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低内在维度数据上支持向量机的自适应学习速率AI 解读

We derive improved regression and classification rates for support vector machines using Gaussian kernels under the assumption that the data has some low-dimensional intrinsic structure that is described by the box-counting dimension. Under some standard regularity assumptions for regression and classification, we prove learning rates, in which the dimension of the ambient space is replaced by the box-counting dimension of the support of the data generating distribution. In…

固定域渐近下具有 Matérn 协方差的各向同性高斯随机场参数估计AI 解读

A method is proposed for estimating the microergodic parameters (including the smoothness parameter) of stationary Gaussian random fields on Rd with isotropic Matérn covariance functions using irregularly spaced data. This approach uses higher-order quadratic variations and is applied to three designs, namely stratified sampling design, randomized sampling design and deformed lattice design. Microergodic parameter estimators are constructed for each of the designs. Under…

多峰似然函数下的在线推断AI 解读

Let (Yt)t≥1 be a sequence of i.i.d. observations and {fθ,θ∈Rd} be a parametric model. We introduce a new online algorithm for computing a sequence (θˆt)t≥1, which is shown to converge almost surely to argmaxθ∈RdE[logfθ(Y1)] at rate O(log(t)(1+ε)/2t−1/2), with ε>0 a user specified parameter. This convergence result is obtained under standard conditions on the statistical model and, most notably, we allow the mapping θ↦E[logfθ(Y1)] to be multi-modal. However, the computational…

通过分组—排序—融合程序估计有限混合模型的成分数AI 解读

Estimation of the number of components (or order) of a finite mixture model is a long standing and challenging problem in statistics. We propose the Group-Sort-Fuse (GSF) procedure—a new penalized likelihood approach for simultaneous estimation of the order and mixing measure in multidimensional finite mixture models. Unlike methods which fit and compare mixtures with varying orders using criteria involving model complexity, our approach directly penalizes a continuous…

In transfer learning, we wish to make inference about a target population when we have access to data both from the distribution itself, and from a different but related source distribution. We introduce a flexible framework for transfer learning in the context of binary classification, allowing for covariate-dependent relationships between the source and target distributions that are not required to preserve the Bayes decision boundary. Our main contributions are to derive…

Expectiles define a least squares analogue of quantiles. They have been the focus of a substantial quantity of research in the context of actuarial and financial risk assessment over the last decade. The behaviour and estimation of unconditional extreme expectiles using independent and identically distributed heavy-tailed observations have been investigated in a recent series of papers. We build here a general theory for the estimation of extreme conditional expectiles in…

协变量偏移中的边际奇异性与标签收益AI 解读

Transfer Learning addresses common situations in Machine Leaning where little or no labeled data is available for a target prediction problem—corresponding to a distribution Q, but much labeled data is available from some related but different data distribution P. This work is concerned with the fundamental limits of transfer, that is, the limits in target performance in terms of (1) sample sizes from P and Q, and (2) differences in data distributions P, Q. In particular, we…

一种简单的条件依赖度量AI 解读

We propose a coefficient of conditional dependence between two random variables Y and Z given a set of other variables X1,…,Xp, based on an i.i.d. sample. The coefficient has a long list of desirable properties, the most important of which is that under absolutely no distributional assumptions, it converges to a limit in [0,1], where the limit is 0 if and only if Y and Z are conditionally independent given X1,…,Xp, and is 1 if and only if Y is equal to a measurable function…

延迟接受随机游走 Metropolis 算法的效率AI 解读

Delayed-acceptance Metropolis–Hastings and delayed-acceptance pseudo-marginal Metropolis–Hastings algorithms can be applied when it is computationally expensive to calculate the true posterior or an unbiased stochastic approximation thereof, but a computationally cheap deterministic approximation is available. An initial accept–reject stage uses the cheap approximation for computing the Metropolis–Hastings ratio; proposals which are accepted at this stage are subjected to a…

用 Wasserstein 距离度量贝叶斯非参数模型的依赖性AI 解读

The proposal and study of dependent Bayesian nonparametric models has been one of the most active research lines in the last two decades, with random vectors of measures representing a natural and popular tool to define them. Nonetheless, a principled approach to understand and quantify the associated dependence structure is still missing. We devise a general, and not model-specific, framework to achieve this task for random measure based models, which consists in: (a)…

海量数据的分布式统计推断AI 解读

This paper considers distributed statistical inference for general symmetric statistics in the context of massive data with efficient computation. Estimation efficiency and asymptotic distributions of the distributed statistics are provided, which reveal different results between the nondegenerate and degenerate cases, and show the number of the data subsets plays an important role. Two distributed bootstrap methods are proposed and analyzed to approximation the underlying…

Willem van Zwet 的研究贡献AI 解读

Willem van Zwet made deep and influential contributions to probability and statistics, which we review in this paper. Bickel and Götze collaborated with him on his major contributions to higher order asymptotics of nonlinear statistics and on resampling and the bootstrap. We relate this work to his remarkable development of the properties of the Hoeffding expansion for symmetric statistics as well as Fourier analytic tools. Fiocco and De Gunst were his students. We describe…

凸约束下选择后推断的整合方法AI 解读

Inference after model selection has been an active research topic in the past few years, with numerous works offering different approaches to addressing the perils of the reuse of data. In particular, major progress has been made recently on large and useful classes of problems by harnessing general theory of hypothesis testing in exponential families, but these methods have their limitations. Perhaps most immediate is the gap between theory and practice: implementing the…

协调高斯似然与 Whittle 似然：频域估计的应用AI 解读

In time series analysis there is an apparent dichotomy between time and frequency domain methods. The aim of this paper is to draw connections between frequency and time domain methods. Our focus will be on reconciling the Gaussian likelihood and the Whittle likelihood. We derive an exact, interpretable, bound between the Gaussian and Whittle likelihood of a second order stationary time series. The derivation is based on obtaining the transformation which is biorthogonal to…

高阶全变差正则化最小二乘的预测界AI 解读

We establish adaptive results for trend filtering: least squares estimation with a penalty on the total variation of (k−1)th order differences. Our approach is based on combining a general oracle inequality for the ℓ1-penalized least squares estimator with “interpolating vectors” to upper bound the “effective sparsity.” This allows one to show that the ℓ1-penalty on the kth order differences leads to an estimator that can adapt to the number of jumps in the (k−1)th order…

两阶段富集试验设计的统计推断AI 解读

Two-stage enrichment designs can be used to target the benefiting population in clinical trials based on patients’ biomarkers. In the case of continuous biomarkers, we show that using a bivariate model that treats biomarkers as random variables more accurately identifies a treatment-benefiting enriched population than assuming biomarkers are fixed. Additionally, we show that under the bivariate model, the maximum likelihood estimators (MLEs) follow a randomly scaled mixture…

函数型数据的经验尾部 CopulaAI 解读

For multivariate distributions in the domain of attraction of a max-stable distribution, the tail copula and the stable tail dependence function are equivalent ways to capture the dependence in the upper tail. The empirical versions of these functions are rank-based estimators whose inflated estimation errors are known to converge weakly to a Gaussian process that is similar in structure to the weak limit of the empirical copula process. We extend this multivariate result to…

非欧几里得响应与预测变量的加性回归AI 解读

Additive regression is studied in a very general setting where both the response and predictors are allowed to be non-Euclidean. The response takes values in a general separable Hilbert space, whereas the predictors take values in general semimetric spaces, which covers a very wide range of nonstandard response variables and predictors. A general framework of estimating additive models is presented for semimetric space-valued predictors. In particular, full details of…

正态模型中光滑泛函的估计：偏差降低与渐近效率AI 解读

Let X1,…,Xn be i.i.d. random variables sampled from a normal distribution N(μ,Σ) in Rd with unknown parameter θ=(μ,Σ)∈Θ:=Rd×C+d, where C+d is the cone of positively definite covariance operators in Rd. Given a smooth functional f:Θ↦R1, the goal is to estimate f(θ) based on X1,…,Xn. Let Θ(a;d):=Rd×{Σ∈C+d:σ(Σ)⊂[1/a,a]},a≥1, where σ(Σ) is the spectrum of covariance Σ. Let θˆ:=(μˆ,Σˆ), where μˆ is the sample mean and Σˆ is the sample covariance, based on the observations…

渐近依赖与独立下的秩估计及空间极值应用AI 解读

Multivariate extreme value theory is concerned with modeling the joint tail behavior of several random variables. Existing work mostly focuses on asymptotic dependence, where the probability of observing a large value in one of the variables is of the same order as observing a large value in all variables simultaneously. However, there is growing evidence that asymptotic independence is equally important in real world applications. Available statistical methodology in the…

用最优决策树自适应估计多元分段多项式与有界变差函数AI 解读

Proposed by Donoho (Ann. Statist. 25 (1997) 1870–1911), Dyadic CART is a nonparametric regression method which computes a globally optimal dyadic decision tree and fits piecewise constant functions in two dimensions. In this article, we define and study Dyadic CART and a closely related estimator, namely Optimal Regression Tree (ORT), in the context of estimating piecewise smooth functions in general dimensions in the fixed design setup. More precisely, these optimal…

高斯过程回归的变量选择一致性AI 解读

Bayesian nonparametric regression under a rescaled Gaussian process prior offers smoothness-adaptive function estimation with near minimax-optimal error rates. Hierarchical extensions of this approach, equipped with stochastic variable selection, are known to also adapt to the unknown intrinsic dimension of a sparse true regression function. But it remains unclear if such extensions offer variable selection consistency, that is, if the true subset of important variables…

基于U统计量置换检验的独立性检验最优速率AI 解读

We study the problem of independence testing given independent and identically distributed pairs taking values in a σ-finite, separable measure space. Defining a natural measure of dependence D(f) as the squared L2-distance between a joint density f and the product of its marginals, we first show that there is no valid test of independence that is uniformly consistent against alternatives of the form {f:D(f)≥ρ2}. We therefore restrict attention to alternatives that impose…

两水平平行平面设计AI 解读

Regular 2n−p designs are also known as single flat designs. Parallel flats designs (PFDs) consisting of three parallel flats (3-PFDs) are the most frequently utilized PFDs, due to their simple structure. Generalizing to f-PFD with f>3 is more challenging. This paper aims to study the general theory for the f-PFD for any f≥3. We propose a method for obtaining the confounding frequency vectors for all nonequivalent f-PFDs, and to find the least G-aberration (or highest…