Subsampling estimates of the Lasso distribution.

48 Subsampling
(ii) lim inf n→∞ inf P ∈P P
(iii) lim n→∞ inf P ∈P P
(
(τ n − τ b )(ˆθ n − θ(P )) ≥
) ˆL
−1
n (α) ≥ 1 − α
(ˆL−1
)
n (α) ≤ (τ n − τ b )(ˆθ n − θ(P )) ≤ ˆL
−1
n (1 − α) = 1 − 2α.
Remark. Following the previous remark, under the additional assumption τ n /τ b → 0,
conclusions i, ii and iii are also valid for a scaling factor τ n in lieu of τ n − τ b .
5.2.2 Proof
Lemma 5.2.2.1. Massart (1990)(Dvoretzky-Kiefer-Wolfowitz inequality) For n ∈
N, let X 1 , . . . , X n be i.i.d. random variables with probability distribution function F . Let
ˆF n be the empirical distribution function, that is
Then for every n ∈ N and ε > 0,
P
(
ˆF n (x) = 1 n∑
1{X i ≤ x}.
n
i=1
sup | ˆF n (x) − F n (x)|
x∈R
〉
(
ε) ≤ 2 exp −2nε 2) (5.2.2.1)
The following inequalities will be useful. Their verification is elementary, hence we omit
the proof.
Lemma 5.2.2.2. (Romano and Shaikh, 2010, Lemma 4.1) Let G and F be (nonrandom)
distribution functions on R, then the following are true :
(i) If sup x∈R {G(x) − F (x) ≤ ε}, then
G −1 (1 − α) ≥ F −1 (1 − (α + ε)) (5.2.2.2)
(ii) If sup x∈R {F (x) − G(x) ≤ ε}, then
G −1 (α) ≥ F −1 ((α + ε)) (5.2.2.3)
Furthermore for a random variable X with probability distribution function F , it holds :
(i) If sup x∈R {G(x) − F (x) ≤ ε}, then
P
(
)
X ≤ G −1 (1 − α) ≥ 1 − (α + ε) (5.2.2.4)
(ii) If sup x∈R {F (x) − G(x) ≤ ε}, then
P
(
)
X ≥ G −1 (α) ≥ 1 − (α + ε) (5.2.2.5)
(iii) If sup x∈R |G(x) − F (x)| ≤ ε, then
P
(
)
G −1 (α) ≤ X ≥ G −1 (1 − α) ≥ 1 − 2(α + ε). (5.2.2.6)