zbMATH — the first resource for mathematics

Variable selection in joint location and scale models of the skew-normal distribution. (English) Zbl 1431.62293
Summary: A regression model with skew-normal errors provides a useful extension for ordinary normal regression models when the data set under consideration involves asymmetric outcomes. Variable selection is an important issue in all regression analyses, and in this paper, we investigate the simultaneously variable selection in joint location and scale models of the skew-normal distribution. We propose a unified penalized likelihood method which can simultaneously select significant variables in the location and scale models. Furthermore, the proposed variable selection method can simultaneously perform parameter estimation and variable selection in the location and scale models. With appropriate selection of the tuning parameters, we establish the consistency and the oracle property of the regularized estimators. Simulation studies and a real example are used to illustrate the proposed methodologies.

62J05 Linear regression; mixed models
62H12 Estimation in multivariate analysis
62-08 Computational methods for problems pertaining to statistics
alr3; GLIM
Full Text: DOI
[1] Azzalini, A. and Capitanio, A.2003. Distributions generate by perturbation of symmetry with emphasis on a multivariate skew-t distribution. J. R. Statist. Soc. Ser. B, 65: 367-389. (doi:10.1111/1467-9868.00391) [Crossref], [Google Scholar] · Zbl 1065.62094
[2] Azzalini, A.1985. A class of distributions which includes the normal ones. Scand. J. Stat, 12: 171-178. [Web of Science ®], [Google Scholar] · Zbl 0581.62014
[3] Azzalini, A. and Capitanio, A.1999. Statistical applications of the multivariate skew normal distribution. J. R. Statist. Soc. Ser. B, 3: 579-602. (doi:10.1111/1467-9868.00194) [Crossref], [Google Scholar] · Zbl 0924.62050
[4] Arnold, B. C. and Beaver, R. J.2002. Skewed multivariate models related to hidden truncation and/or selective reporting (with discussion). Test, 11: 7-54. (doi:10.1007/BF02595728) [Crossref], [Web of Science ®], [Google Scholar] · Zbl 1033.62013
[5] Gupta, R. D. and Gupta, R. C.2008. Analyzing skewed data by power normal model. Test, 17: 197-210. (doi:10.1007/s11749-006-0030-x) [Crossref], [Web of Science ®], [Google Scholar] · Zbl 1148.62008
[6] Xie, F. C., Lin, J. G. and Wei, B. C.2009. Diagnostics for skew-normal nonlinear regression models with AR(1) errors. Comput. Statist. Data Anal, 53: 4403-4416. (doi:10.1016/j.csda.2009.06.010) [Crossref], [Web of Science ®], [Google Scholar] · Zbl 05689188
[7] Xie, F. C., Wei, B. C. and Lin, J. G.2009. Homogeneity diagnostics for skew-normal nonlinear regression models. Statist. Probab. Lett, 79: 821-827. (doi:10.1016/j.spl.2008.11.001) [Crossref], [Web of Science ®], [Google Scholar] · Zbl 1157.62044
[8] Lin, J. G., Xie, F. C. and Wei, B. C.2009. Statistical diagnostics for skew-t-normal nonlinear models. Commun. Statist. Simul. Comput, 38: 2096-2110. (doi:10.1080/03610910903249502) [Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 1182.62150
[9] Gupta, A. K. and Chen, T.2001. Goodness of fit tests for the skew-normal distribution. Commun. Statist. Simul. Comput, 30: 907-930. (doi:10.1081/SAC-100001854) [Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 1008.62606
[10] Cancho, V. G., Lachos, V. H. and Ortega, E. M.2010. A nonlinear regression model with skew-normal errors. Statist. Pap, 51: 547-558. (doi:10.1007/s00362-008-0139-y) [Crossref], [Web of Science ®], [Google Scholar] · Zbl 1247.62160
[11] Park, R. E.1966. Estimation with heteroscedastic error terms. Econometrica, 34: 888 (doi:10.2307/1910108) [Crossref], [Web of Science ®], [Google Scholar]
[12] Harvey, A. C.1976. Estimating regression models with multiplicative heteroscedasticity. Econometrica, 44: 460-465. (doi:10.2307/1913974) [Crossref], [Web of Science ®], [Google Scholar] · Zbl 0333.62040
[13] Aitkin, M.1987. Modelling variance heterogeneity in normal regression using GLIM. Appl. Statist, 36: 332-339. (doi:10.2307/2347792) [Crossref], [Web of Science ®], [Google Scholar]
[14] Verbyla, A. P.1993. Variance heterogeneity: Residual maximum likelihood and diagnostics. J. R. Statist. Soc. Ser. B, 52: 493-508. [Google Scholar] · Zbl 0783.62051
[15] Engel, J. and Huele, A. F.1996. A generalized linear modeling approach to robust design. Technometrics, 38: 365-373. (doi:10.1080/00401706.1996.10484548) [Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 0896.62071
[16] Taylor, J. T. and Verbyla, A. P.2004. Joint modelling of location and scale parameters of the t distribution. Statist. Model, 4: 91-112. (doi:10.1191/1471082X04st068oa) [Crossref], [Web of Science ®], [Google Scholar] · Zbl 1112.62010
[17] Fan, J. Q. and Lv, J. C.2010. A selective overview of variable selection in high dimensional feature space. Statist. Sinica, 20: 101-148. [PubMed], [Web of Science ®], [Google Scholar] · Zbl 1180.62080
[18] Wang, D. R. and Zhang, Z. Z.2009. Variable selection in joint generalized linear models. Chinese J. Appl. Probab. Statist, 25: 245-256. [Google Scholar] · Zbl 1211.62121
[19] Zhang, Z. Z. and Wang, D. R.2011. Simultaneous variable selection for heteroscedastic regression models. Sci. China Math, 54: 515-530. (doi:10.1007/s11425-010-4147-8) [Crossref], [Web of Science ®], [Google Scholar] · Zbl 1216.62104
[20] Fan, J. Q. and Li, R.2001. Variable selection via nonconcave penalized likelihood and its oracle properties. J. Am. Statist. Assoc, 96: 1348-1360. (doi:10.1198/016214501753382273) [Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 1073.62547
[21] Tibshirani, R.1996. Regression shrinkage and selection via the LASSO. J. R. Statist. Soc. Ser. B, 58: 267-288. [Crossref], [Google Scholar] · Zbl 0850.62538
[22] Wang, H., Li, R. and Tsai, C.2007. Tuning parameter selectors for the smoothly clipped absolute deviation method. Biometrika, 94: 553-568. (doi:10.1093/biomet/asm053) [Crossref], [PubMed], [Web of Science ®], [Google Scholar] · Zbl 1135.62058
[23] Antoniadis, A.1997. Wavelets in Statistics: A Review (with Discussion). J. Ital. Stat. Assoc, 6: 97-144. (doi:10.1007/BF03178905) [Crossref], [Google Scholar]
[24] Li, R. and Liang, H.2008. Variable selection in semiparametric regression modeling. Ann. Stat, 36: 261-286. (doi:10.1214/009053607000000604) [Crossref], [PubMed], [Web of Science ®], [Google Scholar] · Zbl 1132.62027
[25] Zhao, P. X. and Xue, L. G.2010. Variable selection for semiparametric varying coefficient partially linear errors-in-variables models. J. Multivariate Anal, 101: 1872-1883. (doi:10.1016/j.jmva.2010.03.005) [Crossref], [Web of Science ®], [Google Scholar] · Zbl 1190.62090
[26] Weisberg, S.1985. Applied Linear Regression, New York: Wiley. [Google Scholar] · Zbl 0646.62058
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.