Subsampling estimates of the Lasso distribution.

Chapter 6
Numerical results
Motivated by the conclusions of Proposition 3.2.1.1 and Theorem 5.2.1.1 , we conduct
numerical simulations to evaluate the finite sample performance of subsampling based
confidence intervals. Control of the type I error is adressed for zero coefficients using
the duality between confidence intervals and hypothesis tests. Also, subsampling based
p-values are constructed for the purpose of multiple testing adjustment. Finally, in the
last section, the method is applied to the adaptive Lasso in a high dimensional setting.
6.1 Low dimensinal setting
For the simulation study in the low dimensional setting we consider data sets generated
from six linears models with random predictors. The six models differ in the correlation
structure between the covariates and the noise level. More precisely, data sets are generated
from a linear model
with regression parameter
Y i = x ′ iβ + ε i , i = 1, . . . , n
β = (1.5, −1.5, 0.75, −1.5, 1.5, −3, 0) ′ ∈ R 20 .
The errors {ε i } i are i.i.d normal with mean zero and standard deviation σ = 1 for the
models A, B and B, σ = 1.5 for the models A, B’ and C’. Rows of X are normally
distributed vectors with mean zero and Toeplitz covariance matrix
The models differ in the value ρ:
C ij = ρ |i−j| .
• Model A and A’: ρ = 0 (orthogonal design)
• Model B and B’: ρ = 0.6
• Model C and B’: ρ = 0.9
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