Subsampling estimates of the Lasso distribution.

Chapter 3
Application to the Lasso estimator
In this chapter, we first apply Theorem 2.1.0.2 and 2.3.0.11 to determine the limit in
probability and in distribution of the Lasso estimator, defined as ˆβ n = arg min φ∈R p Z n (φ)
for
for a linear regression model
Z n (φ) = 1 n∑
(Y i − x ′
n
iφ) 2 + λ n
p∑
|φ j |. (3.0.0.1)
n
i=1
j=1
Y i = x ′ iβ + ε i , i = 1, . . . , n. (3.0.0.2)
Then some continuity properties of the limit distributions are derived and it is shown
under assumption of an orthogonal design that the convergence in distribution is actually
uniform.
Throughout this chapter we make the following assumption on the model 3.0.0.2.
and
C n = 1 n X′ nX n → C (3.0.0.3)
1
n max
1≤i≤n x′ ix i → 0 (3.0.0.4)
3.1 Limit in probability
To determine the limit in probability, the following law of large numbers will be needed.
Theorem 3.1.0.12. (Kallenberg, 2002, Corollary 4.22) Let ξ 1 , ξ 2 , . . . be independent random
variables with mean 0 satisfying
∞∑
n −2c E(ξi 2 ) < ∞
i=1
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