Subsampling estimates of the Lasso distribution.

4.3 Marginal regressors as initial estimates 41
holds, and one easily verifies that σ 2 ∑ n
i=1 vi 2 = 1. So by the Lebesgue’s dominated convergence
theorem it is sufficient to show that
max |v i| = max
1≤i≤n i≤i≤n n−1/2 s −1 ∣
n ∣α nC ′ −1
n1 u i∣ → 0.
This claim follows from assumptions B.5 and B.6 as
max
i≤i≤n n−1/2 s −1 ∣
n
∣α ′ nC −1
n1 u ∣
i
∣ ≤ max
1≤i≤n n−1/2 s −1
n
≤= σ −1 n −1/2 max
1≤i≤n
( )
α ′ nC −1 1/2 ( )
n1 α′ n u ′ iC −1 1/2
n1 u i
( )
u ′ iC −1 1/2
n1 u i
≤ n −1/2 σ −1 ‖C −1
n1 ‖ max
1≤i≤n (u′ iu i ) 1/2
≤ σ −1 τ −1
1 n−1/2 max
1≤i≤n (u′ iu i ) 1/2 → 0.
4.3 Marginal regressors as initial estimates
The validity of both Theorem 4.1.0.9 and 4.2.0.10 requires the existence of an initial
estimator ˜β n which satisfies assumption B.2, that is, is r n -consistent for the estimation
of a proxy η n of the true parameters β 0 . In this section we show that under a weak
correlation assumption between relavant and noise variables, this assumption is satisfied
by marginal regressors which also offer the advantage to be computationally attrative.
The marginal regressor ˜β n is defined as
and the proxies η nj are chosen as
˜β nj = x′ j Y
n , j = 1, . . . , p n (4.3.0.9)
η nj = E( ˜β nj ) = x′ j Xβ 0
n
(4.3.0.10)
The weak correlation assumption is formally stated as:
Assumption C.
C.1 Condition B.1 holds.
C.2 For 1 ≤ j ≤ k n and k n < k ≤ p n , it holds that
∣ 1
n∑ ∣∣∣∣ ∣ x
∣ ij x ik =
1 ∣∣∣
n
∣n x jx k ≤ ρ n , 1 ≤ j ≤ k n , k n < k ≤ p n
i=1
with ρ n satisfying
for some 0 < κ < 1.
(
) ⎛
c n = max |η nj | ⎝
k n