×

zbMATH — the first resource for mathematics

Variable selection in joint mean and variance models of Box-Cox transformation. (English) Zbl 07263611
Summary: In many applications, a single Box-Cox transformation cannot necessarily produce the normality, constancy of variance and linearity of systematic effects. In this paper, by establishing a heterogeneous linear regression model for the Box-Cox transformed response, we propose a hybrid strategy, in which variable selection is employed to reduce the dimension of the explanatory variables in joint mean and variance models, and Box-Cox transformation is made to remedy the response. We propose a unified procedure which can simultaneously select significant variables in the joint mean and variance models of Box-Cox transformation which provide a useful extension of the ordinary normal linear regression models. With appropriate choice of the tuning parameters, we establish the consistency of this procedure and the oracle property of the obtained estimators. Moreover, we also consider the maximum profile likelihood estimator of the Box-Cox transformation parameter. Simulation studies and a real example are used to illustrate the application of the proposed methods.
MSC:
62 Statistics
Software:
GLIM; MINITAB
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aitkin, M. 1987. Modelling variance heterogeneity in normal regression using GLIM. Appl. Stat., 36: 332-339. (doi:10.2307/2347792)
[2] Antoniadis, A. 1997. Wavelets in statistics: A review (with discussion). J. Italian Stat. Assoc., 6: 97-144. (doi:10.1007/BF03178905)
[3] Atkinson, A. C. 1982. Regression diagnostics, transformations and constructed variables (with discussion). J. R. Stat. Soc. Ser. B, 44: 1-36. · Zbl 0508.62058
[4] Box, G. E.P. and Cox, D. R. 1964. An analysis of transformation (with discussion). J. R. Stat. Soc. Ser. B, 26: 211-252.
[5] Carroll, R. J. and Rupert, D. 1988. Transforming and Weighting in Regression, London: Chapman and Hall.
[6] Cook, R. D. and Weisberg, S. 1983. Diagnostics for heteroscedasticity in regression. Biometrika, 70: 1-10. (doi:10.1093/biomet/70.1.1) · Zbl 0502.62063
[7] 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) · Zbl 0896.62071
[8] Fan, J. Q. and Li, R. 2001. Variable selection via nonconcave penalized likelihood and its oracle properties. J. Amer. Statist. Assoc., 96: 1348-1360. (doi:10.1198/016214501753382273) · Zbl 1073.62547
[9] Fan, J. Q. and Lv, J. C. 2010. A selective overview of variable selection in high dimensional feature space. Statist. Sinica, 20: 101-148. · Zbl 1180.62080
[10] Harvey, A. C. 1976. Estimating regression models with multiplicative heteroscedasticity. Econometrica, 44: 460-465. (doi:10.2307/1913974) · Zbl 0333.62040
[11] C.F. Kou and J.X. Pan, Variable selection for joint mean and covariance models via penalized likelihood. Available at http://www.manchester.ac.uk/mims/eprints, MIMS EPrint: 2009.49.
[12] Li, R. and Liang, H. 2008. Variable selection in semiparametric regression modeling. Ann. Statist., 36: 261-286. (doi:10.1214/009053607000000604) · Zbl 1132.62027
[13] Nelder, J. A. and Lee, Y. 1991. Generalized linear models for the analysis of Taguchi-type experiments. Appl. Stoch. Models Data Anal., 7: 107-120. (doi:10.1002/asm.3150070110)
[14] Park, R. E. 1966. Estimation with heteroscedastic error terms. Econometrica, 34: 888 (doi:10.2307/1910108)
[15] Ryan, T. A., Joiner, B. L. and Ryan, B. E.F. 1976. Minitab Student Handbook, North Scituate, MA: Duxbury Press.
[16] Taylor, J. T. and Verbyla, A. P. 2004. Joint modelling of location and scale parameters of the t distribution. Stat. Model., 4: 91-112. (doi:10.1191/1471082X04st068oa) · Zbl 1112.62010
[17] Tibshirani, R. 1996. Regression shrinkage and selection via the LASSO. J. R. Stat. Soc. Ser. B, 58: 267-288. · Zbl 0850.62538
[18] Verbyla, A. P. 1993. Variance heterogeneity: Residual maximum likelihood and diagnostics. J. R. Stat. Soc. Ser. B, 52: 493-508. · Zbl 0783.62051
[19] Wang, D. R. and Zhang, Z. Z. 2009. Variable selection in joint generalized linear models. Chinese J. Appl. Probab. Stat., 25: 245-256. · Zbl 1211.62121
[20] 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) · Zbl 1135.62058
[21] Wei, B. C., Lu, G. B. and Shi, J. Q. 1991. Introduction of Statistical Diagnostics, Nanjing: Southeast University Press.
[22] Wu, L. C. and Li, H. Q. 2012. Variable selection for joint mean and dispersion models of the inverse Gaussian distribution. Metrika, 75: 795-808. (doi:10.1007/s00184-011-0352-x) · Zbl 1410.62132
[23] Xu, D. K. and Zhang, Z. Z. 2011. Regularized REML for estimation in heteroscedastic regression models. Adv. Intell. Soft Comput., 100: 495-502. (doi:10.1007/978-3-642-22833-9_60) · Zbl 06185947
[24] 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) · Zbl 1190.62090
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.