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凸优化、形状约束、复合决策与经验贝叶斯规则AI 解读

Estimation of mixture densities for the classical Gaussian compound decision problem and their associated (empirical) Bayes rules is considered from two new perspectives. The first, motivated by Brown and Greenshtein, introduces a nonparametric maximum likelihood estimator of the mixture density subject to a monotonicity constraint on the resulting Bayes rule. The second, motivated by Jiang and Zhang, proposes a new approach to computing the Kiefer–Wolfowitz nonparametric…

拟凹密度估计AI 解读

Maximum likelihood estimation of a log-concave probability density is formulated as a convex optimization problem and shown to have an equivalent dual formulation as a constrained maximum Shannon entropy problem. Closely related maximum Renyi entropy estimators that impose weaker concavity restrictions on the fitted density are also considered, notably a minimum Hellinger discrepancy estimator that constrains the reciprocal of the square-root of the density to be concave. A…

位置—尺度深度AI 解读

This article introduces a halfspace depth in the location–scale model that is along the lines of the general theory given by Mizera, based on the idea of Rousseeuw and Hubert, and is complemented by a new likelihood-based principle for designing criterial functions. The most tractable version of the proposed depth—the Student depth—turns out to be nothing but the bivariate halfspace depth interpreted in the Poincaré plane model of the Lobachevski geometry. This fact implies…

惩罚三角网模型：二维平滑的全变差正则化AI 解读

Summary Hansen, Kooperberg and Sardy introduced a family of continuous, piecewise linear functions defined over adaptively selected triangulations of the plane as a general approach to statistical modelling of bivariate densities and regression and hazard functions. These triograms enjoy a natural affine equivariance that offers distinct advantages over competing tensor product methods that are more commonly used in statistical applications. Triograms employ basis functions…

线性模型中稳健 M 估计量的崩溃点与变动指数AI 解读

The breakdown point behavior of $M$-estimators in linear models with fixed designs, arising from planned experiments or qualitative factors, is characterized. Particularly, this behavior at fixed designs is quite different from that at designs which can be corrupted by outliers, the situation prevailing in the literature. For fixed designs, the breakdown points of robust $M$-estimators (those with bounded derivative of the score function), depend on the design and the…

独立非同分布随机变量样本中位数弱一致性的充要条件AI 解读

Necessary and sufficient conditions for the weak consistency of the sample median of independent, but not identically distributed random variables are given and discussed.