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Adaptivity and optimality of the monotone least squares estimator

13 May 2008
Eric Cator
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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 f0f_0f0​, 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 f1f_1f1​, such that no other estimator can attain a better rate for both f0f_0f0​ and f1f_1f1​. We also show that under mild conditions the estimator attains the same rate in LqL^qLq sense, and we give general conditions for which we can calculate a (non-standard) limiting distribution for the estimator.

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