On the $\mathbb{L}_p$-error of monotonicity constrained estimators

TL;DR

This paper studies the asymptotic behavior of $ ext{L}_p$-loss for monotonicity constrained estimators, providing a unified approach and extending known results to new models like censored data and Poisson processes.

Contribution

It introduces a unified framework for analyzing the $ ext{L}_p$-error of monotonic estimators and extends results to new models such as failure rates in censored data and Poisson intensities.

Findings

$ ext{L}_p$-loss is asymptotically Gaussian with explicit mean and variance.

Local and global $ ext{L}_p$-risk are of order $n^{-p/3}$.

Results recover and extend known estimators like Grenander and Brunk.

Abstract

We aim at estimating a function $\lambda:[0,1]\to \mathbb {R}$, subject to the constraint that it is decreasing (or increasing). We provide a unified approach for studying the $\mathbb {L}_p$-loss of an estimator defined as the slope of a concave (or convex) approximation of an estimator of a primitive of $\lambda$, based on $n$ observations. Our main task is to prove that the $\mathbb {L}_p$-loss is asymptotically Gaussian with explicit (though unknown) asymptotic mean and variance. We also prove that the local $\mathbb {L}_p$-risk at a fixed point and the global $\mathbb {L}_p$-risk are of order $n^{-p/3}$. Applying the results to the density and regression models, we recover and generalize known results about Grenander and Brunk estimators. Also, we obtain new results for the Huang--Wellner estimator of a monotone failure rate in the random censorship model, and for an estimator of the monotone intensity function of an inhomogeneous Poisson process.