Subsampling estimates of the Lasso distribution.

6.1 Low dimensinal setting 57
5. Determine separately for each j ∈ {1, . . . , p} the following empirical quantiles of L (j)
n,b,·
and L (j)
n,r,·, that is, L (j)
n,b,(·)
and L(j)
n,r,(·)
being the ordered statistics, set
c (j)
n,b
(1 − α) = L(j)
n,b,(⌊(1−α)·B⌋) ,
c (j)
n,b
(α/2) = L(j)
n,b,(⌊α/2·B⌋) ,
and the analogous for L (j)
n,r,(·) .
c (j)
n,b
(1 − α/2) = L(j)
n,b,(⌊(1−α/2)·B⌋) .
6. For each j ∈ {1, . . . , p}, define the confidence intervals
[
I (j)
1 =
[
I (j)
2 =
[
I (j)
3 =
n − √ 1 c (j)
n
β (j)
β (j)
)
n,r(1 − α), ∞
n − √ 1 c (j)
n
n,r(1 − α/2), β n (j) − √ 1 c (j)
n
β (j)
n
−
[
I (j)
4 = β n
(j) −
1
√ n −
√
b
c (j)
n,b (1 − α), ∞ )
1
√ √ c (j) (1 − α/2), β(j) −
n − b
n,b
n
]
n,r(α/2)
1
√ n −
√
b
c (j)
n,b (α/2) ]
Estimated confidence intervals are illustrated in figures 6.1, 6.2 and 6.3
.
6.1.2 Hypothesis testing
For j ∈ {1, . . . , p}, consider the problem of testing the null hypothesis
against the alternative
H 0,j : β j = 0
H A,j : β j ≠ 0.
Definition 6.1.2.1. Let α ∈ (0, 1). A function φ j : R n → {0, 1} said to reject H 0,j
when it takes value 1 and to accept H 0,j when it takes value zero, is called a test for the
hypothesis H 0,j to the level α if it satisfies
E βj =0
( )
φ (j) (Z (n) ) ≤ α.
According to the duality Lemma, if I (j) is a confidence interval to the level α for β j , then
a test is given by
{
φ (j) (Z (n) 0 if 0 ∈ I
) =
(j) (Z (n) )
1 else