Subsampling estimates of the Lasso distribution.

44 Subsampling
The consistency of ˆL n,b as an estimator to J n (·, P ) is derived under
Assumption D.
J(P ) as n → ∞.
There exists a limit law J(P ) such that J n (P ) converges weakly to
Theorem 5.1.0.12. (Politis et al., 1999, Theorem 2.2.1) Assume that assumption D
holds. Also assume τ b /τ n → n, b → ∞, and b/n → 0 as n → ∞. Then the following are
true:
(i) If x is a continuity point of J(·, P ), then
ˆL n,b (x) → P J(x, P )
(ii) If J(·, P ) is continuous, then
∣
sup ∣ˆL n,b (x) − J n (x, P ) ∣ → P 0
(iii) Let
Correspondingly, define
x∈R
c n,b (1 − α) = inf{x ∈ R|ˆL n,b (x) ≥ 1 − α}.
c(1 − α, P ) = inf{x ∈ R|J(x, P ) ≥ 1 − α}.
Then
(
1 − α − ∆J(c(1 − α, P )) ≤ lim inf P n→∞
≤ lim sup P
n→∞
≤ 1 − α.
)
τ n (ˆθ n − θ(P ) ≤ c n,b (1 − α))
)
(
τ n (ˆθ n − θ(P )) ≤ c n,b (1 − α)
If J(·, P ) is continuous at c(1 − α, P ), then
(
)
lim P τ n (ˆθ n − θ(P )) ≤ c n,b (1 − α)
n→∞
(iv) Assume that τ b (ˆθ n − θ(P )) → 0 almost surely and that
∞∑
exp{−d(n/b)} < ∞,
n=1
= 1 − α
for every d > 0, then the convergence in i and ii hold with probability one.
Proof. Define
U n (x) = U n (x, P ) = N −1
n
N n
∑
i=1
{
) }
1 τ b
(ˆθn,b,i − θ(P ) ≤ x
To prove i, we show that U n (x) converges in probablity to J(x, P ) for every continuity
point x of J(x, P ). Note that
ˆL n,b (x) = N −1
n
N n
∑
i=1
{
1 τ b (ˆθ n,b,i − θ(P )) + τ b (θ(P ) − ˆθ
}
n ) .