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对数凹分布逼近及其回归应用AI 解读

We study the approximation of arbitrary distributions P on d-dimensional space by distributions with log-concave density. Approximation means minimizing a Kullback–Leibler-type functional. We show that such an approximation exists if and only if P has finite first moments and is not supported by some hyperplane. Furthermore we show that this approximation depends continuously on P with respect to Mallows distance D1(⋅, ⋅). This result implies consistency of the maximum…

多维对数凹密度的最大似然估计AI 解读

Summary Let X1,…,Xn be independent and identically distributed random vectors with a (Lebesgue) density f. We first prove that, with probability 1, there is a unique log-concave maximum likelihood estimator f^n of f. The use of this estimator is attractive because, unlike kernel density estimation, the method is fully automatic, with no smoothing parameters to choose. Although the existence proof is non-constructive, we can reformulate the issue of computing f^n in terms of…

球对称分布均值的小体积置信集AI 解读

Summary Suppose that X has a k-variate spherically symmetric distribution with mean vector θ and identity covariance matrix. We present two spherical confidence sets for θ, both centred at a positive part Stein estimator TS+(X). In the first, we obtain the radius by approximating the upper α-point of the sampling distribution of ‖TS+(X)−θ‖2 by the first two non-zero terms of its Taylor series about the origin. We can analyse some of the properties of this confidence set and…