# Technical report: Adaptivity and optimality of the monotone least squares estimator for four different models - Mathematics > Statistics Theory

Technical report: Adaptivity and optimality of the monotone least squares estimator for four different models - Mathematics > Statistics Theory - Download this document for free, or read online. Document in PDF available to download.

Abstract: In this paper we will consider the estimation of a monotone regression ordensity function in a fixed point by the least squares Grenander estimator.We will show that this estimator is fully adaptive, in the sense that theattained rate is given by a functional relation using the underlying function$f 0$, and not by some smoothness parameter, and that this rate is optimal whenconsidering the class of all monotone functions, in the sense that there existsa sequence of alternative monotone functions $f 1$, such that no otherestimator can attain a better rate for both $f 0$ and $f 1$. We also show thatunder mild conditions the estimator attains the same rate in $L^q$ sense, andwe give general conditions for which we can calculate a non-standard limitingdistribution for the estimator.

Author: ** Eric Cator**

Source: https://arxiv.org/