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

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

2007-08-16

arxiv.org/abs/0708.2219v1

Annals of Statistics 2007, Vol. 35, No. 3, 1080-1104

Mathematics

Statistics Theory

Published at http://dx.doi.org/10.1214/009053606000001497 in the Annals of Statistics (http://www.imstat.org/aos/) by the Inst

Scientific paper

10.1214/009053606000001497

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.

Affiliated with

Also associated with

No associations

Rating

On the $\mathbb{L}_p$-error of monotonicity constrained estimators does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On the $\mathbb{L}_p$-error of monotonicity constrained estimators, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the $\mathbb{L}_p$-error of monotonicity constrained estimators will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-332424