On efficiency of the plug-in principle for estimating smooth integrated functionals of a nonincreasing density

Abstract

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. To the best of our knowledge, this is the first time that such a large class of integrated functionals based on the Grenander estimator is treated in a unified fashion. Compared to the previous results on this topic (see e.g., Nickl (2008), Giné and Nickl (2016, Chapter 7), Jankowski (2014), and Söhl (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.