Journal of Applied Probability

Improved approximation of the sum of random vectors by the skew normal distribution

Marcus C. Christiansen and Nicola Loperfido

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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.

Article information

J. Appl. Probab., Volume 51, Number 2 (2014), 466-482.

First available in Project Euclid: 12 June 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case)
Secondary: 15A69: Multilinear algebra, tensor products 62E17: Approximations to distributions (nonasymptotic)

Central limit theorem Cramer's condition order of convergence skew normal skewness third-order tensor


Christiansen, Marcus C.; Loperfido, Nicola. Improved approximation of the sum of random vectors by the skew normal distribution. J. Appl. Probab. 51 (2014), no. 2, 466--482. doi:10.1239/jap/1402578637.

Export citation


  • \item[] Adcock, C. J. (2007). Extensions of Stein's lemma for the skew-normal distribution. Commun. Statist. Theory Meth. 36, 1661–1671.
  • \item[] Adcock, C. J. (2010). Asset pricing and portfolio selection based on the multivariate extended skew-Student-$t$ distribution. Ann. Operat. Res. 176, 221–234.
  • \item[] Adcock, C. J. and Shutes, K. (2012). On the multivariate extended skew-normal, normal-exponential and normal-gamma distributions. J. Statist. Theory Practice 6, 636–664.
  • \item[] Arnold, B. C. and Beaver, R. J. (2002). Skewed multivariate models related to hidden truncation and/or selective reporting. Test 11, 7–54.
  • \item[] Azzalini, A. (1985) A class of distributions which includes the normal ones. Scand. J. Statist. 12, 171–178.
  • \item[] Azzalini, A. (2005). The skew-normal distribution and related multivariate families (with discussion). Scand. J. Statist. 32, 159–200.
  • \item[] Azzalini, A. (2006). Some recent developments in the theory of distributions and their applications. Atti della XLIII Riunione Scientifica della Società Italiana di Statistica, 51–64.
  • \item[] Azzalini, A. and Dalla Valle, A. (1996). The multivariate skew-normal distribution. Biometrika 83, 715–726.
  • \item[] Bartoletti, S. and Loperfido, N. (2010). Modelling air pollution data by the skew-normal distribution. Stoch. Environ. Res. Risk Assess. 24, 513–517.
  • \item[] Bhattacharya, R. N. and Ranga Rao, R. (1986). Normal Approximation and Asymptotic Expansions. Robert E. Krieger, Malabar, FL.
  • \item[] Brachat, J., Comon, P., Mourrain, B. and Tsigaridas, E. (2010). Symmetric tensor decomposition. Linear Algebra Appl. 433, 1851–1872.
  • \item[] Braman, K. (2010). Third-order tensors as linear operators on a space of matrices. Linear Algebra Appl. 433,, 1241–1253.
  • \item[] Brunekreef, B. and Holgate, S. T. (2002). Air pollution and health. Lancet 360, 1233–1242.
  • \item[] Chang, C.-H., Lin, J.-J., Pal, N. and Chiang, M.-C. (2008). A note on improved approximation of the binomial distribution by the skew-normal distribution. Amer. Statistician 62, 167–170.
  • \item[] Comon, P., Golub, G., Lim, L.-H. and Mourrain, B. (2008). Symmetric tensors and symmetric tensor rank. SIAM J. Matrix Anal. Appl. 30, 1254–1279.
  • \item[] Cont, R. (2001). Empirical properties of asset returns: stylized facts and statistical issues. Quantitative Finance 1, 223–236.
  • \item[] De Luca, G. and Loperfido, N. (2012). Modelling multivariate skewness in financial returns: a SGARCH approach. European J. Finance, DOI:10.1080/1351847X.2011.640342.
  • \item[] De Luca, G., Genton, M. and Loperfido, N. (2006). A multivariate skew-GARCH Model. In Econometric Analysis of Economic and Financial Time Series, Part A (Adv. Econometrics 20), eds T. B. Formby, D. Terrell and R. Carter Hill, JAI Press Inc., Bingley, UK, pp. 33–56.
  • \item[] Field, C. A. and Ronchetti, E. (1990). Small Sample Asymptotics. Institute of Mathematical Statistics – Monograph Series, Hayward, CA.
  • \item[] Franceschini, C. and Loperfido, N. (2010). A skewed GARCH-type model for multivariate financial time series. In Mathematical and Statistical Methods for Actuarial Sciences and Finance XII, eds M. Corazza and C. Pizzi, Springer, Milan, pp. 143–152.
  • \item[] Gupta, A. K. and Kollo, T. (2003). Density expansions based on the multivariate skew-normal distribution. Sankhya 65, 821–835.
  • \item[] Haas, M., Mittnik, S. and Paolella, M. S. (2009). Asymmetric multivariate normal mixture GARCH. Comput. Statist. Data Anal. 53, 2129–2154.
  • \item[] Kilmer, M. E. and Martin, C. D. (2011). Factorization strategies for third-order tensors. Linear Algebra Appl. 435, 641–658.
  • \item[] Kofidis, E. and Regalia, P. A. (2002). On the best rank-1 approximation of higher-order supersymmetric tensors. SIAM J. Matrix Anal. Appl. 23, 863–884.
  • \item[] Kolda, T. G. and Bader, B. W. (2009). Tensor decompositions and applications. SIAM Rev. 51, 455–500.
  • \item[] Kotz, S. and Vicari, D. (2005). Survey of developments in the theory of continuous skewed distributions. Metron 63, 225–261.
  • \item[] Loperfido, N. and Guttorp, P. (2008). Network bias in air quality monitoring design. Environmetrics 19, 661–671.
  • \item[] Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979). Multivariate Analysis. Academic Press, London.
  • \item[] Mateu-Figueras, G., Puig, P. and Pewsey, A. (2007). Goodness-of-fit tests for the skew-normal distribution when the parameters are estimated from the data. Commun. Statist. Theory Meth. 36, 1735–1755.
  • \item[] McCullagh, P. (1987). Tensor Methods in Statistics. Chapman and Hall, London.
  • \item[] McLachlan, G. and Peel, D. (2000). Finite Mixture Models. John Wiley, New York.
  • \item[] Qi, L. (2011). The best rank-one approximation ratio of a tensor space. SIAM J. Matrix Anal. Appl. 32, 430–442.
  • \item[] Qi, L., Sun, W. and Wang, Y. (2007). Numerical multilinear algebra and its applications. Frontiers Math. China 2, 501–526.
  • \item[] Rydberg, T. H. (2000). Realistic statistical modelling of financial data. Internat. Statist. Rev. 68, 233–258.
  • \item[] Serfling, R. J. (2006). Multivariate symmetry and asymmetry. In Encyclopedia of Statistical Sciences, 2nd edn, eds S. Kotz, C. B. Read, N. Balakrishnan and B. Vidakovic, Wiley, New York.
  • \item[] Van Hulle, M. M. (2005). Edgeworth approximation of multivariate differential entropy. Neural Computation 17, 1903–1910.
  • \item[] Wang, H. and Ahuja, N. (2004). Compact representation of multidimensional data using tensor rank-onedecomposition. In Pattern Recognition, 2004. (Proc. 17th Internat. Conf. Pattern Recognition 1), IEEE, Red Hook, NY., pp. 44–47.
  • \item[] Zhang, T. and Golub, G. H. (2001). Rank-one approximation to high order tensors. SIAM J. Matrix Anal. Appl. 23, 534–550. \endharvreferences