Subsampling estimates of the Lasso distribution.

More documents

Recommendations

Info

vi CONTENTS Contents Notation viii 1 Introduction 1 2 Minimizers of convex processes 3 2.1 Convergence in probability . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Convergence in distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.1 Weak convergence in metric spaces . . . . . . . . . . . . . . . . . . . 7 2.2.2 Bounded and locally bounded functions . . . . . . . . . . . . . . . . 10 2.3 A continous mapping theorem for argmin functionals . . . . . . . . . . . . . 15 3 Application to the Lasso estimator 21 3.1 Limit in probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 Limit in distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2.1 Limiting distribution of components . . . . . . . . . . . . . . . . . . 24 3.2.2 Uniform convergence in the orthogonal case . . . . . . . . . . . . . . 27 4 The adaptive Lasso in a high dimensional setting 31 4.1 Variable selection consistency . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.2 Partial asymptotic normality . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.3 Marginal regressors as initial estimates . . . . . . . . . . . . . . . . . . . . . 41 5 Subsampling 43 5.1 Pointwise consistency for distribution estimation . . . . . . . . . . . . . . . 43 5.2 Uniform consistency for quantiles appproximation . . . . . . . . . . . . . . . 46 5.2.1 Statement of the general result . . . . . . . . . . . . . . . . . . . . . 47 5.2.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 6 Numerical results 55 6.1 Low dimensinal setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 6.1.1 Confidence intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 6.1.2 Hypothesis testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.1.3 F.W.E.R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 6.2 High dimensional setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 7 Concluding remarks 71 Bibliography 72 A R Codes 75 A.1 Simulation code for the Lasso in a low dimensional setting . . . . . . . . . . 75 A.2 Simulation code for the adaptive Lasso in a high dimensional setting . . . . 79
LIST OF FIGURES vii List of Figures 3.1 Monte Carlo estimates of the distribution of the root √ n( ˆβ j − β j ), j = 7, . . . , 10, with penalization parameter λ n = 2 √ n. . . . . . . . . . . . . . . . 29 5.1 In red, the subsampling distribution estimates for the roots √ n( ˆβ j − β j ), j = 1, . . . , 9. n = 10000, b = n 0.65 ≈ 400, B = 2000 with penalization parameter λ n = 2 √ n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 6.1 Subsampling confidence intervals for single scenarios of the models A and A’ (n= 250). Red triangles stand for the true parameters. . . . . . . . . . . 58 6.2 Subsampling confidence intervals for single scenarios of the models B and B’ (n= 250). Red triangles stand for the true parameters. . . . . . . . . . . 59 6.3 Subsampling confidence intervals for single scenarios of the models C and C’ (n= 250). Red triangles stand for the true parameters. . . . . . . . . . . 60 6.4 Coverage rates of the two sided confidence intervals I 2 for the adaptive Lasso in high dimension. Green triangles correspond to relevant variables. Black dots correspond to noise variables. . . . . . . . . . . . . . . . . . . . . 69
Page 1: ✄✄ ✄ ✄ ✄ ✄✄ ✄ ✄
Page 4 and 5: iv Abstract
Page 8 and 9: viii Notation List of Tables 6.1 Mo
Page 10 and 11: Chapter 1 Introduction In this thes
Page 12 and 13: Chapter 2 Minimizers of convex proc
Page 14 and 15: 2.1 Convergence in probability 5 fo
Page 16 and 17: 2.2 Convergence in distribution 7 a
Page 18 and 19: 2.2 Convergence in distribution 9 L
Page 20 and 21: 2.2 Convergence in distribution 11
Page 22 and 23: 2.2 Convergence in distribution 13
Page 24 and 25: 2.3 A continous mapping theorem for
Page 30 and 31: Chapter 3 Application to the Lasso
Page 32 and 33: 3.2 Limit in distribution 23 almost
Page 34 and 35: 3.2 Limit in distribution 25 We als
Page 36 and 37: 3.2 Limit in distribution 27 3.2.2
Page 38 and 39: 3.2 Limit in distribution 29 Figure
Page 40 and 41: Chapter 4 The adaptive Lasso in a h
Page 42 and 43: 33 B.3 (Adaptive irrepresentable co
Page 44 and 45: 35 exists a constant K d depending
Page 46 and 47: 4.1 Variable selection consistency
Page 48 and 49: 4.2 Partial asymptotic normality 39
Page 50 and 51: 4.3 Marginal regressors as initial
Page 52 and 53: Chapter 5 Subsampling In his semina
Page 54 and 55: 5.1 Pointwise consistency for distr
Page 56 and 57:
5.2 Uniform consistency for quantil
Page 58 and 59:
Page 60 and 61:
Page 62 and 63:
Page 64 and 65:
Chapter 6 Numerical results Motivat
Page 66 and 67:
6.1 Low dimensinal setting 57 5. De
Page 68 and 69:
6.1 Low dimensinal setting 59 Figur
Page 70 and 71:
6.2 High dimensional setting 61 3.
Page 72 and 73:
6.2 High dimensional setting 63 (d)
Page 74 and 75:
6.2 High dimensional setting 65 Mod
Page 76 and 77:
6.2 High dimensional setting 67 Mod
Page 78 and 79:
6.2 High dimensional setting 69 Fig
Page 80 and 81:
Chapter 7 Concluding remarks We rev
Page 82 and 83:
Bibliography Andersen, P. and R. Gi
Page 84 and 85:
Appendix A R Codes A.1 Simulation c
Page 86 and 87:
A.1 Simulation code for the Lasso i
Page 88 and 89:
A.2 Simulation code for the adaptiv
Page 90 and 91:
A.2 Simulation code for the adaptiv
show all

Subsampling estimates of the Lasso distribution.

Create successful ePaper yourself

Delete template?

Save as template?