The Asymptotic Properties of Scad Penalized Generalized Linear Models with Adaptive Designs
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The Asymptotic Properties of Scad Penalized Generalized Linear Models with Adaptive Designs∗ GAO Qibing · ZHU Chunhua · DU Xiuli · ZHOU Xingcai · YIN Dingxin
DOI: 10.1007/s11424-020-9134-8 Received: 21 April 2019 c The Editorial Office of JSSC & Springer-Verlag GmbH Germany 2020 Abstract This paper discusses the asymptotic properties of the SCAD (smoothing clipped absolute deviation) penalized quasi-likelihood estimator for generalized linear models with adaptive designs, which extend the related results for independent observations to dependent observations. Under certain conditions, the authors proved that the SCAD penalized method correctly selects covariates with nonzero coefficients with probability converging to one, and the penalized quasi-likelihood estimators of non-zero coefficients have the same asymptotic distribution they would have if the zero coefficients were known in advance. That is, the SCAD estimator has consistency and oracle properties. At last, the results are illustrated by some simulations. Keywords
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Adaptive designs, generalized linear models, oracle properties, SCAD penalty function.
Introduction
The generalized linear models (GLMs) is an important extension of the classical linear models and has been widely applied to many fields since Nelder and Wedderburm[1] introduced such models, which is suitable for continuous response data and especially for analyzing discrete response data. For the classical GLMs, the distribution of the response variable is assumed to belong to an exponential family. The idea of quasi-likelihood proposed by Wedderburm[2] GAO Qibing School of Mathematics Science, Nanjing Normal University, Nanjing 210023, China. Email: [email protected]. ZHU Chunhua School of Statistics and Mathematics, Nanjing Audit University, Nanjing 211815, China. Email: [email protected]. DU Xiuli School of Mathematics Science, Nanjing Normal University, Nanjing 210023, China. ZHOU Xingcai School of Statistics and Mathematics, Nanjing Audit University, Nanjing 211815, China. YIN Dingxin School of Mathematics Science, Nanjing Normal University, Nanjing 210023, China. ∗ This research was supported by the National Social Science Foundation of China under Grant No. 18BTJ040. This paper was recommended for publication by Editor ZHU Liping.
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GAO QIBING, et al.
further extended GLMs by only specifying mean and variance functions. For the case that the variance function is unknown but the specification of mean function is correct, further researches indicate that the quasi-likelihood method is still valid. Some related works can be found in Chen, et al.[3] , Chen and Chen[4] , Yin and Zhao[5] , and Gao, et al.[6] and etc. For more details of the generalized linear models and the quasi-likelihood methods, we propose to refer to McCullagh and Nelder[7] . Variable selection is a hot topic in modern regression analysis. In practice, such as in the economic and biological areas, researchers usually collected a large number of variables at the initial stage of modelling, but only a few variabl
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