Abstract
Variable selection plays an important role in the high dimensionality data analysis, the Dantzig selector performs variable selection and model fitting for linear and generalized linear models. In this paper we focus on variable selection and parametric estimation for partially linear models via the Dantzig selector. Large sample asymptotic properties of the Dantzig selector estimator are studied when sample size n tends to infinity while p is fixed. We see that the Dantzig selector might not be consistent. To remedy this drawback, we take the adaptive Dantzig selector motivated by Dicker and Lin (submitted). Moreover, we obtain that the adaptive Dantzig selector estimator for the parametric component of partially linear models has the oracle properties under some appropriate conditions. As generalizations of the Dantzig selector, both the adaptive Dantzig selector and the Dantzig selector optimization can be implemented by the efficient algorithm DASSO proposed by James et al. (J R Stat Soc Ser B 71:127–142, 2009). Choices of tuning parameter and bandwidth are also discussed.
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References
Bickel PJ, Ritov Y, Tsybakov A (2009) Simultaneous analysis of Lasso and Dantzig selector. Ann Stat 37: 1705–1732
Candès E, Tao T (2007) The Dantzig selector: statistical estimation when p is much larger than n. Ann Stat 35: 2313–2351
Candès E, Tao T (2007) Rejoinder—the Dantzig selector: statistical estimation when p is much larger than n. Ann Stat 35: 2392–2404
Chen H (1988) Convergence rates for parametric components in a partly linear model. Ann Stat 16: 136–146
Craven P, Wahba G (1979) Smoothing noisy data with spline functions: estimating the correct degree of smoothing by the method of generalized cross-validation. Numer Math 31: 377–403
Dicker L, Lin X (2009) A large sample analysis of the Dantzig selector and extensions (submitted)
Efron B, Hastie T, Tibshirani R (2004) Least angle regression (with discussion). Ann Stat 32: 407–451
Engle RF, Granger CWJ, Rice J, Weiss A (1986) Semiparametric estimates of the relation between weather and electricity sales. J Am Stat Assoc 81: 310–320
Fan J, Li R (2001) Variable selection via nonconcave penalized likelihood and its oracle properties. J Am Stat Assoc 96: 1348–1360
Fan J, Li R (2004) New estimation and model selection procedures for semiparametric modeling in longitudinal data analysis. J Am Stat Assoc 99: 710–723
Frank IE, Friedman JH (1993) A statistical view of some chemometrics regression tools (with discussion). Technometrics 35: 109–148
Fu WJ (1998) Penalized regressions: the bridge versus the lasso. J Comput Graph Stat 7: 397–416
Gao JT (1992) Large sample theory in semiparametric regression models. PhD Thesis, Graduate School, University of Science & Technology of China, Hefei, P.R. China
Härdle W, Liang H, Gao JT (2000) Partially linear models. Springer, Heidelberg
Heckman NE (1986) Spline smoothing in partly linear models. J R Stat Soc Ser B 48: 244–248
Hong SY (1991) Estimation theory of a class of semiparametric regression models. Sci China Ser A 12: 1258–1272
James GM, Radchenko P (2009) A generalized Dantzig selector with shrinkage tuning. Biometrika 96: 323–337
James GM, Radchenkob P, Lv JC (2009) Dasso: connections between the Dantzig selector and lasso. J R Stat Soc Ser B 71: 127–142
Knight K, Fu WJ (2000) Asymptotics for Lasso-type estimators. Ann Stat 28: 1356–1378
Liang H (1992) Asymptotic efficiency in semiparametric models and related topics. Thesis, Institute of Systems Science, Chinese Academy of Sciences, Beijing, P.R. China
Liang H (2006) Estimation in partially linear models and numerical comparisons. Comput Stat Data Anal 50: 675–687
Liang H, Li RZ (2009) Variable selection for partially linear models with measurement errors. J Am Stat Assoc 104: 234–248
Meinshausen N, Rocha G, Yu B (2007) A tale of three cousins: Lasso, L2Boosging, and Dantzig (discussion on Candes and Tao’s Dantzig selector paper). Ann Stat 35: 2372–2384
Moyeed RA, Diggle PJ (1994) Rates of convergence in semiparametric modeling of longitudinal data. Biometrika 86: 691–702
Ni X, Zhang HH, Zhang D (2009) Automatic model selection for partially linear models. J Multivar Anal 100: 2100–2111
Rice J (1986) Convergence rates for partially splined models. Stat Probab Lett 4: 203–208
Robinson PM (1988) Root-n-consistent semiparametric regression. Econometrica 56: 931–954
Ruppert D, Sheather SJ, Wand MP (1995) An effective bandwidth selector for local least squares regression. J Am Stat Assoc 90: 1257–1270
Ruppert D, Wand MP, Carroll RJ (2003) Semiparametric regression. Cambridge University Press, Cambridge
Speckman P (1988) Kernel smoothing in partial linear models. J R Stat Soc Ser B 50: 413–436
Tibshirani R (1996) Regression shrinkage and selection via the lasso. J R Stat Soc Ser B 58: 267–288
Wang H, Li R, Tsai C (2007) Tuning parameter selectors for the smoothly clipped absolute deviation method. Biometrika 94: 553–568
Wang Lie, Brown LD, Cai TT (2011) A difference based approach to the semiparametric partially linear model. Electron J Stat 5: 619–641
Xie HL, Huang J (2009) SCAD-penalized regression in high-dimensional partially linear models. Ann Stat 37: 673–696
Yatchew A (1997) An elementary estimator for the partially linear model. Econ Lett 57: 135–143
Zou H, Hastie T (2005) Regularization and variable selection via the elastic net. J R Stat Soc Ser B 67: 301–320
Zou H, Hastie T, Tibshirani R (2007) On the degrees of freedom of the lasso. Ann Stat 35: 2173–2192
Zou H, Li R (2008) One-step sparse estimates in nonconcave penalized likelihood models. Ann Stat 36: 1509–1533
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Li, F., Lin, L. & Su, Y. Variable selection and parameter estimation for partially linear models via Dantzig selector. Metrika 76, 225–238 (2013). https://doi.org/10.1007/s00184-012-0384-x
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DOI: https://doi.org/10.1007/s00184-012-0384-x