Subsampling estimates of the Lasso distribution.

5.2 Uniform consistency for quantiles appproximation 47
5.2.1 Statement of the general result
Theorem 5.2.1.1. (Uniform asymptotic validity of subsampling) Let X (n) =
(X 1 , . . . , X n ) be an i.i.d. sequence of random variables with distribution P ∈ P. Denote
by J n (x, P ) the distribution of a real-valued root R n = R n (X (n) , P ) under P . Let
b = b n < n be a sequence of positive integers tending to infinity, but satisfying b/n → 0.
Let N n = ( b
n
) and
L n (x, P ) = 1
N n
∑
1≤i≤N n
1{R b (X n,(b),i , P ) ≤ x}, (5.2.1.1)
where X n,(b),i denotes the i-th subset of data of size b. Then, the following statements are
true for every α ∈ (0, 1) :
(i) If lim sup n→∞ sup P ∈P sup x∈R {J b (x, P ) − J n (x, P )} ≤ 0, then
lim inf
n→∞
(
)
inf P R n ≤ L −1
n (1 − α, P ) ≥ 1 − α (5.2.1.2)
P ∈P
(ii) If lim sup n→∞ sup P ∈P sup x∈R {J n (x, P ) − J b (x, P )} ≤ 0, then
lim inf
n→∞
(
)
inf P R n ≥ L −1
n (α, P ) ≥ 1 − α (5.2.1.3)
P ∈P
(iii) If lim sup n→∞ sup P ∈P sup x∈R |J b (x, P ) − J n (x, P )| = 0, then 5.2.1.2 and 5.2.1.3 hold
with the lim inf n→∞ and ≥ replaced by lim n→∞ and =, respectively. Moreover,
(
)
lim inf P L −1
n→∞
n (α, P ) ≤ R n ≤ L −1
n (1 − α, P ) = 1 − 2α (5.2.1.4)
P ∈P
Remark. Consider again the root R n of the previous section:
R n (X (n) , P ) = τ n (ˆθ n − θ(P )) (5.2.1.5)
where ˆθ n = ˆθ n (X (n) ) is an estimate of a real-valued parameter θ(P ) and tau n > 0 is
a normalizing sequence tending to infinity. For the feasible estimate ˆL n (x) of J n (x, P )
defined as
we have
L −1
n (1 − α, P ) = inf
ˆL n (x) = 1
N n
⎧
⎨
1
x∈R ⎩N n
⎧
⎨ 1
= inf
x∈R ⎩N n
=
∑
1≤i≤N n
1
(
τ b (X n,(b),i − ˆθ
)
n )
⎫
∑
)
⎬
1{τ b
(ˆθb (X n,(b),i ) − θ(P ) ≤ x} ≥ 1 − α
⎭
1≤i≤N n
⎫
∑
1{τ b
(ˆθb (X n,(b),i ) − ˆθ
)
⎬
n ≤ x − τ b (ˆθ n − θ(P ))}
⎭
1≤i≤N n
ˆL
−1
n (1 − α) + τ b (ˆθ n − θ(P ))
Hence, under the conditions of Theorem 5.2.1.1, we obtain for this particular root
(i) lim inf n→∞ inf P ∈P P
(
)
(τ n − τ b )(ˆθ n − θ(P )) ≤ ˆL
−1
n (1 − α) ≥ 1 − α