  Suppose one has a collection of parameters indexed by a (possibly infinite
dimensional) set. Given data generated from some distribution, the objective is
to estimate the maximal parameter in this collection evaluated at this
distribution. This estimation problem is typically non-regular when the
maximizing parameter is non-unique, and as a result standard asymptotic
techniques generally fail in this case. We present a technique for developing
parametric-rate confidence intervals for the quantity of interest in these
non-regular settings. We show that our estimator is asymptotically efficient
when the maximizing parameter is unique so that regular estimation is possible.
We apply our technique to a recent example from the literature in which one
wishes to report the maximal absolute correlation between a prespecified
outcome and one of p predictors. The simplicity of our technique enables an
analysis of the previously open case where p grows with sample size.
Specifically, we only require that log(p) grows slower than the square root of
n, where n is the sample size. We show that, unlike earlier approaches, our
method scales to massive data sets: the point estimate and confidence intervals
can be constructed in O(np) time.
