Variance estimation in high dimensional regression models

We treat the problem of variance estimation of the least squares estimate of the parameter in high dimensional linear regression models by using the Uncor- related Weights Bootstrap ( UBS ). We find a representation of the UBS dispersion matrix and show that the bootstrap estimator is consistent if p 2 /n → 0w herep is the dimension of the parameter and n is the sample size. For fixed dimension we show that the UBS belongs to the R-class as defined in Liu and Singh (1992).

[1]  D. Hinkley.  Jackknifing in Unbalanced Situations , 1977 .

[2]  Regina Y. Liu,et al.  Efficiency and robustness in resampling , 1992 .

[3]  Changbao Wu,et al.  Jackknife, Bootstrap and Other Resampling Methods in Regression Analysis , 1986 .

[4]  Raul Cano.  On The Bayesian Bootstrap , 1992 .

[5]  Jun Shao,et al.  Heteroscedasticity-Robustness of Jackknife Variance Estimators in Linear Models , 1987 .

[6]  Snigdhansu Chatterjee.  Another look at the jackknife: further examples of generalized bootstrap , 1998 .

[7]  A. Lo.  Bayesian bootstrap clones and a biometry function , 1991 .

[8]  B. Efron.  The jackknife, the bootstrap, and other resampling plans , 1987 .

[9]  J. Fox.  Bootstrapping Regression Models , 2002 .

[10]  John F. Monahan,et al.  Bootstrap methods using prior information , 1986 .

[11]  C. Weng,et al.  On a Second-Order Asymptotic Property of the Bayesian Bootstrap Mean , 1989 .

[12]  Anil K. Bera,et al.  ARCH Models: Properties, Estimation and Testing , 1993 .

[13]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[14]  P. Bertail,et al.  The Weighted Bootstrap , 1995 .

[15]  Z Zheng,et al.  RANDOM WEIGHTING METHOD IN REGRESSION MODELS , 1988 .

[16]  E. Mammen.  Bootstrap and Wild Bootstrap for High Dimensional Linear Models , 1993 .

[17]  James Stephen Marron,et al.  BOOTSTRAP SIMULTANEOUS ERROR BARS FOR NONPARAMETRIC REGRESSION , 1991 .

[18]  J. Wellner,et al.  Exchangeably Weighted Bootstraps of the General Empirical Process , 1993 .

[19]  Albert Y. Lo,et al.  A large sample study of the Bayesian bootstrap , 1987 .

[20]  B. Efron,et al.  The Jackknife: The Bootstrap and Other Resampling Plans. , 1983 .