Due to advances in technologies, modern statistical studies often encounter linear models with high-dimension, where the number of explanatory variables is larger than the sample size. Estimation in these high-dimensional problems with deterministic covariates or designs is very different from those in the case of random covariates, due to the identifiability of the high-dimensional semiparametric regression parameters. In this paper, we consider ridge estimators and propose preliminary test, shrinkage and its positive rule ridge estimators in the restricted semiparametric regression model when the errors are dependent under a multicollinear setting, in high-dimension. The asymptotic risk expressions in addition to biases are exactly derived for the estimators under study. For our proposal, a real data analysis about production of vitamin B2 and a Monté–Carlo simulation study are considered to illustrate the efficiency of the proposed estimators. In this regard, kernel smoothing and cross-validation methods for estimating the optimum ridge parameter and nonparametric function are used.
Generalized restricted ridge estimator High-dimension Kernel smoothing Linear restriction Multicollinearity Semiparametric regression model Shrinkage estimator Sparsity
Mathematics Subject Classification
Primary: 62G08 62J05 Secondary: 62J07 62G20
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We would like to thank two anonymous reviewers for their valuable comments and suggestions on the earlier version of this article which significantly improved the presentation. First author’s research is supported by Shahrood University of Technology (Grant No. 23088), Iran. The second author’s research is supported in part by a grant 94811069 from the Iran National Science Foundation (INSF) and Research Council of Semnan University.
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