Universal pointwise selection rule in multivariate function estimation

The Partial Entropy Decomposition: Decomposing multivariate entropy and mutual information via pointwise common surprisal

Obtaining meaningful quantitative descriptions of the statistical dependence within multivariate systems is a difficult open problem. Recently, the Partial Information Decomposition (PID) was proposed to decompose mutual information (MI) about a target variable into components which are redundant, unique and synergistic within different subsets of predictor variables. Here, we propose to apply the elegant formalism of the PID to multivariate entropy, resulting in a Partial Entropy Decomposition (PED). We implement the PED with an entropy redundancy measure based on pointwise common surprisal; …

Adaptive pointwise estimation of conditional density function

In this paper we consider the problem of estimating $f$, the conditional density of $Y$ given $X$, by using an independent sample distributed as $(X,Y)$ in the multivariate setting. We consider the estimation of $f(x,.)$ where $x$ is a fixed point. We define two different procedures of estimation, the first one using kernel rules, the second one inspired from projection methods. Both adapted estimators are tuned by using the Goldenshluger and Lepski methodology. After deriving lower bounds, we show that these procedures satisfy oracle inequalities and are optimal from the minimax point of view…

Normalized Maximum Likelihood with Luckiness for Multivariate Normal Distributions

The normalized maximum likelihood (NML) is one of the most important distribution in coding theory and statistics. NML is the unique solution (if exists) to the pointwise minimax regret problem. However, NML is not defined even for simple family of distributions such as the normal distributions. Since there does not exist any meaningful minimax-regret distribution for such case, it is pointed out that NML with luckiness (LNML) can be employed as an alternative to NML. In this paper, we develop the closed form of LNMLs for multivariate normal distributions.

On convergence and mass distributions of multivariate Archimedean copulas and their interplay with the Williamson transform

Motivated by a recently established result saying that within the class of bivariate Archimedean copulas standard pointwise convergence implies weak convergence of almost all conditional distributions this contribution studies the class $\mathcal{C}_{ar}^d$ of all $d$-dimensional Archimedean copulas with $d \geq 3$ and proves the afore-mentioned implication with respect to conditioning on the first $d-1$ coordinates. Several proper\-ties equivalent to pointwise convergence in $\mathcal{C}_{ar}^d$ are established and - as by-product of working with conditional distributions (Markov kernels) - a…

Multivariate density estimation from privatised data: universal consistency and minimax rates

We revisit the classical problem of nonparametric density estimation but impose local differential privacy constraints. Under such constraints, the original multivariate data $X_1,\ldots,X_n \in \mathbb{R}^d$ cannot be directly observed, and all estimators are functions of the randomised output of a suitable privacy mechanism. The statistician is free to choose the form of the privacy mechanism, and in this work we propose to add Laplace distributed noise to a discretisation of the location of a vector $X_i$. Based on these randomised data, we design a novel estimator of the density function, …

A robust, adaptive M-estimator for pointwise estimation in heteroscedastic regression

We introduce a robust and fully adaptive method for pointwise estimation in heteroscedastic regression. We allow for noise and design distributions that are unknown and fulfill very weak assumptions only. In particular, we do not impose moment conditions on the noise distribution. Moreover, we do not require a positive density for the design distribution. In a first step, we study the consistency of locally polynomial M-estimators that consist of a contrast and a kernel. Afterwards, minimax results are established over unidimensional Hölder spaces for degenerate design. We then choose the cont…

Posterior Contraction and Testing for Multivariate Isotonic Regression

We consider the nonparametric regression problem with multiple predictors and an additive error, where the regression function is assumed to be coordinatewise nondecreasing. We propose a Bayesian approach to make an inference on the multivariate monotone regression function, obtain the posterior contraction rate, and construct a universally consistent Bayesian testing procedure for multivariate monotonicity. To facilitate posterior analysis, we set aside the shape restrictions temporarily, and endow a prior on blockwise constant regression functions with heights independently normally distribu…

Adaptive estimation under single-index constraint in a regression model

The problem of adaptive multivariate function estimation in the single-index regression model with random design and weak assumptions on the noise is investigated. A novel estimation procedure that adapts simultaneously to the unknown index vector and the smoothness of the link function by selecting from a family of specific kernel estimators is proposed. We establish a pointwise oracle inequality which, in its turn, is used to judge the quality of estimating the entire function (``global'' oracle inequality). Both the results are applied to the problems of pointwise and global adaptive estima…

Convex programming approach to robust estimation of a multivariate Gaussian model

Multivariate Gaussian is often used as a first approximation to the distribution of high-dimensional data. Determining the parameters of this distribution under various constraints is a widely studied problem in statistics, and is often considered as a prototype for testing new algorithms or theoretical frameworks. In this paper, we develop a nonasymptotic approach to the problem of estimating the parameters of a multivariate Gaussian distribution when data are corrupted by outliers. We propose an estimator---efficiently computable by solving a convex program---that robustly estimates the popu…

使用带贝叶斯自适应带宽的扩展β核有效估计多元密度函数

An effective estimation of multivariate density functions using extended-beta kernels with Bayesian adaptive bandwidths

针对多元密度函数估计问题，提出带贝叶斯自适应带宽的多重扩展β核（MEBK）估计器，建立其渐近性质并提供带宽选择方法，经模拟与真实数据验证，在灵活性和通用性上优于高斯、伽马核估计器。

Universal Approximation on the Hypersphere

It is well known that any continuous probability density function on $\mathbb{R}^m$ can be approximated arbitrarily well by a finite mixture of normal distributions, provided that the number of mixture components is sufficiently large. The von-Mises-Fisher distribution, defined on the unit hypersphere $S^m$ in $\mathbb{R}^{m+1}$, has properties that are analogous to those of the multivariate normal on $\mathbb{R}^{m+1}$. We prove that any continuous probability density function on $S^m$ can be approximated to arbitrary degrees of accuracy by a finite mixture of von-Mises-Fisher distributions.

超越一致性：深度非参数Cox模型中相对风险函数的推断

Beyond Consistency: Inference for the Relative risk functional in Deep Nonparametric Cox Models

本文研究深度非参数Cox模型中相对风险函数的推断问题，提出新的渐近分布理论，解决梯度优化误差传播、点估计偏差控制及不确定性量化等挑战。