针对自适应设计广义线性模型,研究自适应 Lasso惩罚最小二乘变量选择方法。在一定条件下,得到自适应 Lasso 惩罚最小二乘估计的相合性和 Oracle 性质,该结果将固定设计广义线性模型相关结果推广到自适应设计广义线性模型中。通过模拟可知,自适应 Lasso 惩罚方法优于 Lasso 惩罚方法。
For generalized linear models with adaptive designs, the variable selection method based on the adaptive Lasso penalized least squared method is considered. Under certain conditions, the consistency and oracle properties of the adaptive Lasso penalized least squared estimator are established, which extend the corresponding results from the fixed designs to the adaptive designs for generalized linear models. The simulation results show that the adaptive Lasso penalty method is better than Lasso punishment.
[1] NELDER J A,WEDDERBURN R W M.Generalized linear models[J].Journal of the Royal Statistical Society Series A (General),1972,135(3):370.
[2] CHANG Y C I.Strong consistency of maximum quasi-likelihood estimate in generalized linear models via a last time[J].Statistics & Probability Letters,1999,45(3):237-246.
[3] CHANG Y C I.Sequential confidence regions of generalized linear models with adaptive designs[J].Journal of Statistical Planning and Inference,2001,93(1/2):277-293.
[4] CHEN K N,HU I,YING Z L.Strong consistency of maximum quasi-likelihood estimators in generalized linear models with fixed and adaptive designs[J].The Annals of Statistics,1999,27(4):1155-1163.
[5] YUE L.Rate of strong consistency of quasi maximum likelihood estimate in generalized linear models[J].Science in China Series A,2004,47(6):882.
[6] 陈夏,陈希孺.广义线性模型参数的自适应拟似然估计[J].中国科学(A辑),2005,35(4):463-480.
CHEN X, CHEN X R.Adaptive quasi-likelihood estimate in generalized linear models[J].Science in China(Series A), 2005,35(4):829-846.
[7] YIN C M,ZHAO L C,WEI C D.Asymptotic normality and strong consistency of maximum quasi-likelihood estimates in generalized linear models[J].Science in China Series A,2006,49(2):145-157.
[8] DING J L,CHEN X R.Asymptotic properties of the maximum likelihood estimate in generalized linear models with stochastic regressors[J].Acta Mathematica Sinica,2006,22(6):1679-1686.
[9] GAO Q B,LIN J G,ZHU C H,et al.Asymptotic properties of maximum quasi-likelihood estimators in generalized linear models with adaptive designs[J].Statistics,2012,46(6):833-846.
[10] TIBSHIRANI R.Regression shrinkage and selection via the lasso[J].Journal of the Royal Statistical Society:Series B (Methodological),1996,58(1):267-288.
[11] FAN J Q,LI R Z.Variable selection via nonconcave penalized likelihood and its oracle properties[J].Journal of the American Statistical Association,2001,96(456):1348-1360.
[12] FRANK I E,FRIEDMAN J H.A statistical view of some chemometrics regression tools:response[J].Technometrics,1993,35(2):143.
[13] ZOU H,HASTIE T.Regularization and variable selection via the elastic net[J].Journal of the Royal Statistical Society:Series B (Statistical Methodology),2005,67(2):301-320.
[14] ZHANG C H.Nearly unbiased variable selection under minimax concave penalty[J].The Annals of Statistics,2010,38(2):894-942.
[15] ZOU H.The adaptive Lasso and its oracle properties[J].Journal of the American Statistical Association,2006,101(476):1418-1429.
[16] WANG M Q,SONG L X,WANG X G.Bridge estimation for generalized linear models with a diverging number of parameters[J].Statistics & Probability Letters,2010,80(21/22):1584-1596.
[17] WANG L C,YOU Y,LIAN H.Convergence and sparsity of Lasso and group Lasso in high-dimensional generalized linear models[J].Statistical Papers,2015,56(3):819-828.
[18] CUI X H,CHEN X,YAN L.Quasi-likelihood Bridge estimators for high-dimensional generalized linear models[J].Communications in Statistics-Simulation and Computation,2017,46(10):8190-8204.
[19] CUI Y,CHEN X,YAN L.Adaptive Lasso for generalized linear models with a diverging number of parameters[J].Communications in Statistics-Theory and Methods,2017,46(23):11826-11842.
[20] 陈夏,崔艳.高维广义线性模型的拟似然自适应Lasso估计[J].陕西师范大学学报(自然科学版),2019,47(2):1-9.
CHEN X, CUI Y.Quasi-likelihood adaptive Lasso estimators for high-dimensional generalized linear models[J].Journal of Shaanxi Normal University (Natural Science Edition), 2019, 47(2):1-9.
[21] LAI T L,WEI C Z.Least squares estimates in stochastic regression models with applications to identification and control of dynamic systems[J].The Annals of Statistics,1982,10(1):154-166.
[22] HALL P,HEYDE C C.Martingale Limit Theory and its Application[M].Amsterdam:Elsevier,1980:9-11.
