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Knockoff 方法的稳健推断AI 解读

We consider the variable selection problem, which seeks to identify important variables influencing a response $Y$ out of many candidate features $X_{1},\ldots ,X_{p}$. We wish to do so while offering finite-sample guarantees about the fraction of false positives—selected variables $X_{j}$ that in fact have no effect on $Y$ after the other features are known. When the number of features $p$ is large (perhaps even larger than the sample size $n$), and we have no prior…

多维扩散过程漂移向量场的非参数统计推断AI 解读

The problem of determining a periodic Lipschitz vector field $b=(b_{1},\ldots ,b_{d})$ from an observed trajectory of the solution $(X_{t}:0\le t\le T)$ of the multi-dimensional stochastic differential equation \begin{equation*}dX_{t}=b(X_{t})\,dt+dW_{t},\quad t\geq 0,\end{equation*} where $W_{t}$ is a standard $d$-dimensional Brownian motion, is considered. Convergence rates of a penalised least squares estimator, which equals the maximum a posteriori (MAP) estimate…

直接插值：无岭惩罚核回归也能泛化AI 解读

In the absence of explicit regularization, Kernel “Ridgeless” Regression with nonlinear kernels has the potential to fit the training data perfectly. It has been observed empirically, however, that such interpolated solutions can still generalize well on test data. We isolate a phenomenon of implicit regularization for minimum-norm interpolated solutions which is due to a combination of high dimensionality of the input data, curvature of the kernel function and favorable…

发散尖峰特征值与最大非尖峰样本协方差特征值的极限规律AI 解读

We study the asymptotic distributions of the spiked eigenvalues and the largest nonspiked eigenvalue of the sample covariance matrix under a general covariance model with divergent spiked eigenvalues, while the other eigenvalues are bounded but otherwise arbitrary. The limiting normal distribution for the spiked sample eigenvalues is established. It has distinct features that the asymptotic mean relies on not only the population spikes but also the nonspikes and that the…

多项式时间内实现次高斯速率的均值估计AI 解读

We study polynomial time algorithms for estimating the mean of a heavy-tailed multivariate random vector. We assume only that the random vector $X$ has finite mean and covariance. In this setting, the radius of confidence intervals achieved by the empirical mean are large compared to the case that $X$ is Gaussian or sub-Gaussian. We offer the first polynomial time algorithm to estimate the mean with sub-Gaussian-size confidence intervals under such mild assumptions. Our…

连续函数 Banach 空间中的函数型数据分析AI 解读

Functional data analysis is typically conducted within the $L^{2}$-Hilbert space framework. There is by now a fully developed statistical toolbox allowing for the principled application of the functional data machinery to real-world problems, often based on dimension reduction techniques such as functional principal component analysis. At the same time, there have recently been a number of publications that sidestep dimension reduction steps and focus on a fully functional $L^{2}$…

抛物型随机偏微分方程的高频分析AI 解读

We consider the problem of estimating stochastic volatility for a class of second-order parabolic stochastic PDEs. Assuming that the solution is observed at high temporal frequency, we use limit theorems for multipower variations and related functionals to construct consistent nonparametric estimators and asymptotic confidence bounds for the integrated volatility process. As a byproduct of our analysis, we also obtain feasible estimators for the regularity of the spatial…

β 混合厚尾时间序列中 Lasso 的理论保证AI 解读

Many theoretical results for lasso require the samples to be i.i.d. Recent work has provided guarantees for lasso assuming that the time series is generated by a sparse Vector Autoregressive (VAR) model with Gaussian innovations. Proofs of these results rely critically on the fact that the true data generating mechanism (DGM) is a finite-order Gaussian VAR. This assumption is quite brittle: linear transformations, including selecting a subset of variables, can lead to the…

部分成对比较估计的最坏情形与平均情形设计AI 解读

Pairwise comparison data arises in many domains, including tournament rankings, web search and preference elicitation. Given noisy comparisons of a fixed subset of pairs of items, we study the problem of estimating the underlying comparison probabilities under the assumption of strong stochastic transitivity (SST). We also consider the noisy sorting subclass of the SST model. We show that when the assignment of items to the topology is arbitrary, these permutation-based…

Archimax Copula 的统计推断AI 解读

Archimax copula models can account for any type of asymptotic dependence between extremes and at the same time capture joint risks at medium levels. An Archimax copula is characterized by two functional parameters: the stable tail dependence function $\ell $, and the Archimedean generator $\psi $ which distorts the extreme-value dependence structure. This article develops semiparametric inference for Archimax copulas: a nonparametric estimator of $\ell $ and a moment-based…

多项 Logistic 模型的 D 最优设计AI 解读

We consider optimal designs for general multinomial logistic models, which cover baseline-category, cumulative, adjacent-categories and continuation-ratio logit models, with proportional odds, nonproportional odds or partial proportional odds assumption. We derive the corresponding Fisher information matrices in three different forms to facilitate their calculations, determine the conditions for their positive definiteness, and search for optimal designs. We conclude that,…

样本协方差矩阵特征向量经验谱分布的收敛AI 解读

The eigenvector empirical spectral distribution (VESD) is a useful tool in studying the limiting behavior of eigenvalues and eigenvectors of covariance matrices. In this paper, we study the convergence rate of the VESD of sample covariance matrices to the deformed Marčenko–Pastur (MP) distribution. Consider sample covariance matrices of the form $\Sigma ^{1/2}XX^{*}\Sigma ^{1/2}$, where $X=(x_{ij})$ is an $M\times N$ random matrix whose entries are independent random…

利用数据子集证明最大似然估计一致性及其在多元混合模型中的应用AI 解读

We present new results for consistency of maximum likelihood estimators with a focus on multivariate mixed models. Our theory builds on the idea of using subsets of the full data to establish consistency of estimators based on the full data. It requires neither that the data consist of independent observations, nor that the observations can be modeled as a stationary stochastic process. Compared to existing asymptotic theory using the idea of subsets, we substantially weaken…

具有统计保证的 α 变分推断AI 解读

We provide statistical guarantees for a family of variational approximations to Bayesian posterior distributions, called $\alpha $-VB, which has close connections with variational approximations of tempered posteriors in the literature. The standard variational approximation is a special case of $\alpha $-VB with $\alpha =1$. When $\alpha \in (0,1]$, a novel class of variational inequalities are developed for linking the Bayes risk under the variational approximation to the…

Hurst 函数估计AI 解读

This paper considers a wide range of issues concerning the estimation of the Hurst function of a multifractional Brownian motion when the process is observed on a regular grid. A theoretical lower bound for the minimax risk of this inference problem is established for a wide class of smooth Hurst functions. We also propose a new nonparametric estimator and show that it is rate optimal. Implementation issues of the estimator including how to overcome the presence of a…

高维数据下正则化校准估计的处理效应模型辅助推断AI 解读

Consider the problem of estimating average treatment effects when a large number of covariates are used to adjust for possible confounding through outcome regression and propensity score models. We develop new methods and theory to obtain not only doubly robust point estimators for average treatment effects, which remain consistent if either the propensity score model or the outcome regression model is correctly specified, but also model-assisted confidence intervals, which…

Ising 模型参数的联合估计AI 解读

We study joint estimation of the inverse temperature and magnetization parameters $(\beta ,B)$ of an Ising model with a nonnegative coupling matrix $A_{n}$ of size $n\times n$, given one sample from the Ising model. We give a general bound on the rate of consistency of the bi-variate pseudo-likelihood estimator. Using this, we show that estimation at rate $n^{-1/2}$ is always possible if $A_{n}$ is the adjacency matrix of a bounded degree graph. If $A_{n}$ is the scaled…

模型搜索后的预测误差AI 解读

Estimation of the prediction error of a linear estimation rule is difficult if the data analyst also uses data to select a set of variables and constructs the estimation rule using only the selected variables. In this work, we propose an asymptotically unbiased estimator for the prediction error after model search. Under some additional mild assumptions, we show that our estimator converges to the true prediction error in $L^{2}$ at the rate of $O(n^{-1/2})$, with $n$ being…

高斯位置混合密度的非参数极大似然估计及其高斯去噪应用AI 解读

We study the nonparametric maximum likelihood estimator (NPMLE) for estimating Gaussian location mixture densities in $d$-dimensions from independent observations. Unlike usual likelihood-based methods for fitting mixtures, NPMLEs are based on convex optimization. We prove finite sample results on the Hellinger accuracy of every NPMLE. Our results imply, in particular, that every NPMLE achieves near parametric risk (up to logarithmic multiplicative factors) when the true…

面向预测的树结构 Ising 模型学习AI 解读

We study the problem of learning a tree Ising model from samples such that subsequent predictions made using the model are accurate. The prediction task considered in this paper is that of predicting the values of a subset of variables given values of some other subset of variables. Virtually all previous work on graphical model learning has focused on recovering the true underlying graph. We define a distance (“small set TV” or ssTV) between distributions $P$ and $Q$ by…

存在干扰的网络中处理效应估计的实验设计AI 解读

In this paper, we introduce new, easily implementable designs for drawing causal inference from randomized experiments on networks with interference. Inspired by the idea of matching in observational studies, we introduce the notion of considering a treatment assignment as a “quasi-coloring” on a graph. Our idea of a perfect quasi-coloring strives to match every treated unit on a given network with a distinct control unit that has identical number of treated and control…

指数族中的多维多尺度扫描：极限理论与统计结论AI 解读

We consider the problem of finding anomalies in a $d$-dimensional field of independent random variables $\{Y_{i}\}_{i\in \{1,\ldots,n\}^{d}}$, each distributed according to a one-dimensional natural exponential family $\mathcal{F}=\{F_{\theta }\}_{\theta \in \Theta }$. Given some baseline parameter $\theta _{0}\in \Theta $, the field is scanned using local likelihood ratio tests to detect from a (large) given system of regions $\mathcal{R}$ those regions $R\subset \{1,\ldots,n\}^{d}$…

单调分段常数信号的估计AI 解读

Consider a sequence of real data points $X_{1},\ldots ,X_{n}$ with underlying means $\theta ^{*}_{1},\dots ,\theta ^{*}_{n}$. This paper starts from studying the setting that $\theta ^{*}_{i}$ is both piecewise constant and monotone as a function of the index $i$. For this, we establish the exact minimax rate of estimating such monotone functions, and thus give a nontrivial answer to an open problem in the shape-constrained analysis literature. The minimax rate under the…

高维最大值统计量的自助法：弱方差衰减下的近参数速率及应用AI 解读

In recent years, bootstrap methods have drawn attention for their ability to approximate the laws of “max statistics” in high-dimensional problems. A leading example of such a statistic is the coordinatewise maximum of a sample average of $n$ random vectors in $\mathbb{R}^{p}$. Existing results for this statistic show that the bootstrap can work when $n\ll p$, and rates of approximation (in Kolmogorov distance) have been obtained with only logarithmic dependence in $p$.…