7,19]), spatial statistics ( [20][21][22]) and nonrandom sampling ( [23]).…”