3 research outputs found
Variance function estimation in high-dimensions
We consider the high-dimensional heteroscedastic regression model, where the
mean and the log variance are modeled as a linear combination of input
variables. Existing literature on high-dimensional linear regres- sion models
has largely ignored non-constant error variances, even though they commonly
occur in a variety of applications ranging from biostatis- tics to finance. In
this paper we study a class of non-convex penalized pseudolikelihood estimators
for both the mean and variance parameters. We show that the Heteroscedastic
Iterative Penalized Pseudolikelihood Optimizer (HIPPO) achieves the oracle
property, that is, we prove that the rates of convergence are the same as if
the true model was known. We demonstrate numerical properties of the procedure
on a simulation study and real world data.Comment: Appearing in Proceedings of the 29 th International Conference on
Machine Learning, Edinburgh, Scotland, UK, 201
From sparse to dense functional data in high dimensions: Revisiting phase transitions from a non-asymptotic perspective
Nonparametric estimation of the mean and covariance functions is ubiquitous
in functional data analysis and local linear smoothing techniques are most
frequently used. Zhang and Wang (2016) explored different types of asymptotic
properties of the estimation, which reveal interesting phase transition
phenomena based on the relative order of the average sampling frequency per
subject to the number of subjects , partitioning the data into three
categories: ``sparse'', ``semi-dense'' and ``ultra-dense''. In an increasingly
available high-dimensional scenario, where the number of functional variables
is large in relation to , we revisit this open problem from a
non-asymptotic perspective by deriving comprehensive concentration inequalities
for the local linear smoothers. Besides being of interest by themselves, our
non-asymptotic results lead to elementwise maximum rates of convergence
and uniform convergence serving as a fundamentally important tool for further
convergence analysis when grows exponentially with and possibly .
With the presence of extra terms to account for the high-dimensional
effect, we then investigate the scaled phase transitions and the corresponding
elementwise maximum rates from sparse to semi-dense to ultra-dense functional
data in high dimensions. Finally, numerical studies are carried out to confirm
our established theoretical properties