Technical report: Adaptivity and optimality of the monotone least squares estimator for four different models

2008/05/13 by Eric Cator, Cator, Eric
Engineering · Mathematics · #62G07 #62G08 #Advanced Statistical Methods and Models #Control Systems and Identification #FOS: Mathematics #Statistical Methods and Inference #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.0805.1855

openalex publication_date 2008/05/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we will consider the estimation of a monotone regression (or density) function in a fixed point by the least squares (Grenander) estimator. We will show that this estimator is fully adaptive, in the sense that the attained rate is given by a functional relation using the underlying function f₀, and not by some smoothness parameter, and that this rate is optimal when considering the class of all monotone functions, in the sense that there exists a sequence of alternative monotone functions f₁, such that no other estimator can attain a better rate for both f₀ and f₁. We also show that under mild conditions the estimator attains the same rate in L^q sense, and we give general conditions for which we can calculate a (non-standard) limiting distribution for the estimator.

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