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

TL;DR

This paper establishes the asymptotic distribution and efficiency of a plug-in estimator for smooth integrated functionals of a nonincreasing density, extending previous results to broader classes of functionals and less restrictive conditions.

Contribution

It provides the first exact asymptotic distribution results for a broad class of functionals of monotone densities using a simple, tuning parameter-free plug-in estimator.

Findings

The estimator is always $ oot n$-consistent and asymptotically normal.

It achieves semiparametric efficiency for certain functionals.

Explicit asymptotic distribution characterized for uniform distribution case.

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. 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.