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

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 (2008), Jankowski (2014), and Sohl (2015)) our results hold under minimal 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.