Subsampling estimates of the Lasso distribution.

More documents

Recommendations

Info

viii Notation List of Tables 6.1 Model A and A’. Empirical coverage/false positive rates for the two sided interval I 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.2 Model B nd B’. Empirical coverage/false positive rates for the two sided interval I 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 6.3 Model C and C’. Empirical coverage/false positive rates for the two sided interval I 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 6.4 Model A and A’. FWER and empirical power for nonzero coefficients. . . . 67 6.5 Model B and B’. FWER and empirical power for nonzero coefficients. . . . 68 6.6 Model C and C’. FWER and empirical power for nonzero coefficients. . . . 68
Notation Spaces and sets A A σ-algebra. B o The Borel σ-algebra on [0, 1]. B loc (R p ) Set of locally bounded functions on R p . B(x, ε) Open ball of radius ε around a point x in a metric space. C b (D) Bounded continous functions on a metric space D. C min (R p ) Set of continuous functions uniquely minimized on R p . K δ δ-enlargement of a compact set K in a metric space. l ∞ (T ) Space of uniformly bounded functions on a set T . l ∞ (T 2 , T 2 , . . . ) Space of functions uniformly bounded on each T i , i ∈ N. R The extended real line. Ω Sample space. Modes of convergence → → P → as∗ → au Deterministic or almost sure convergence. Convergence in probability. Outer almost sure convergence. Almost uniform convergence. Weak convergence. O notation X n = o P (a n ) Convergence in probability of {X n /a n } n to zero. X n = O P (a n ) Uniform tightness of {X n /a n } n . Further notation ∆J(x) Jump height at x of a right continous function J with left limit. E, E ∗ , E ∗ Expectation, outer expectation and inner expection, respectively. f |T Restriction of a function f on a subset T . P, P ∗ , P ∗ Probability, outer probability and inner probability, respectively. T ∗ Minimal measurable majorant of a random map T . T ∗ Maximal measurable minorant of a random map T . ‖X‖ ψd Orlicz norm of a random variable X. ix
Page 1: ✄✄ ✄ ✄ ✄ ✄✄ ✄ ✄
Page 4 and 5: iv Abstract
Page 6 and 7: vi CONTENTS Contents Notation viii
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:
5.2 Uniform consistency for quantil
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?