Subsampling estimates of the Lasso distribution.

62 Numerical results
• C i,j = 0.8 |i−j| , i, j = 1, . . . , 800 in model E’.
Models D and E respect the partial orthogonality condition between relevant and irrelevant
parameters while D’ and E’ violate it.
Estimation procedure
For a sample (Y i , x i ) n i=1 of simulated observations, we proceed as follows to construct
confidence intervals for the coefficinents:
(i) Center and scale the observations, that is, consider
Ỹ i = Y i − n −1
⎛
n ∑
˜x ij = ⎝x ij − n −1
Y l , i = 1, . . . , n
l=1
⎞
∑ n (
⎠ n −1
n ∑
(x lj −
lj
l=1 k=1
)
n∑ −1
x kj ) 2 , j = 1, . . . , p n , i = 1, . . . , n.
(ii) Set weights w j as
w j = n −1
n ∑
i=1
Ỹ i˜x ij , j = 1, . . . , p n .
(iii) Compute the Lasso path for the data set with scaled covariates, that is, for
(Ỹi, diag(w 1 , . . . , w pn )˜x i ) n i=1.
Then choose the penalization parameter by K-fold cross validation (we choose K=10)
based on (Ỹi, diag(w 1 , . . . , w pn )x i ) n i=1 . Denote it by λ n,CV . Let ˜β n be the Lasso solution
corresponding to (Ỹi, diag(w 1 , . . . , w pn )x i ) n i=1 and to the penalization parameter
λ n,CV . Finally, set the adaptive Lasso solution to
ˆβ n = diag(w −1
1 , . . . , w−1 p n
) ˜β n .
(iv) Repeat the folowing steps for m = 1, . . . , B:
(a) Generate a random subsample I m ⊂ {1, . . . , n} of size b by drawing without
replacement.
(b) Center and scale observations with index i ∈ I m to obtain a data set (Ỹ (m)
i , x (m)
i ) i∈Im .
(m)
(c) Compute the Lasso solution path for the data set (Ỹi , diag(w1 , . . . , w pn )x (m)
i
with covariates scaled by weights obtained in step ii. Let ˜β (m)
b be the Lasso solution
corresponding to (Ỹi , diag(w1 , . . . , w pn )x (m)
i ) i∈Im and to the rescaled
(m)
penalization parameter λ b,CV = λ n,CV (b/n) 0.4 .
) i∈Im