Subsampling estimates of the Lasso distribution.

56 Numerical results
6.1.1 Confidence intervals
We recall the definition of a confidence interval.
Definition 6.1.1.1. Let α ∈ (0, 1). An interval valued functions I (j) (Z (n) ) is called confidence
interval for the parameter β j if it satisfies
(
)
P β j ∈ I (j) (Z (n) ) ≥ (1 − α).
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 coefficients:
1. Compute the Lasso solution path for the scaled and centered observations, that is, for
Ỹ i = Y i − n −1
˜x ij =
(
n ∑
l=1
x ij − n −1
Y l , i = 1, . . . , n
∑ n ) (
x lj n −1
l=1
n ∑
l=1
(x lj −
)
n∑ −1
x kj ) 2 , j = 1, . . . , p, i = 1, . . . , n.
2. Choose the penalization parameter by K-fold cross validation (we choose K = 10) on
the whole data set (Ỹi, ˜x i ) n i=1 . Denote it by λ n,CV .
k=1
3. Set ˆβ n as the Lasso solution to the data set (Ỹi, ˜x i ) n i=1 corresponding to the parameter
λ n,CV .
4. Repeat the following steps for m = 1, . . . , B :
(a) Generate a random subsample I m ⊂ {1, . . . , n} of size b by drawing without replacement.
(b) Compute the Lasso solution path for the scaled and centered data set with indices
i ∈ I m , that is for
= Y i − b −1 ∑
Ỹ (m)
i
˜x (m)
ij =
Y l , i ∈ I m
l∈I m
⎛
⎞ ⎛
⎝x ij − b −1 ∑
x lj
⎠ ⎝b −1 ∑
l∈I m
⎞
(x lj − ∑
x kj ) 2 ⎠
l∈I m k∈I m
−1
, j = 1, . . . , p, i ∈ I m .
(c) Set ˆβ (m)
b as the Lasso solution to the data set
√
(Ỹ (m) , ˜X (m) ) corresponding to the
rescaled penalization parameter λ b,CV = λ n,CV b/n. Set Ln,b,m and L n,r,m to
L n,b,m = √ ( )
b ˆβ (m)
b − ˆβ n
and
L n,r,m = √ ( )
r ˆβ (m)
b − ˆβ n
respectively. Here, r = b/(1 − b/n) is the finite sample corrected subsample size,
cf. (Politis et al., 1999, Section 10.3.1).