Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-27T03:24:07.316Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Probability theory on algebraic and topological structures Distribution theory Limit theorems Basic linear algebra

Published online by Cambridge University Press:  19 February 2016

Marcus C. Christiansen  and
Nicola Loperfido
Marcus C. Christiansen*
Affiliation:
University of Ulm
Nicola Loperfido*
Affiliation:
Università degli Studi di Urbino Carlo Bo
*
∗ Postal address: Institute of Insurance Science, University of Ulm, 89081 Ulm, Germany. Email address: [email protected].
∗∗ Postal address: Dipartimento di Economia, Politica e Società, Università degli Studi di Urbino Carlo Bo, Via Saffi 42, 61029 Urbino, Italy.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

Central limit theoremCramer's conditionorder of convergenceskew normalskewnessthird-order tensor

MSC classification

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)
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 2 , June 2014 , pp. 466 - 482
Copyright
© Applied Probability Trust 

References

Adcock, C. J. (2007). Extensions of Stein's lemma for the skew-normal distribution. Commun. Statist. Theory Meth. 36, 1661–1671.CrossRefGoogle Scholar
Adcock, C. J. (2010). Asset pricing and portfolio selection based on the multivariate extended skew-Student-t distribution. Ann. Operat. Res. 176, 221–234.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Arnold, B. C. and Beaver, R. J. (2002). Skewed multivariate models related to hidden truncation and/or selective reporting. Test 11, 7–54.Google Scholar
Azzalini, A. (1985) A class of distributions which includes the normal ones. Scand. J. Statist. 12, 171–178.Google Scholar
Azzalini, A. (2005). The skew-normal distribution and related multivariate families (with discussion). Scand. J. Statist. 32, 159–200.CrossRefGoogle Scholar
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.Google Scholar
Azzalini, A. and Dalla Valle, A. (1996). The multivariate skew-normal distribution. Biometrika 83, 715–726.CrossRefGoogle Scholar
Bartoletti, S. and Loperfido, N. (2010). Modelling air pollution data by the skew-normal distribution. Stoch. Environ. Res. Risk Assess. 24, 513–517.CrossRefGoogle Scholar
Bhattacharya, R. N. and Ranga Rao, R. (1986). Normal Approximation and Asymptotic Expansions. Robert E. Krieger, Malabar, FL.Google Scholar
Brachat, J., Comon, P., Mourrain, B. and Tsigaridas, E. (2010). Symmetric tensor decomposition. Linear Algebra Appl. 433, 1851–1872.CrossRefGoogle Scholar
Braman, K. (2010). Third-order tensors as linear operators on a space of matrices. Linear Algebra Appl. 433, 1241–1253.CrossRefGoogle Scholar
Brunekreef, B. and Holgate, S. T. (2002). Air pollution and health. Lancet 360, 1233–1242.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Cont, R. (2001). Empirical properties of asset returns: stylized facts and statistical issues. Quantitative Finance 1, 223–236.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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 Formby, T. B., Terrell, D. and Carter Hill, R., JAI Press Inc., Bingley, UK, pp. 33–56.CrossRefGoogle Scholar
Field, C. A. and Ronchetti, E. (1990). Small Sample Asymptotics. Institute of Mathematical Statistics – Monograph Series, Hayward, CA.CrossRefGoogle Scholar
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 Corazza, M. and Pizzi, C., Springer, Milan, pp. 143–152.CrossRefGoogle Scholar
Gupta, A. K. and Kollo, T. (2003). Density expansions based on the multivariate skew-normal distribution. Sankhya 65, 821–835.Google Scholar
Haas, M., Mittnik, S. and Paolella, M. S. (2009). Asymmetric multivariate normal mixture GARCH. Comput. Statist. Data Anal. 53, 2129–2154.Google Scholar
Kilmer, M. E. and Martin, C. D. (2011). Factorization strategies for third-order tensors. Linear Algebra Appl. 435, 641–658.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Kolda, T. G. and Bader, B. W. (2009). Tensor decompositions and applications. SIAM Rev. 51, 455–500.CrossRefGoogle Scholar
Kotz, S. and Vicari, D. (2005). Survey of developments in the theory of continuous skewed distributions. Metron 63, 225–261.Google Scholar
Loperfido, N. and Guttorp, P. (2008). Network bias in air quality monitoring design. Environmetrics 19, 661–671.CrossRefGoogle Scholar
Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979). Multivariate Analysis. Academic Press, London.Google Scholar
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.CrossRefGoogle Scholar
McCullagh, P. (1987). Tensor Methods in Statistics. Chapman and Hall, London.Google Scholar
McLachlan, G. and Peel, D. (2000). Finite Mixture Models. John Wiley, New York.CrossRefGoogle Scholar
Qi, L. (2011). The best rank-one approximation ratio of a tensor space. SIAM J. Matrix Anal. Appl. 32, 430–442.CrossRefGoogle Scholar
Qi, L., Sun, W. and Wang, Y. (2007). Numerical multilinear algebra and its applications. Frontiers Math. China 2, 501–526.CrossRefGoogle Scholar
Rydberg, T. H. (2000). Realistic statistical modelling of financial data. Internat. Statist. Rev. 68, 233–258.CrossRefGoogle Scholar
Serfling, R. J. (2006). Multivariate symmetry and asymmetry. In Encyclopedia of Statistical Sciences, 2nd edn, eds Kotz, S., Read, C. B., Balakrishnan, N. and Vidakovic, B., Wiley, New York.Google Scholar
Van Hulle, M. M. (2005). Edgeworth approximation of multivariate differential entropy. Neural Computation 17, 1903–1903.CrossRefGoogle ScholarPubMed
Wang, H. and Ahuja, N. (2004). Compact representation of multidimensional data using tensor rank-one decomposition. In Pattern Recognition, 2004. (Proc. 17th Internat. Conf. Pattern Recognition 1), IEEE, Red Hook, NY., pp. 44–47.Google Scholar
Zhang, T. and Golub, G. H. (2001). Rank-one approximation to high order tensors. SIAM J. Matrix Anal. Appl. 23, 534–550.CrossRefGoogle Scholar