  We consider the problem of estimating smooth integrated functionals of a
monotone nonincreasing density $f$ on $[0,\infty)$ using the nonparametric
maximum likelihood based plug-in estimator. We find the exact asymptotic
distribution of this natural (tuning parameter-free) plug-in estimator,
properly normalized. In particular, we show that the simple plug-in estimator
is always $\sqrt{n}$-consistent, and is additionally asymptotically normal with
zero mean and the semiparametric efficient variance for estimating a subclass
of integrated functionals. Compared to the previous results on this topic (see
e.g., Nickl (2007), Gine and Nickl (2008), Jankowski (2014), and Sohl (2015))
our results hold for a much larger class of functionals (which include linear
and non-linear functionals) under less restrictive assumptions on the
underlying $f$ --- we do not require $f$ to be (i) smooth, (ii) bounded away
from $0$, or (iii) compactly supported. Further, when $f$ is the uniform
distribution on a compact interval we explicitly characterize the asymptotic
distribution of the plug-in estimator --- which now converges at a non-standard
rate --- thereby extending the results in Groeneboom and Pyke (1983) for the
case of the quadratic functional.
