Subsampling estimates of the Lasso distribution.

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6.1 Low dimensinal setting 59 Figure 6.2: Subsampling confidence intervals for single scenarios of the models B and B’ (n= 250). Red triangles stand for the true parameters. at least one false rejection. This is formalized by the Family Wise Error Rate (FEWR) defined as follows Definition 6.1.3.1. Let H 1 , · · · , H s be a family of hypotheses. The corresponding FWER is defined as the probability to make at least one false rejection among the hypotheses H i , i.e. ( s ) ⋃ F W ER = P H1 ,...,H s {H i is rejected } i=1 (6.1.3.1) Our goal is to test the hypotheses H 1 , . . . , H s while controling the FWER to a given level α. That is, F W ER ≤ α. We focus on the Bonferonni-Holm procedure. It is a so called step-down procedure based on p-values for individual hypotheses. We recall the definition of a p-value,
60 Numerical results Figure 6.3: Subsampling confidence intervals for single scenarios of the models C and C’ (n= 250). Red triangles stand for the true parameters. Definition 6.1.3.2. Suppose that for a random variable X and an hypothesis H, one has a familiy {S α (X); α ∈ (0, 1)} of rejection regions satisfying S α ⊂ S α ′ whenever α < α ′ . Then a p-value is defined as ˆp = ˆp(X) = inf{α : X ∈ S α } that is, the smallest significance level at which one would reject the hypothesis H. Given p-values ˆp 1 , . . . , ˆp s associated to hypothesis tests H 1 , . . . , H s , the Holm procedure consits int he following steps: 1. Consider the ordered realized p-values ˆp (1) ≤ . . . , ˆp (s) and the corresponding hypotheses H (1) , . . . , H (s) . 2. If ˆp (1) ≥ α/s, accept H 1 , ·, H s and stop. If ˆp (1) < α/s reject H (1) and test the remaining s − 1 hypotheses at level α/(s − 1).
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iv Abstract
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vi CONTENTS Contents Notation viii
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viii Notation List of Tables 6.1 Mo
Page 10 and 11:
Chapter 1 Introduction In this thes
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Chapter 2 Minimizers of convex proc
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2.1 Convergence in probability 5 fo
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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 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
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Subsampling estimates of the Lasso distribution.

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