Abstract
In this paper we are concerned with the problems of variable selection and estimation in double generalized linear models in which both the mean and the dispersion are allowed to depend on explanatory variables. We propose a maximum penalized pseudo-likelihood method when the number of parameters diverges with the sample size. With appropriate selection of the tuning parameters, the consistency of the variable selection procedure and asymptotic properties of the resulting estimators are established. We also carry out simulation studies and a real data analysis to assess the finite sample performance of the proposed variable selection procedure, showing that the proposed variable selection method works satisfactorily.
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References
Bergman B, Hynén A (1997) Dispersion effects from unreplicated designs in the 2k-p series. Technometrics 39: 191–198
Breiman L (1995) Better subset selection using nonnegative garrote. Technometrics 37: 373–384
Cao CZ, Lin JG, Zhu LX (2010) Heteroscedasticity and/or autocorrelation diagnostics in nonlinear models with AR(1) and symmetrical errors. Stat Papers 51: 813–836
Davidian M, Carroll RJ (1988) A note on extended quasi-likelihood. J R Stat Soc B 50: 74–82
Efron B, Hastie T, Johnstone I, Tibshirani R (2004) Least angle regression. Ann Stat 32: 407–451
Engel J, Huele AF (1996) A generalized linear modeling approach to robust design. Technometrics 38: 365–373
Fan J, Li R (2001) Variable selection via nonconcave penalized likelihood and its oracle properties. J Am Stat Assoc 96: 1348–1360
Fan J, Peng H (2004) Nonconcave penalized likelihood with diverging number of parameters. Ann Stat 32: 928–961
Fu WJ (1998) Penalized regression: the bridge versus the LASSO. J Comput Graph Stat 7: 397–416
Gijbels I, Prosdocimi I, Claeskens G (2010) Nonparametric estimation of mean and dispersion functions in extended generalized linear models test. doi:10.1007/s11749-010-0187-1
Huang J, Horowitz JL, Ma SG (2006) Asymptotic properties of bridge estimators in sparse high-dimensional regression models. Ann Stat 36: 587–613
Kou CF, Pan JX (2009) Variable selection for joint mean and covariance models via penalized likelihood. http://www.manchester.ac.uk/mims/eprints,MIMSEPrint:2009.49
Li R, Liang H (2008) Variable selection in semiparametric regression modeling. Ann Stat 36: 261–286
Lin L (2004) Generalized quasi-likelihood. Stat Papers 45: 529–544
McCullagh P, Nelder JA (1989) Generalized linear models. Chapman and Hall, London
Nair VN, Pregibon D (1986) A data analysis strategy for quality engineering experiments. AT&T Tech J 65: 73–84
Nair VN, Pregibon D (1988) Analyzing dispersion effects from replicated factorial experiments. Technometrics 30: 247–257
Nelder JA (2000) Quasi-likelihood and pseudo-likelihood are not the samething. J Appl Stat 27: 1007–1011
Nelder JA, Pregibon D (1987) An extended quasi-likelihood function. Biometrika 74: 221–232
Nelder JA, Wedderburn RWM (1972) Generalized linear models. J R Stat Soc A 135: 370–384
Rigby RA, Stasinopolous DM (1996) Mean and dispersion additive models. In: Hardle W, Schimek MG (eds) Statical theory and computational aspects of smoothing. Physica-Verlag, Berlin, pp 215–230
Smyth GK (1989) Generalized linear models with varying dispersion. J R Stat Soc B 51: 47–60
Tibshirani R (1996) Regression shrinkage and selection via the LASSO. J R Stat Soc B 58: 267–288
Van der Vaart AW (1998) Asymptotic statics. Cambridge University Press, Cambridge
Wang DR, Zhang ZZ (2009) Variable selection in joint generalized linear models. Chin J Appl Probab Stat 25: 245–256
Wang H, Li R, Tsai C (2007) Tuning parameter selectors for the smoothly clipped absolute deviation method. Biometrika 94: 553–568
Wang H, Li B, Leng C (2009) Shrinkage tuning parameter selection with a diverging number of parameters. J R Stat Soc B 71: 671–683
Wedderburn RWM (1974) Quasi-likelihood function, generalized linear models and the Gauss-Newton method. Biometrika 61: 439–447
Zhang ZZ, Wang DR (2011) Simultaneous variable selection for heteroscedastic regression models. Sci China Math 54(3): 515–530
Zhou HB, You JH, Zhou B (2010) Statical inference for fixed-effects partially linear regression models with errors in variables. Stat Papers 51: 629–650
Zou H (2006) The adaptive lasso and its oracle properties. J Am Stat Assoc 101: 1418–1429
Zou H, Li R (2008) One-step sparse estimates in nonconcave penalized likelihood models. Ann Stat 36: 1509–1533
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Xu, D., Zhang, Z. & Wu, L. Variable selection in high-dimensional double generalized linear models. Stat Papers 55, 327–347 (2014). https://doi.org/10.1007/s00362-012-0481-y
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DOI: https://doi.org/10.1007/s00362-012-0481-y