Subsampling estimates of the Lasso distribution.

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6.2 High dimensional setting 65 Model A (σ = 1) Model A’ (σ = 1.5) n 100 250 500 1000 2000 100 250 500 1000 2000 Rates of coverage β 1 0.95 0.96 0.94 0.93 0.95 0.96 0.95 0.94 0.93 0.95 β 2 0.94 0.94 0.92 0.95 0.95 0.95 0.96 0.92 0.95 0.95 β 3 0.95 0.95 0.95 0.95 0.93 0.91 0.96 0.96 0.94 0.93 β 4 0.96 0.95 0.96 0.95 0.95 0.96 0.95 0.96 0.96 0.95 β 5 0.96 0.94 0.96 0.95 0.94 0.95 0.95 0.96 0.95 0.93 β 6 0.95 0.94 0.94 0.95 0.95 0.94 0.95 0.97 0.95 0.95 False rejection rates β 7 0.02 0.02 0.01 0.02 0.02 0.02 0.02 0.01 0.02 0.02 β 8 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 β 9 0.01 0.02 0.02 0.03 0.02 0.01 0.02 0.02 0.03 0.02 β 10 0.02 0.02 0.02 0.02 0.03 0.02 0.02 0.02 0.02 0.03 β 11 0.03 0.02 0.03 0.03 0.03 0.03 0.02 0.03 0.03 0.03 β 12 0.04 0.03 0.03 0.02 0.02 0.04 0.03 0.03 0.02 0.02 β 13 0.01 0.03 0.02 0.03 0.03 0.01 0.03 0.02 0.03 0.03 β 14 0.03 0.04 0.02 0.03 0.02 0.03 0.04 0.02 0.03 0.02 β 15 0.03 0.02 0.02 0.03 0.02 0.03 0.02 0.02 0.03 0.02 β 16 0.02 0.02 0.03 0.01 0.02 0.02 0.02 0.03 0.01 0.02 β 17 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 β 18 0.02 0.01 0.02 0.02 0.03 0.02 0.01 0.02 0.02 0.03 β 19 0.03 0.01 0.03 0.03 0.02 0.03 0.01 0.03 0.03 0.02 β 20 0.02 0.02 0.03 0.03 0.02 0.02 0.02 0.03 0.03 0.02 Table 6.1: Model A and A’. Empirical coverage/false positive rates for the two sided interval I 2 .
66 Numerical results Model B (σ = 1) Model B’ (σ = 1.5) n 100 250 500 1000 2000 100 250 500 1000 2000 Rates of coverage β 1 0.97 0.96 0.94 0.94 0.94 0.97 0.96 0.95 0.94 0.93 β 2 0.95 0.93 0.92 0.92 0.94 0.93 0.94 0.91 0.93 0.95 β 3 0.90 0.95 0.94 0.93 0.91 0.73 0.92 0.94 0.93 0.91 β 4 0.94 0.93 0.95 0.92 0.94 0.92 0.93 0.94 0.94 0.94 β 5 0.94 0.94 0.95 0.92 0.91 0.94 0.94 0.94 0.92 0.91 β 6 0.94 0.93 0.94 0.94 0.94 0.95 0.94 0.95 0.94 0.93 False rejection rates β 7 0.02 0.02 0.02 0.03 0.02 0.03 0.02 0.02 0.03 0.02 β 8 0.02 0.02 0.04 0.03 0.04 0.02 0.02 0.03 0.03 0.04 β 9 0.01 0.02 0.03 0.04 0.03 0.02 0.02 0.03 0.04 0.03 β 10 0.02 0.03 0.02 0.03 0.03 0.02 0.03 0.02 0.02 0.03 β 11 0.01 0.02 0.03 0.05 0.02 0.02 0.02 0.04 0.05 0.02 β 12 0.03 0.03 0.04 0.03 0.02 0.03 0.03 0.03 0.03 0.02 β 13 0.02 0.03 0.05 0.04 0.03 0.01 0.03 0.04 0.04 0.03 β 14 0.03 0.03 0.03 0.04 0.03 0.02 0.04 0.03 0.04 0.03 β 15 0.03 0.03 0.04 0.03 0.03 0.02 0.03 0.04 0.03 0.03 β 16 0.03 0.02 0.05 0.03 0.03 0.02 0.02 0.03 0.03 0.03 β 17 0.02 0.03 0.03 0.03 0.03 0.02 0.03 0.02 0.03 0.03 β 18 0.03 0.03 0.03 0.02 0.02 0.03 0.03 0.03 0.02 0.02 β 19 0.03 0.03 0.03 0.04 0.03 0.03 0.03 0.03 0.04 0.03 β 20 0.02 0.02 0.04 0.04 0.03 0.03 0.02 0.03 0.04 0.03 Table 6.2: Model B nd B’. Empirical coverage/false positive rates for the two sided interval I 2 .
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✄✄ ✄ ✄ ✄ ✄✄ ✄ ✄
Page 4 and 5:
iv Abstract
Page 6 and 7:
vi CONTENTS Contents Notation viii
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 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 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
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Subsampling estimates of the Lasso distribution.

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