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• W2963694441 abstract "In this paper, we consider estimators for an additive functional of φ, which is defined as θ(P; φ) = Σ <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>k</sup> <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>i=1</sub> φ(p <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>i</sub> ), from n i.i.d. random samples drawn from a discrete distribution P = (p <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>1</sub> ,..., p <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>k</sub> ) with alphabet size k. We propose a minimax optimal estimator for the estimation problem of the additive functional. We reveal that the minimax optimal rate is characterized by the divergence speed of the fourth derivative of φ if the divergence speed is high. As a result, we show there is no consistent estimator if the divergence speed of the fourth derivative of φ is larger than p <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>-4</sup> . Furthermore, if the divergence speed of the fourth derivative of φ is p <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>4-α</sup> for α ϵ (0,1), the minimax optimal rate is obtained within a universal multiplicative constant as k <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>2</sup> /(n ln n) <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>2α</sup> + k <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>2-2α</sup> /n." @default.
• W2963694441 created "2019-07-30" @default.
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• W2963694441 date "2017-06-01" @default.
• W2963694441 modified "2023-10-18" @default.
• W2963694441 title "Minimax optimal estimators for additive scalar functionals of discrete distributions" @default.
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• W2963694441 doi "https://doi.org/10.1109/isit.2017.8006900" @default.
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