  The $\ell_1$-penalized method, or the Lasso, has emerged as an important tool
for the analysis of large data sets. Many important results have been obtained
for the Lasso in linear regression which have led to a deeper understanding of
high-dimensional statistical problems. In this article, we consider a class of
weighted $\ell_1$-penalized estimators for convex loss functions of a general
form, including the generalized linear models. We study the estimation,
prediction, selection and sparsity properties of the weighted
$\ell_1$-penalized estimator in sparse, high-dimensional settings where the
number of predictors $p$ can be much larger than the sample size $n$. Adaptive
Lasso is considered as a special case. A multistage method is developed to
apply an adaptive Lasso recursively. We provide $\ell_q$ oracle inequalities, a
general selection consistency theorem, and an upper bound on the dimension of
the Lasso estimator. Important models including the linear regression, logistic
regression and log-linear models are used throughout to illustrate the
applications of the general results.
