一个凸离散分布的估计

用于结合协方差估计的多元回归的凸重参数化与自协调算法

Convex Reparameterization and Self-Concordant Algorithms for Multivariate Regression with Covariance Estimation

该研究提出结合协方差估计的多元回归的凸重参数化方法，证明其缩放高斯损失为标准自协调，推导两种算法，模拟及蛋白质表达应用显示其预测误差相近且稠密精度矩阵时速度显著提升。

Piecewise Convex Function Estimation: Representations, Duality and Model Selection

We consider spline estimates which preserve prescribed piecewise convex properties of the unknown function. A robust version of the penalized likelihood is given and shown to correspond to a variable halfwidth kernel smoother where the halfwidth adaptively decreases in regions of rapid change of the unknown function. When the convexity change points are prescribed, we derive representation results and smoothness properties of the estimates. A dual formulation is given which reduces the estimate is reduced to a finite dimensional convex optimization in the dual space.

Piecewise Convex Function Estimation and Model Selection

Given noisy data, function estimation is considered when the unknown function is known apriori to consist of a small number of regions where the function is either convex or concave. When the regions are known apriori, the estimate is reduced to a finite dimensional convex optimization in the dual space. When the number of regions is unknown, the model selection problem is to determine the number of convexity change points. We use a pilot estimator based on the expected number of false inflection points.

凸函数极小点与极小值的估计与推断：最优性、自适应性与不确定性原理

Estimation and inference for minimizer and minimum of convex functions: optimality, adaptivity and uncertainty principles

该研究在非渐近局部极小极大框架下探究凸回归函数极小点与极小值的估计推断，提出自适应高效算法并给出下界，还建立了二者同时估计的不确定性原理。

Mean and variance estimation in high-dimensional heteroscedastic models with non-convex penalties

Despite its prevalence in statistical datasets, heteroscedasticity (non-constant sample variances) has been largely ignored in the high-dimensional statistics literature. Recently, studies have shown that the Lasso can accommodate heteroscedastic errors, with minor algorithmic modifications (Belloni et al., 2012; Gautier and Tsybakov, 2013). In this work, we study heteroscedastic regression with linear mean model and log-linear variances model with sparse high-dimensional parameters. In this work, we propose estimating variances in a post-Lasso fashion, which is followed by weighted-least squa…

Consistent estimation of distribution functions under increasing concave and convex stochastic ordering

A random variable $Y_1$ is said to be smaller than $Y_2$ in the increasing concave stochastic order if $\mathbb{E}[ϕ(Y_1)] \leq \mathbb{E}[ϕ(Y_2)]$ for all increasing concave functions $ϕ$ for which the expected values exist, and smaller than $Y_2$ in the increasing convex order if $\mathbb{E}[ψ(Y_1)] \leq \mathbb{E}[ψ(Y_2)]$ for all increasing convex $ψ$. This article develops nonparametric estimators for the conditional cumulative distribution functions $F_x(y) = \mathbb{P}(Y \leq y \mid X = x)$ of a response variable $Y$ given a covariate $X$, solely under the assumption that the conditiona…

Signal Processing and Piecewise Convex Estimation

Many problems on signal processing reduce to nonparametric function estimation. We propose a new methodology, piecewise convex fitting (PCF), and give a two-stage adaptive estimate. In the first stage, the number and location of the change points is estimated using strong smoothing. In the second stage, a constrained smoothing spline fit is performed with the smoothing level chosen to minimize the MSE. The imposed constraint is that a single change point occurs in a region about each empirical change point of the first-stage estimate. This constraint is equivalent to requiring that the third d…

从正则变化随机向量的凸组合估计谱测度

Estimation of the Spectral Measure from ConvexCombinations of Regularly Varying RandomVectors

本文提出利用正则变化随机向量分量凸组合的极值行为构造谱向量组合矩的新估计量，证明其渐近正态性，并通过子采样 bootstrap 的 plug-in 估计实现极值系数的最小渐近方差。

广义广义线性模型：凸估计与在线界

Generalized generalized linear models: Convex estimation and online bounds

本研究提出基于单调算子变分不等式的GGLM参数估计框架，解决时空数据下模型参数估计的非凸问题并给出恢复保证，同时提出在线界，经模拟和野火数据验证有效。