Minimax Optimal Estimators for Additive Scalar Functionals of Discrete Distributions

Minimax Optimal Additive Functional Estimation with Discrete Distribution: Slow Divergence Speed Case

“…We previously analyze the minimax optimal rate for the additive functional with characterization of the divergence speed in Fukuchi et al [1]. Under the condition that the divergence speed of φ is p α , we revealed the optimal rate for α ∈ (0, 1).…”

“…In the line of literature [1], [9]- [11], the basic construction of the optimal estimator for the additive functional has a common part. Here, we describe the common methodology to construct the optimal estimator.…”

“…We previously derived the minimax optimal estimator for general φ ∈ C 4 [0, 1] such that φ (4) (p) is finite for p ∈ (0, 1], where C m [0, 1] denotes a class of mth times differentiable functions from [0, 1] to R and φ (m) is mth derivative of φ [1]. We revealed that the minimax optimal risk can be characterized by the divergence speed of the fourth derivative of φ, or referring it to the forth divergence speed of φ in this manuscript.…”

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This paper addresses an estimation problem of an additive functional of φ, which is defined as θ(P ; φ) = k i=1 φ(pi), given n i.i.d. random samples drawn from a discrete distribution P = (p1, ..., p k ) with alphabet size k. We have revealed in the previous paper [1] that the minimax optimal rate of this problem is characterized by the divergence speed of the fourth derivative of φ in a range of fast divergence speed. In this paper, we prove this fact for a more general range of the divergence speed.

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On estimation of nonsmooth functionals of sparse normal means

“…In recent years, there has been a growing interest in statistical estimation of non-smooth functionals (cf. Cai and Low (2011); Jiao et al (2015); Wu andYang (2015, 2016); Han et al (2017a,b); Carpentier and Verzelen (2017); Fukuchi and Sakuma (2017)). Some of these papers deal with the normal means model (cf.…”