zbMATH — the first resource for mathematics

Regularized estimation for the least absolute relative error models with a diverging number of covariates. (English) Zbl 06918566
Summary: This paper considers the variable selection for the least absolute relative error (LARE) model, where the dimension of model, \(p_n\), is allowed to increase with the sample size \(n\). Under some mild regular conditions, we establish the oracle properties, including the consistency of model selection and the asymptotic normality for the estimator of non-zero parameter. An adaptive weighting scheme is considered in the regularization, which admits the adaptive Lasso, SCAD and MCP penalties by linear approximation. The theoretical results allow the dimension diverging at the rate \(p_n = o(n^{1 / 2})\) for the consistency and \(p_n = o(n^{1 / 3})\) for the asymptotic normality. Furthermore, a practical variable selection procedure based on least squares approximation (LSA) is studied and its oracle property is also provided. Numerical studies are carried out to evaluate the performance of the proposed approaches.

62 Statistics
hgam; KELLEY
Full Text: DOI
[1] Belsley, D. A.; Kuh, E.; Welsch, R. E., Regression diagnostics: identifying influential data and sources of collinearity, (1980), Wiley New Jersey · Zbl 0479.62056
[2] Breheny, P.; Huang, J., Group descent algorithms for nonconvex penalized linear and logistic regression models with grouped predictors, Stat. Comput., 25, 173-187, (2015) · Zbl 1331.62359
[3] Chen, K.; Guo, S.; Lin, Y.; Ying, Z., Least absolute relative error estimation, J. Amer. Statist. Assoc., 105, 1104-1112, (2010) · Zbl 1390.62117
[4] Chen, K., Lin, Y., Wang, Z., Ying, Z., 2013. Least product relative error estimation. arXiv:1309.0220. · Zbl 1328.62146
[5] Chung, K. L., A course in probability theory, (2001), Academic Press
[6] Efron, B.; Hastie, T.; Johnstone, I.; Tibshirani, R., Least angle regression, Ann. Statist., 32, 407-499, (2004) · Zbl 1091.62054
[7] Fan, J.; Fan, Y.; Barut, E., Adaptive robust variable selection, Ann. Statist., 42, 324-351, (2014) · Zbl 1296.62144
[8] Fan, J.; Li, R., Variable selection via nonconcave penalized likelihood and its oracle properties, J. Amer. Statist. Assoc., 96, 1348-1360, (2001) · Zbl 1073.62547
[9] Fan, J.; Ma, Y.; Dai, W., Nonparametric independent screening in sparse ultra-high dimensional varying coefficient models, J. Amer. Statist. Assoc., 109, 1270-1284, (2014) · Zbl 1368.62095
[10] Fan, J.; Peng, H., Nonconcave penalized likelihood with a diverging number of parameters, Ann. Statist., 32, 928-961, (2004) · Zbl 1092.62031
[11] Harrison, D.; Rubinfeld, D. L., Hedonic prices and the demand for Clean air, J. Environ. Econ. Manag., 5, 81-102, (1978) · Zbl 0375.90023
[12] Huang, J.; Horowitz, J. L.; Wei, F., Variable selection in nonparametric additive models, Ann. Statist., 38, 2282-2313, (2010) · Zbl 1202.62051
[13] Huang, J.; Ma, S., Variable selection in the accelerated failure time model via the bridge method, Lifetime Data Anal., 16, 176-195, (2010) · Zbl 1322.62189
[14] Huang, J.; Ma, S.; Xie, H., Regularized estimation in the accelerated failure time model with high-dimensional covariates, Biometrics, 62, 813-820, (2006) · Zbl 1111.62090
[15] Jin, Z.; Lin, D. Y.; Wei, L. J.; Ying, Z., Rank-based inference for the accelerated failure time model, Biometrika, 90, 341-353, (2003) · Zbl 1034.62103
[16] Kelley, C. T., Iterative methods for optimization, (1999), SIAM · Zbl 0934.90082
[17] Khoshgoftaar, T. M.; Bhattacharyya, B. B.; Richardson, G. D., Predicting software errors, during development, using nonlinear regression models: a comparative study, IEEE Trans. Reliab., 41, 390-395, (1992) · Zbl 0825.68223
[18] Li, Z.; Lin, Y.; Zhou, G.; Zhou, W., Empirical likelihood for least absolute relative error regression, Test, 23, 86-99, (2014) · Zbl 1297.62072
[19] Makridakis, S.; Andersen, A.; Carbone, R.; Fildes, R.; Hibon, M.; Lewandowski, R.; Newton, J.; Parzen, E.; Winkler, R., The forecasting accuracy of major time series methods, (1984), Wiley New York
[20] Meier, L.; Van de Geer, S.; Bühlmann, P., High-dimensional additive modeling, Ann. Statist., 37, 3779-3821, (2009) · Zbl 1360.62186
[21] Narula, S. C.; Wellington, J. F., Prediction, linear regression and the minimum sum of relative errors, Technometrics, 19, 185-190, (1977) · Zbl 0377.62054
[22] Park, H.; Stefanski, L. A., Relatvie-error prediction, Statist. Probab. Lett., 40, 227-236, (1998) · Zbl 0959.62063
[23] Tibshirani, R., Regression shrinkage and selection via the lasso, J. R. Stat. Soc. Ser. B, 58, 267-288, (1996) · Zbl 0850.62538
[24] Wang, H.; Leng, C., Unified lasso estimation by least squares approximation, J. Amer. Statist. Assoc., 102, 1039-1048, (2007) · Zbl 1306.62167
[25] Wang, H.; Li, G.; Jiang, G., Robust regression shrinkage and consistent variable selection through the LAD-lasso, J. Bus. Econom. Statist., 25, 347-355, (2007)
[26] Wang, H.; Li, B.; Leng, C., Shrinkage tuning parameter selection with a diverging number of parameters, J. R. Stat. Soc. Ser. B, 71, 671-683, (2009) · Zbl 1250.62036
[27] Wang, L.; Wu, Y.; Li, R., Quantile regression for analyzing heterogeneity in ultra-high dimension, J. Amer. Statist. Assoc., 107, 214-222, (2012) · Zbl 1328.62468
[28] Ye, J., Price models and the value relevance of accounting information. technical report, (2007), Baruch College Stan Ross Dept. of Accountancy, City University of New York
[29] Zhang, C., Nearly unbiased variable selection under minimax concave penalty, Ann. Statist., 38, 894-942, (2010) · Zbl 1183.62120
[30] Zhang, Q.; Wang, Q., Local least absolute relative error estimating approach for partially linear multiplicative model, Statist. Sinica, 23, 1091-1116, (2013) · Zbl 06202699
[31] Zou, H., The adaptive lasso and its oracle properties, J. Amer. Statist. Assoc., 110, 1418-1429, (2006) · Zbl 1171.62326
[32] Zou, H.; Li, R., One-step sparse estimates in nonconcave penalized likelihood models, Ann. Statist., 36, 1509-1566, (2008) · Zbl 1282.62112
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.