The skew-normal distribution and related multivariate families. (English) Zbl 1091.62046
If \(f_0\) is a \(d\)-dimensional density with \(f_0(x)=f_0(-x)\), \(G\) is a one-dimensional CDF with symmetric about 0 PDF and \(w: R^d\to R\) is any function with \(w(-x)=-w(x)\), then \(f(z)=2f_0(z)G(w(z))\) is a density function on \(R^d\). With \(f_0\sim N(0,\Omega)\), \(G\sim N(0,1)\), and a linear function \(w\) one obtains \(f\) being a skew-normal (SN) multivariate distribution. An overview of properties of univariate and multivariate SN distributions is given. Statistical inference for i.i.d. SN observations based on the maximum likelihood is described. Skew-elliptical distributions with elliptical \(f_0\) (specially skew \(t\)-distribution) and distributions with polynomial \(w\) are discussed. Such distributions arise in selective sampling problems, stochastic frontier models and some models of financial markets. In the discussion some semiparametric and nonparametric problems related to such distributions are described.
Reviewer: R. E. Maiboroda (Kyïv)
MSC:
| 62H10 | Multivariate distribution of statistics |
| 62H12 | Estimation in multivariate analysis |
| 62E15 | Exact distribution theory in statistics |
| 62F10 | Point estimation |
Keywords:
flexible parametric family; graphical model; hidden truncation model; selective sampling; skew-elliptical distribution; stochastic frontier modelSoftware:
snReferences:
| [1] | Adcock C., Skew-elliptical distributions and their applications pp 191– (2004) |
| [2] | DOI: 10.1016/0304-4076(77)90052-5 · Zbl 0366.90026 · doi:10.1016/0304-4076(77)90052-5 |
| [3] | Aitchison J., The statistical analysis of compositional data (1986) · Zbl 0688.62004 · doi:10.1007/978-94-009-4109-0 |
| [4] | DOI: 10.1093/biomet/86.2.351 · Zbl 0931.62009 · doi:10.1093/biomet/86.2.351 |
| [5] | DOI: 10.1023/B:MATG.0000002983.12476.89 · Zbl 1031.62051 · doi:10.1023/B:MATG.0000002983.12476.89 |
| [6] | Andel J., Kybernetika 20 pp 89– (1984) |
| [7] | R. Arellano-Valle, and A. Azzalini (2004 ). On the unification of families of skew-normal distributions . Submitted. · Zbl 1117.62051 |
| [8] | Arellano-Valle R. B., J. Data Sci. 3 (2005) |
| [9] | DOI: 10.1016/S0167-7152(02)00088-3 · Zbl 1045.62046 · doi:10.1016/S0167-7152(02)00088-3 |
| [10] | Arellano-Valle R. B., J. Multivariate Anal. (2005) |
| [11] | Arellano-Valle R. B., J. Multivariate Anal. (2005) |
| [12] | Arnold B. C., Sankhya, series A 62 pp 22– (2000) |
| [13] | DOI: 10.1016/S0167-7152(00)00059-6 · Zbl 0969.62037 · doi:10.1016/S0167-7152(00)00059-6 |
| [14] | Arnold B. C., Amer. J. Math. Manage. Sci. 20 pp 27– (2000) · Zbl 1189.62087 · doi:10.1080/01966324.2000.10737499 |
| [15] | Arnold B. C., Psychometrika 58 pp 471– (1993) |
| [16] | Azzalini A., Scand. J. Statist. 12 pp 171– (1985) |
| [17] | Azzalini A., Statistica pp 199– (1986) · Zbl 0964.62001 |
| [18] | Azzalini A., Metron pp 27– (2001) |
| [19] | DOI: 10.1111/1467-9868.00194 · Zbl 0924.62050 · doi:10.1111/1467-9868.00194 |
| [20] | DOI: 10.1111/1467-9868.00391 · Zbl 1065.62094 · doi:10.1111/1467-9868.00391 |
| [21] | DOI: 10.1093/biomet/83.4.715 · Zbl 0885.62062 · doi:10.1093/biomet/83.4.715 |
| [22] | Birnbaum Z. W., Ann. Math. Statist. 21 pp 272– (1950) |
| [23] | DOI: 10.1006/jmva.2000.1960 · Zbl 0992.62047 · doi:10.1006/jmva.2000.1960 |
| [24] | DOI: 10.1111/1467-9469.00322 · Zbl 1035.60008 · doi:10.1111/1467-9469.00322 |
| [25] | Cappuccio N., Studies nonlinear dyn. econometrics 8 (2004) |
| [26] | Chiogna M., Notes on estimation problems with scalar skew-normal distributions (1997) |
| [27] | Chiogna M., J. Ital. Statist. Soc. 7 pp 1– (1998) · doi:10.1007/BF03178918 |
| [28] | Cook R. D., An introduction to regression graphics (1994) · Zbl 0925.62287 · doi:10.1002/9780470316863 |
| [29] | DOI: 10.1111/1467-9868.00055 · doi:10.1111/1467-9868.00055 |
| [30] | De Luca G., Skew-elliptical distributions and their applications pp 205– (2004) |
| [31] | DOI: 10.1198/016214504000000359 · Zbl 1117.62318 · doi:10.1198/016214504000000359 |
| [32] | Dominguez-Molina J. A., Skew-elliptical distributions and their applications pp 223– (2004) |
| [33] | DOI: 10.1016/S0378-3758(02)00155-6 · Zbl 1041.62036 · doi:10.1016/S0378-3758(02)00155-6 |
| [34] | Ellison B. E., J. Am. Statist. Assoc. 59 pp 89– (1964) |
| [35] | DOI: 10.1016/S0047-259X(03)00054-X · Zbl 1042.60005 · doi:10.1016/S0047-259X(03)00054-X |
| [36] | Fang K.-T., Symmetric multivariate and related distributions (1990) · doi:10.1007/978-1-4899-2937-2 |
| [37] | Genton M. G., Skew-elliptical distributions and their applications: a journey beyond normality (2004) · Zbl 1069.62045 |
| [38] | Genton M. G., Skew-elliptical distributions and their applications pp 81– (2004) · Zbl 1069.62045 · doi:10.1201/9780203492000 |
| [39] | DOI: 10.1016/S0167-7152(00)00164-4 · Zbl 0972.62031 · doi:10.1016/S0167-7152(00)00164-4 |
| [40] | Genton M. G., Ann. Inst. Statist. Math. (2005) |
| [41] | DOI: 10.1016/j.jspi.2003.09.008 · Zbl 1076.62052 · doi:10.1016/j.jspi.2003.09.008 |
| [42] | Gonzalez-Farias G., Skew-elliptical distributions and their applications pp 25– (2004) |
| [43] | S. Goria (1999 ). Uso di generalizzazioni della distribuzione normale nei modelli per serie storiche. Tesi di laurea , Facolta di Scienze Statistiche, Universita di Padova, Padova, Italia. |
| [44] | Gualtierotti A., 8th World Multiconference on Systemics, Cybernetics and Informatics (SCI 2004) pp 88– (2004) |
| [45] | Gupta A. K., Statistics 37 pp 359– (2003) |
| [46] | Gupta A. K., Random Operators Stochastic Eqs 10 pp 133– (2002) |
| [47] | DOI: 10.1016/S0022-247X(02)00270-6 · Zbl 1005.62058 · doi:10.1016/S0022-247X(02)00270-6 |
| [48] | Healy M. J. R., Appl. Stat. 17 pp 157– (1968) |
| [49] | Henze N., Scand. J. Statist. 13 pp 271– (1986) |
| [50] | Hill M. A., Biometrics 38 pp 377– (1982) |
| [51] | DOI: 10.1016/S0167-7152(03)00121-4 · Zbl 1116.62357 · doi:10.1016/S0167-7152(03)00121-4 |
| [52] | DOI: 10.1016/S0378-3758(02)00501-3 · Zbl 1038.62027 · doi:10.1016/S0378-3758(02)00501-3 |
| [53] | Lange K. L., J. Am. Statist. Assoc. 84 pp 881– (1989) |
| [54] | Liseo B., Statistica pp 59– (1990) |
| [55] | Liseo B., Skew-elliptical distributions and their applications pp 153– (2004) |
| [56] | DOI: 10.1016/S0167-7152(02)00398-X · Zbl 1101.62342 · doi:10.1016/S0167-7152(02)00398-X |
| [57] | Liseo B., Proceedings of the 2003 Meeting of the Italian Statistical Society (2003) |
| [58] | Liseo B., J. Statist. Plann. Inference (2004) |
| [59] | DOI: 10.1016/S0167-7152(01)00103-1 · Zbl 1002.62039 · doi:10.1016/S0167-7152(01)00103-1 |
| [60] | DOI: 10.1016/S0167-7152(01)00052-9 · Zbl 0994.62012 · doi:10.1016/S0167-7152(01)00052-9 |
| [61] | Loperfido N. M. R., Skew-elliptical distributions and their applications pp 65– (2004) |
| [62] | DOI: 10.1111/j.1467-9469.2004.03_007.x · Zbl 1063.62079 · doi:10.1111/j.1467-9469.2004.03_007.x |
| [63] | Ma Y., Skew-elliptical distributions and their applications pp 339– (2004) |
| [64] | G. Mateu-Figueras (2003 ).Models de distribucio sobre el simplex. PhD thesis , Universitat Politecnica de Catalunya, Barcelona. |
| [65] | Monti A. C., Metron pp 205– (2003) |
| [66] | Mukhopadhyay S., J. R. Stat. Soc. series D 44 pp 389– (1995) |
| [67] | Muliere P., Metron pp 21– (2002) |
| [68] | DOI: 10.1016/j.spl.2003.07.013 · Zbl 1048.62014 · doi:10.1016/j.spl.2003.07.013 |
| [69] | Nelson L. S., Technometrics 6 pp 469– (1964) |
| [70] | O’Hagan A., Biometrika 63 pp 201– (1976) |
| [71] | Owen D. B., Ann. Math. Statist. 27 pp 1075– (1956) |
| [72] | DOI: 10.1080/02664760050120542 · Zbl 1076.62514 · doi:10.1080/02664760050120542 |
| [73] | Pewsey A., Commun. Statist. - Theory Methods 29 pp 2459– (2000) |
| [74] | E. Pietrobon (2003 ). Modelli ARCH, GARCH e SV con componenti di errore aventi distribuzionetasimmetrica. Tesi di laurea , Facolta di Scienze Statistiche, Universita di Padova, Padova, Italia. |
| [75] | Roberts C., J. Am. Statist. Assoc. 61 pp 1184– (1966) |
| [76] | Rotnitzky A., Bernoulli 6 pp 243– (2000) |
| [77] | Sahu K., Can. J. Statist. 31 pp 129– (2003) |
| [78] | Salvan A., Atti della XXXIII Riunione Scientifica della Societa Italiana di Statistica pp 173– (1986) |
| [79] | Sartori N., J. Statist. Plann. Inference (2005) |
| [80] | Serfling R., Encyclopedia of statistical sciences (2004) |
| [81] | Subbotin M. F., Matematicheskii Sbornik 31 pp 296– (1923) |
| [82] | Tancredi A., Accounting for heavy tails in stochastic frontier models (2002) |
| [83] | Wang J., Statistica Sinica 14 pp 1259– (2004) |
| [84] | Weinstein M. A., Technometrics 6 pp 104– (1964) |
| [85] | Zacks S., Parametric statistical inference (1981) · Zbl 0452.62020 |
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