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Extended differential geometric LARS for high-dimensional GLMs with general dispersion parameter. (English) Zbl 1384.62270

Summary: A large class of modeling and prediction problems involves outcomes that belong to an exponential family distribution. Generalized linear models (GLMs) are a standard way of dealing with such situations. Even in high-dimensional feature spaces GLMs can be extended to deal with such situations. Penalized inference approaches, such as the \(\ell_1\) or SCAD, or extensions of least angle regression, such as dgLARS, have been proposed to deal with GLMs with high-dimensional feature spaces. Although the theory underlying these methods is in principle generic, the implementation has remained restricted to dispersion-free models, such as the Poisson and logistic regression models. The aim of this manuscript is to extend the differential geometric least angle regression method for high-dimensional GLMs to arbitrary exponential dispersion family distributions with arbitrary link functions. This entails, first, extending the predictor-corrector (PC) algorithm to arbitrary distributions and link functions, and second, proposing an efficient estimator of the dispersion parameter. Furthermore, improvements to the computational algorithm lead to an important speed-up of the PC algorithm. Simulations provide supportive evidence concerning the proposed efficient algorithms for estimating coefficients and dispersion parameter. The resulting method has been implemented in our R package (which will be merged with the original dglars package) and is shown to be an effective method for inference for arbitrary classes of GLMs.

MSC:

62J12 Generalized linear models (logistic models)
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[1] Aho, K; Derryberry, D; Peterson, T, Model selection for ecologists: the worldviews of AIC and BIC, Ecology, 95, 631-636, (2014)
[2] Akaike, H, A new look at the statistical model identification, IEEE Trans. Autom. Control, 19, 716-723, (1974) · Zbl 0314.62039
[3] Allgower, E., Georg, K.: Introduction to Numerical Continuation Methods. Society for Industrial and Applied Mathematics, New York (2003) · Zbl 1036.65047
[4] Arlot, S; Celisse, A, A survey of cross-validation procedures for model selection, Stat. Surv., 4, 40-79, (2010) · Zbl 1190.62080
[5] Augugliaro, L; Mineo, AM; Wit, EC, Differential geometric least angle regression: a differential geometric approach to sparse generalized linear models, J. R. Stat. Soc. B, 75, 471-498, (2013) · Zbl 1411.62214
[6] Augugliaro, L; Mineo, AM; Wit, EC, Dglars: an R package to estimate sparse generalized linear models, J. Stat. Softw., 59, 1-40, (2014)
[7] Augugliaro, L.: dglars: Differential Geometric LARS (dgLARS) Method. R package version 1.0.5. http://CRAN.R-project.org/package=dglars (2014b) · Zbl 0042.38403
[8] Augugliaro, L; Mineo, AM; Wit, EC, A differential geometric approach to generalized linear models with grouped predictors, Biometrika, 103, 563-593, (2016) · Zbl 07072138
[9] Augugliaro, L., Pazira, H.: dglars: Differential Geometric Least Angle Regression. R package version 2.0.0. http://CRAN.R-project.org/package=dglars (2017)
[10] Burnham, K.P., Anderson, D.R.: Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, \(2^{{\rm nd}}\) edn. Springer, New York (2002)
[11] Candes, EJ; Tao, T, The Dantzig selector: statistical estimation when \(p\) is much larger than \(n\), Ann. Stat., 35, 2313-2351, (2007) · Zbl 1139.62019
[12] Chen, Y; Du, P; Wang, Y, Variable selection in linear models, Wiley Interdiscip. Rev. Comput. Stat., 6, 1-9, (2014)
[13] Cordeiro, GM; McCullagh, P, Bias correction in generalized linear models, J. R. Stat. Soc. B, 53, 629-643, (1991) · Zbl 0800.62432
[14] Efron, B; Hastie, T; Johnstone, I; Tibshirani, R, Least angle regression, Ann. Stat., 32, 407-499, (2004) · Zbl 1091.62054
[15] Fan, J; Li, R, Variable selection via nonconcave penalized likelihood and its oracle properties, J. Am. Stat. Assoc., 96, 1348-1360, (2001) · Zbl 1073.62547
[16] Fan, J; Lv, J, Sure independence screening for ultrahigh dimensional feature space, J. R. Stat. Soc. B, 70, 849-911, (2008) · Zbl 1411.62187
[17] Fan, J; Guo, S; Hao, N, Variance estimation using refitted cross-validation in ultrahigh dimensional regression, J. R. Stat. Soc. B, 74, 37-65, (2012) · Zbl 1411.62199
[18] Farrington, CP, On assessing goodness of fit of generalized linear model to sparse data, J. R. Stat. Soc. B, 58, 349-360, (1996) · Zbl 0866.62040
[19] Friedman, J., Hastie, T., RTibshirani: glmnet: Lasso and Elastic-Net Regularized Generalized Linear Models. R Package Version 1.1-5. http://CRAN.R-project.org/package=glmnet (2010b)
[20] Hastie, T., Efron, B.: lars: Least Angle Regression, Lasso and Forward Stagewise. R Package Version 1.2. http://CRAN.R-project.org/package=lars (2013)
[21] Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer, New York (2009) · Zbl 1273.62005
[22] Hoerl, AE; Kennard, R, Ridge regression: biased estimation for nonorthogonal problems, Technometrics, 12, 55-67, (1970) · Zbl 0202.17205
[23] Ishwaran, H; Kogalur, UB; Rao, J, Spikeslab: prediction and variable selection using spike and slab regression, R J., 2, 68-73, (2010)
[24] Ishwaran, H., Kogalur, U.B., Rao, J.: spikeslab: prediction and variable selection using spike and slab regression. R package version 1.1.2. http://CRAN.R-project.org/package=spikeslab (2010b) · Zbl 1073.62547
[25] James, G; Radchenko, P, A generalized Dantzig selector with shrinkage tuning, Biometrika, 96, 323-337, (2009) · Zbl 1163.62054
[26] Jorgensen, B, Exponential dispersion models, J. R. Stat. Soc. B, 49, 127-162, (1987) · Zbl 0662.62078
[27] Jorgensen, B.: The Theory of Dispersion Models. Chapman & Hall, London (1997) · Zbl 0928.62052
[28] Kullback, S; Leibler, RA, On information and sufficiency, Ann. Math. Stat., 22, 79-86, (1951) · Zbl 0042.38403
[29] Li, K.C.: Asymptotic optimality for \(c_p\), \(c_l\), cross-validation and generalized cross-validation: discrete index set. Ann. Stat. 15, 958-975 (1987) · Zbl 0653.62037
[30] Littell, R.C., Stroup, W.W., Feund, R.J.: SAS for Linear Models, 4th edn. Sas Institute Inc., Cary (2002)
[31] McCullagh, P., Nelder, J.A.: Generalized Liner Models. Chapman & Hall, London (1989) · Zbl 0744.62098
[32] McQuarrie, A.D.R., Tsai, C.L.: Regression and Time Series Model Selection, 1st edn. World Scientific Publishing Co. Pte. Ltd, Singapore (1998) · Zbl 0907.62095
[33] Meng, R.: Estimation of dispersion parameters in glms with and without random effects. Master’s thesis, Stockholm University (2004) · Zbl 0314.62039
[34] Park, M.Y., Hastie, T.: glmpath: \(L_1\) Regularization Path for Generalized Linear Models and Cox Proportional Hazards Model. R Package Version 0.94. http://CRAN.R-project.org/package=glmpath (2007b) · Zbl 1091.62054
[35] Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T.: Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd edn. Cambridge University Press, England (1992) · Zbl 0778.65002
[36] Schwarz, G, Estimating the dimension of a model, Ann. Stat., 6, 461-464, (1978) · Zbl 0379.62005
[37] Shao, J, An asymptotic theory for linear model selection, Stat. Sin., 7, 221-264, (1997) · Zbl 1003.62527
[38] Shibata, R, An optimal selection of regression variables, Biometrika, 68, 45-54, (1981) · Zbl 0464.62054
[39] Shibata, R, Approximation efficiency of a selection procedure for the number of regression variables, Biometrika, 71, 43-49, (1984) · Zbl 0543.62053
[40] Stone, M, Asymptotics for and against cross-validation, Biometrika, 64, 29-35, (1977) · Zbl 0368.62046
[41] Tibshirani, R, Regression shrinkage and selection via the lasso, J. R. Stat. Soc. B, 58, 267-288, (1996) · Zbl 0850.62538
[42] Ultricht, J., Tutz, G.: Combining quadratic penalization and variable selection via forward boosting. Tech. Rep., Department of Statistics, Munich University, Technical Reports No. 99 (2011)
[43] Vos, PW, A geometric approach to detecting influential cases, Ann. Stat., 19, 1570-1581, (1991) · Zbl 0741.62067
[44] Whittaker, E.T., Robinson, G.: The Calculus of Observations: An Introduction to Numerical Analysis, 4th edn. Dover Publications, New York (1967)
[45] Wood, S.N.: Generalized Additive Models: An Introduction with R. Chapman & Hall/CRC, Boca Raton (2006) · Zbl 1087.62082
[46] Zhang, CH, Nearly unbiased variable selection under minimax concave penalty, Ann. Stat., 38, 894-942, (2010) · Zbl 1183.62120
[47] Zou, H; Hastie, T, Regularization and variable selection via the elastic net, J. R. Stat. Soc. B, 67, 301-320, (2005) · Zbl 1069.62054
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