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We consider the problem of estimating smooth integrated functionals of amonotone nonincreasing density $f$ on $[0,\infty)$ using the nonparametricmaximum likelihood based plug-in estimator. We find the exact asymptoticdistribution of this natural (tuning parameter-free) plug-in estimator,properly normalized. In particular, we show that the simple plug-in estimatoris always $\sqrt{n}$-consistent, and is additionally asymptotically normal withzero mean and the semiparametric efficient variance for estimating a subclassof integrated functionals. Compared to the previous results on this topic (seee.g., Nickl (2007), Gine and Nickl (2008), Jankowski (2014), and Sohl (2015))our results hold for a much larger class of functionals (which include linearand non-linear functionals) under less restrictive assumptions on theunderlying $f$ --- we do not require $f$ to be (i) smooth, (ii) bounded awayfrom $0$, or (iii) compactly supported. Further, when $f$ is the uniformdistribution on a compact interval we explicitly characterize the asymptoticdistribution of the plug-in estimator --- which now converges at a non-standardrate --- thereby extending the results in Groeneboom and Pyke (1983) for thecase of the quadratic functional.

On Efficiency of the Plug-in Principle for Estimating Smooth Integrated Functionals of a Nonincreasing Density