Subsampling estimates of the Lasso distribution.

6.2 High dimensional setting 63
(d) Set the adaptive Lasso solution to
ˆβ (m)
b
and then set L n,b,m and L n,r,m to
= diag(w1 −1 , . . . , w−1 p n
) ˜β (m)
b
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).
(v) Determine separately for each j ∈ {1, . . . , p n } 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⌋) .
(vi) 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) ]
Remark. Note that the procedure above does not follow the subsampling scheme in the
strict sense since the adaptive Lasso weights are not recomputed over subsamples, however
we could note that using weights obtained on the whole sample to compute subsamples
estimates actually yields better results.
Coverage rates of the two sided confidence interval I (·)
2 are illustrated in Figure 6.4. First,
we note that the results are quite robust to violation of the partial orthogonality condition.
Then we see that the coverage rate for relevant coefficients are slightly below the nominal
level and that the false positive rates for zero coefficients are conservative. A possible
reason for the former is the bias introduced by the Lasso. The conservative false rejection