Subsampling estimates of the Lasso distribution.

32 The adaptive Lasso in a high dimensional setting
where β 1 is a k n × 1 vector, has each of its components different from zero and β 2 is a
m n × 1 vector equal to zero. Let x i = (x i1 , . . . , x ipn ) ′ be the covariate vector of the i-th
observation, denote by u i its first part, corresponding to non-zero coefficients, and by z i
the part corresponding to zero coefficients, i.e.
The empirical covariance matrix
x i = (u ′ i, z ′ i) ′ .
C n = 1 n X′ nX n
is also partitioned according to zero and non-zero coefficients. For
X ′ n1 = (u 1 , · · · , u kn ) ′ ,
set
C n1 = 1 n X′ n1X n1 ,
a k n ×k n matrix. Finally let ρ n1 and τ n1 be the smallest respectively the largest eigenvalue
of τ n1 of C n1 .
Results on the variable selection consistency and the partial normality of the adaptive
Lasso will be derived under the following assumptions:
Assumption B.
B.1 {ε i } i is a sequence of i.i.d. random variables with mean zero andc there are some
constants 1 ≤ d ≤ 2, C > 0 and K > 0, such that the tail probability satisfies
for every x ≥ 0.
P (|ε i | > x) ≤ K exp
(−Cx d)
B.2 The initial estimators ˜β nj are r n -consistent for the estimation of unknown constants
η nj depending on β, that is
r n
The constants η nj satisfy
∣ ∣ ∣∣ ∣∣
max ˜βnj − η nj = OP (1), r n → ∞.
1≤j≤p n
max
k n