TITLE:
Extended Oracle Properties of Adaptive Lasso Estimators
AUTHORS:
Lorenzo Camponovo
KEYWORDS:
Adaptive Lasso, Asymptotics, Oracle Properties
JOURNAL NAME:
Open Journal of Statistics,
Vol.12 No.2,
April
18,
2022
ABSTRACT: We study the asymptotic properties of
adaptive lasso estimators when some components of the parameter of interest β
are strictly different than zero, while other components may be zero or may converge
to zero with rate n-δ, with δ>0, where n denotes the sample size. To achieve this objective, we analyze the
convergence/divergence rates of each term in the first-order conditions of adaptive
lasso estimators. First, we derive conditions that allow selecting tuning parameters
in order to ensure that adaptive lasso estimates of n-δ-components indeed collapse
to zero. Second, in this case, we also derive asymptotic distributions of adaptive
lasso estimators for nonzero components. When δ>1/2, we obtain the usual n1/2-asymptotic
normal distribution, while when 0δ ≤ 1/2, we show nδ-consistency combined with
(biased) n1/2-δ-asymptotic normality
for nonzero components. We call these properties, Extended Oracle Properties.
These results allow practitioners to exclude in their model the asymptotically negligible
variables and make inferences on the asymptotically relevant variables.