Titlebook: Lectures on the Nearest Neighbor Method; Gérard Biau,Luc Devroye Book 2015 Springer International Publishing Switzerland 2015 Density Esti

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Weighted ,-nearest neighbor density estimatesThere are different ways to weigh or smooth the .-nearest neighbor density estimate. Some key ideas are surveyed in this chapter. For some of them, consistency theorems are stated.

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Pointwise consistencyTheorem 11.1 below is a slight extension of a theorem due to Devroye (1981a). It offers sufficient conditions on the probability weight vector guaranteeing that the (raw) nearest neighbor estimate (8.2) satisfies, for all . ≥ 1.

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Uniform consistencyThe supremum creates two problems—first of all, by moving . about ., the data ordering changes. We will count the number of possible data permutations in the second section. Second, we need a uniform condition on the “noise” . − .(.) so that the averaging done by the weights .. is strong enough. This is addressed in the third section.

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Advanced properties of uniform order statisticsVarious properties of ., uniform [0, 1] order statistics, will be needed in the analysis that follows. These are collected in the present chapter. The first group of properties is directly related to .. (1 ≤ . ≤ .), while the second group deals with random linear combinations of them.

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