Asymptotic optimality of the nonnegative garrote estimator under heteroscedastic errors. (English) Zbl 1448.62112
Summary: This paper proposes the Nonnegative Garrote (NG) estimator for linear model with heteroscedastic errors. On the other hand, under some regularity conditions, the authors show the asymptotic optimality of the NG estimator by referring to the idea of the asymptotic optimality of the model average estimator. Simulation results and a real data analysis are reported for testing the results obtained previously. These results provide a stronger theoretical basis for the use of NG estimator by strengthening existing findings.
MSC:
| 62J05 | Linear regression; mixed models |
| 62G07 | Density estimation |
| 62G20 | Asymptotic properties of nonparametric inference |
| 62P12 | Applications of statistics to environmental and related topics |
| 86A10 | Meteorology and atmospheric physics |
Keywords:
nonnegative garrote estimator; asymptotic optimality; coefficient shrinkage; heteroscedastic errorsSoftware:
alr3References:
| [1] | Breiman, L., Better subset regression using the Nonnegative Garrote, Technometrics, 37, 373-384 (1995) · Zbl 0862.62059 · doi:10.1080/00401706.1995.10484371 |
| [2] | Tibshirani, R., Regression shrinkage and selection via the Lasso: A retrospective, Journal of the Royal Statistical Society, Series B, 58, 1, 267-288 (1996) · Zbl 0850.62538 |
| [3] | Zou, H.; Hastie, T., Regularization and variable selection via the elastic net, Journal of the Royal Statistical Society, Series B, 67, 2, 301-320 (2005) · Zbl 1069.62054 · doi:10.1111/j.1467-9868.2005.00503.x |
| [4] | Zhang, Z. Z.; Wang, D. R., Simultaneous variable selection for heteroscedastic regression models, Science China Mathematics, 54, 3, 515-530 (2011) · Zbl 1216.62104 · doi:10.1007/s11425-010-4147-8 |
| [5] | Wu, L. C.; Zhang, Z. Z.; Xu, D. K., Variable selection in joint location and scale models of the skew-normal distribution, Journal of Statistical Computation and Simulation, 83, 7, 1266-1278 (2013) · Zbl 1431.62293 · doi:10.1080/00949655.2012.657198 |
| [6] | Li, H. Q.; Wu, L. C.; Ma, T., Variable selection in joint location, scale and skewness models of the skew-normal distribution, Journal of Systems Science and Complexity, 30, 3, 694-709 (2017) · Zbl 1369.62166 · doi:10.1007/s11424-016-5193-2 |
| [7] | Xu, D. K.; Zhang, Z. Z.; Wu, L. C., Variable selection in high-dimensional double generalized linear models, Statistical Papers, 55, 2, 327-347 (2014) · Zbl 1297.62127 · doi:10.1007/s00362-012-0481-y |
| [8] | Breiman, L., Heuristics of instability and stabilization in model selection, The Annals of Statistics, 24, 2350-2383 (1996) · Zbl 0867.62055 · doi:10.1214/aos/1032181158 |
| [9] | Yuan, M.; Lin, Y., Model selection and estimation in regression with grouped variables, Journal of the Royal Statistical Society, Series B, 68, 1, 49-68 (2006) · Zbl 1141.62030 · doi:10.1111/j.1467-9868.2005.00532.x |
| [10] | Golub, G. H.; Heath, M.; Wahba, G., Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics, 21, 215-223 (1979) · Zbl 0461.62059 · doi:10.1080/00401706.1979.10489751 |
| [11] | Akaike, H., Maximum likelihood identification of Gaussian autoregressive moving average models, Biometrika, 60, 255-265 (1973) · Zbl 0318.62075 · doi:10.1093/biomet/60.2.255 |
| [12] | Schwarz, G., Estimating the dimension of a model, The Annals of Statistics, 6, 2, 461-464 (1978) · Zbl 0379.62005 · doi:10.1214/aos/1176344136 |
| [13] | Xiong, S. F., Some notes on the nonnegative garrote, Technometrics, 52, 3, 349-361 (2010) · doi:10.1198/TECH.2010.09148 |
| [14] | Mallows, C. L., Some comments on CP, Technometrics, 15, 4, 661-675 (1973) · Zbl 0269.62061 |
| [15] | Chen, X. P.; Cai, G. H.; Gao, Y., Asymptotic optimality of NG estimator for linear regression model based on mallows criterion, Journal of Systems Science and Mathematical Sciences, 37, 11, 2260-2270 (2017) · Zbl 1399.62044 |
| [16] | Hoeting, J. A.; Madigan, D.; Raftery, A. E., Bayesian model averaging: A tutorial, Statistical Science, 14, 4, 382-401 (1999) · Zbl 1059.62525 · doi:10.1214/ss/1009212519 |
| [17] | Hansen, B. E., Least squares model averaging, Econometrica, 75, 4, 1175-1189 (2007) · Zbl 1133.91051 · doi:10.1111/j.1468-0262.2007.00785.x |
| [18] | Hansen, B. E., Least-squares forecast averaging, Journal of Econometrics, 146, 2, 342-350 (2008) · Zbl 1429.62421 · doi:10.1016/j.jeconom.2008.08.022 |
| [19] | Wan, A. T K.; Zhang, X. Y.; Zou, G. H., Least squares model averaging by mallows criterion, Journal of Econometrics, 156, 2, 277-283 (2010) · Zbl 1431.62291 · doi:10.1016/j.jeconom.2009.10.030 |
| [20] | Liang, H.; Zou, G. H.; Wan, A. T K., Optimal weight choice for frequentist model average estimators, Journal of the American Statistical Association, 106, 495, 1053-1066 (2011) · Zbl 1229.62090 · doi:10.1198/jasa.2011.tm09478 |
| [21] | Hansen, B. E.; Racine, J. S., Jackknife model averaging, Journal of Econometrics, 167, 1, 38-46 (2012) · Zbl 1441.62721 · doi:10.1016/j.jeconom.2011.06.019 |
| [22] | Zhang, X. Y.; Wan, A. T K.; Zou, G. H., Model averaging by Jackknife criterion in models with dependent data, Journal of Econometrics, 174, 2, 82-94 (2013) · Zbl 1283.62059 · doi:10.1016/j.jeconom.2013.01.004 |
| [23] | Liu, Q. F.; Okui, R., Heteroscedasticity-robust Cp model averaging, The Econometrics Journal, 16, 3, 463-472 (2013) · Zbl 1521.62117 · doi:10.1111/ectj.12009 |
| [24] | Zhang, X. Y.; Zou, G. H.; Carroll, R. J., Model averaging based on kullback-leibler distance, Statistica Sinica, 25, 4, 1583-1598 (2015) · Zbl 1377.62089 |
| [25] | Gao, Y.; Zhang, X. Y.; Wang, S. Y., Model averaging based on leave-subject-out cross-validation, Journal of Econometrics, 192, 1, 139-151 (2016) · Zbl 1419.62084 · doi:10.1016/j.jeconom.2015.07.006 |
| [26] | Li, K. C., Asymptotic optimality for Cp, CL, cross-validation and generalized cross-validations: Discrete index set, Annals of Statistics, 15, 3, 958-975 (1987) · Zbl 0653.62037 · doi:10.1214/aos/1176350486 |
| [27] | Andrews, D. W K., Asymptotic optimality of generalized CL, cross-validation, and generalized cross-validation in regression with heteroskedastic errors, Journal of Econometrics, 47, 2, 359-377 (1991) · Zbl 0734.62069 · doi:10.1016/0304-4076(91)90107-O |
| [28] | Weisberg, S., Applied Linear Regression (2005), Hoboken NJ: John Wiley, Hoboken NJ · Zbl 1068.62077 |
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