Characteristic functions of scale mixtures of multivariate skew-normal distributions. (English) Zbl 1221.60020
Summary: We obtain the characteristic function of scale mixtures of skew-normal distributions both in the univariate and multivariate cases. The derivation uses the simple stochastic relationship between skew-normal distributions and scale mixtures of skew-normal distributions. In particular, we describe the characteristic function of skew-normal, skew-\(t\), and other related distributions.
MSC:
| 60E10 | Characteristic functions; other transforms |
| 62E10 | Characterization and structure theory of statistical distributions |
| 60E05 | Probability distributions: general theory |
Keywords:
characteristic function; mixing; multivariate; scale mixture; selection; skew-normal; skew-\(t\)Software:
snReferences:
| [1] | Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (1972), Dover Publications, Inc.: Dover Publications, Inc. New York · Zbl 0543.33001 |
| [2] | Arellano-Valle, R. B.; Azzalini, A., On the unification of families of skew-normal distributions, Scandinavian Journal of Statistics, 33, 561-574 (2006) · Zbl 1117.62051 |
| [3] | Arellano-Valle, R. B.; Branco, M. D.; Genton, M. G., A unified view on skewed distributions arising from selections, The Canadian Journal of Statistics, 34, 581-601 (2006) · Zbl 1121.60009 |
| [4] | Arellano-Valle, R. B.; Genton, M. G., On fundamental skew distributions, Journal of Multivariate Analysis, 96, 93-116 (2005) · Zbl 1073.62049 |
| [5] | Arellano-Valle, R. B.; Genton, M. G., An invariance property of quadratic forms in random vectors with a selection distribution, with application to sample variogram and covariogram estimators, Annals of the Institute of Statistical Mathematics, 62, 363-381 (2010) · Zbl 1440.62178 |
| [6] | Azzalini, A., A class of distributions which includes the normal ones, Scandinavian Journal of Statistics, 12, 171-178 (1985) · Zbl 0581.62014 |
| [7] | Azzalini, A., The skew-normal distribution and related multivariate families (with discussion), Scandinavian Journal of Statistics, 32, 159-188 (2005), C/R 189-200 · Zbl 1091.62046 |
| [8] | Azzalini, A.; Capitanio, A., Statistical applications of the multivariate skew normal distribution, Journal of the Royal Statistical Society, Series B, 61, 579-602 (1999) · Zbl 0924.62050 |
| [9] | Azzalini, A.; Capitanio, A., Distributions generated by perturbation of symmetry with emphasis on a multivariate skew \(t\) distribution, Journal of the Royal Statistical Society, Series B, 65, 367-389 (2003) · Zbl 1065.62094 |
| [10] | Azzalini, A.; Dalla Valle, A., The multivariate skew-normal distribution, Biometrika, 83, 715-726 (1996) · Zbl 0885.62062 |
| [11] | Azzalini, A.; Genton, M. G., Robust likelihood methods based on the skew-\(t\) and related distributions, International Statistical Review, 76, 106-129 (2008) · Zbl 1206.62102 |
| [12] | Bisgaard, T. M.; Sasvári, Z., Characteristic Functions and Moment Sequences (2000), Nova Science Publishers, Inc.: Nova Science Publishers, Inc. New York · Zbl 1064.60027 |
| [13] | Branco, M. D.; Dey, D. K., A general class of multivariate skew-elliptical distributions, Journal of Multivariate Analysis, 79, 99-113 (2001) · Zbl 0992.62047 |
| [14] | Buckle, D. J., Bayesian inference for stable distributions, Journal of the American Statistical Association, 90, 605-613 (1995) · Zbl 0826.62020 |
| [15] | Genton, M. G., Skew-Elliptical Distributions and their Applications: A Journey Beyond Normality, Edited Volume (2004), Chapman & Hall, CRC: Chapman & Hall, CRC Boca Raton, FL · Zbl 1069.62045 |
| [16] | Henze, N., A probabilistic representation of the skew-normal distribution, Scandinavian Journal of Statistics, 13, 271-275 (1986) · Zbl 0648.62016 |
| [17] | S. Hurst, The characteristic function of the Student \(t\); S. Hurst, The characteristic function of the Student \(t\) |
| [18] | Kotz, S.; Nadarajah, S., Multivariate t Distributions and their Applications (2004), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1100.62059 |
| [19] | Kozubowski, T. J.; Podgórski, K., Skew Laplace distributions. I. Their origins and inter-relations, Mathematical Scientist, 33, 22-34 (2008) · Zbl 1167.62014 |
| [20] | Lachos, V. H.; Dey, D. K.; Cancho, V. G., Robust linear mixed models with skew-normal independent distributions from a Bayesian perspective, Journal of Statistical Planning and Inference, 139, 4098-4110 (2009) · Zbl 1183.62048 |
| [21] | Lukacs, E., Characteristic Functions (1970), Griffin: Griffin London · Zbl 0201.20404 |
| [22] | Lukacs, E., Developments in Characteristic Function Theory (1983), Macmillan Publishing Co., Inc.: Macmillan Publishing Co., Inc. New York · Zbl 0515.60022 |
| [23] | Nelson, P. R., An approximation for the complex normal probability integral, BIT Numerical Mathematics, 22, 94-100 (1982) · Zbl 0486.30023 |
| [24] | Pewsey, A., The wrapped skew-normal distribution on the circle, Communications in Statistics—Theory and Methods, 29, 2459-2472 (2000) · Zbl 0992.62048 |
| [25] | Samorodnitsky, G.; Taqqu, M. S., Stable Non-Gaussian Random Processes (1994), Chapman and Hall: Chapman and Hall New York · Zbl 0925.60027 |
| [26] | Steck, G. P.; Owen, B., A note on the equicorrelated multivariate normal distribution, Biometrika, 49, 269-271 (1962) · Zbl 0112.11110 |
| [27] | Wang, J.; Genton, M. G., The multivariate skew-slash distribution, Journal of Statistical Planning and Inference, 136, 209-220 (2006) · Zbl 1081.60013 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
