Subsampling estimates of the Lasso distribution.

5.2 Uniform consistency for quantiles appproximation 49
If Ĝ is a random function R and δ > 0, then we further have that:
)
(i) If P
(sup {Ĝ(x) − F (x) ≤ ε} x∈R ≥ 1 − δ, then
)
(ii) If P
(sup x∈R {F (x) − Ĝ(x) ≤ ε} ≥ 1 − δ, then
)
(iii) If P
(sup |Ĝ(x) − F (x)| ≤ ε x∈R ≥ 1 − δ, then
P
P
(
)
X ≤ Ĝ−1 (1 − α) ≥ 1 − (α + ε + δ) (5.2.2.7)
P
(
)
X ≥ G −1 (α) ≥ 1 − (α + ε + δ) (5.2.2.8)
(
)
G −1 (α) ≤ X ≥ G −1 (1 − α) ≥ 1 − 2(α + ε) (5.2.2.9)
Lemma 5.2.2.3. Let X (n) = (X 1 , . . . , X n ) be an i.i.d. sequence of random variables with
distribution P . Denote by J n (x, P ) the probability distribution function of a real valued
root R n = R n (X (n) , P ) under P . Let N n = ( b
n) , kn = ⌊n/b⌋ and define L n (x, P ) according
to 5.2.2.5. Then, for every ε > 0 and every 0 < δ < 1, we have :
i.)
ii.)
P
(
P
(
sup |L n (x, P ) − J b (x, P )| > ε
x∈R
sup |L n (x, P ) − J b (x, P )| > ε
x∈R
)
)
≤ 1 ε
√
2π
k n
(5.2.2.10)
≤ δ ε + 2 ε exp(−2k nδ 2 ) (5.2.2.11)
Proof. Define
S n (x, P ; X 1 , . . . , X n ) = 1
k n
∑
1
1≤ik n
{
}
R((X b(i−1) , . . . , X bi ), P ) ≤ x − J b (x, P )
Denote by S n the symmetric group on a set of cardinality n. Note that {X n,(b),i } 1≤i≤Nn =
⋃
π∈S n
{(X π(b(i−1)+1) , . . . , X π(bi) )} 1≤i≤kn , This allows us to express
as
Then we have
1
N n
∑
1≤i≤N n
1{R b (X n,(b),i , P ≤ x)} − J b (x, P )
Z n (x, P ; X 1 , . . . , X n ) = 1 n!
sup
x∈R
∑ (
)
S n x, P ; X π(1) , . . . , X π(n) .
π∈S n
|Z n (x, P ; X 1 , . . . , X n )| ≤ 1 n! sup |S n (x, P ; X π(1) , . . . , X π(n) )|
x∈R