  A general method is presented for deriving the limiting behavior of
estimators that are defined as the values of parameters optimizing an empirical
criterion function. The asymptotic behavior of such estimators is typically
deduced from uniform limit theorems for rescaled and reparametrized criterion
functions. The new method can handle cases where the standard approach does not
yield the complete limiting behavior of the estimator. The asymptotic analysis
depends on a decomposition of criterion functions into sums of components with
different rescalings. The method is explained by examples from Lasso-type
estimation, $k$-means clustering, Shorth estimation and partial linear models.
