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This paper investigates estimation of the mean vector under invariant quadratic loss for a spherically symmetric location family with a residual vector with density of the form $f(x,u)=\eta ^{(p+n)/2}f(\eta \{\|x-\theta \|^{2}+\|u\|^{2}\})$, where $\eta $ is unknown. We show that the natural estimator $x$ is admissible for $p=1,2$. Also, for $p\geq 3$, we find classes of generalized Bayes estimators that are admissible within the class of equivariant estimators of the form $\{1-\xi (x/\|u\|)\}x$…

单调函数非参数推断的统一研究AI 解读

The problem of nonparametric inference on a monotone function has been extensively studied in many particular cases. Estimators considered have often been of so-called Grenander type, being representable as the left derivative of the greatest convex minorant or least concave majorant of an estimator of a primitive function. In this paper, we provide general conditions for consistency and pointwise convergence in distribution of a class of generalized Grenander-type…

基于均值中位数的稳健机器学习：理论与实践AI 解读

Median-of-means (MOM) based procedures have been recently introduced in learning theory (Lugosi and Mendelson (2019); Lecué and Lerasle (2017)). These estimators outperform classical least-squares estimators when data are heavy-tailed and/or are corrupted. None of these procedures can be implemented, which is the major issue of current MOM procedures (Ann. Statist. 47 (2019) 783–794). In this paper, we introduce minmax MOM estimators and show that they achieve the same…

尖峰 Wigner 模型中的检测基本极限AI 解读

We study the fundamental limits of detecting the presence of an additive rank-one perturbation, or spike, to a Wigner matrix. When the spike comes from a prior that is i.i.d. across coordinates, we prove that the log-likelihood ratio of the spiked model against the nonspiked one is asymptotically normal below a certain reconstruction threshold which is not necessarily of a “spectral” nature, and that it is degenerate above. This establishes the maximal region of contiguity…

删失数据下估计方程数量增长的惩罚广义经验似然AI 解读

This article considers simultaneous variable selection and parameter estimation as well as hypothesis testing in censored survival models where a parametric likelihood is not available. For the problem, we utilize certain growing dimensional general estimating equations and propose a penalized generalized empirical likelihood, where the general estimating equations are constructed based on the semiparametric efficiency bound of estimation with given moment conditions. The…

We study weighted particle systems in which new generations are resampled from current particles with probabilities proportional to their weights. This covers a broad class of sequential Monte Carlo (SMC) methods, widely-used in applied statistics and cognate disciplines. We consider the genealogical tree embedded into such particle systems, and identify conditions, as well as an appropriate time-scaling, under which they converge to the Kingman $n$-coalescent in the…

线性变换尖峰模型中的最优预测AI 解读

We consider the linearly transformed spiked model, where the observations $Y_{i}$ are noisy linear transforms of unobserved signals of interest $X_{i}$: \begin{equation*}Y_{i}=A_{i}X_{i}+\varepsilon_{i},\end{equation*} for $i=1,\ldots ,n$. The transform matrices $A_{i}$ are also observed. We model the unobserved signals (or regression coefficients) $X_{i}$ as vectors lying on an unknown low-dimensional space. Given only $Y_{i}$ and $A_{i}$ how should we predict or recover…

主成分线性泛函的有效估计AI 解读

We study principal component analysis (PCA) for mean zero i.i.d. Gaussian observations $X_{1},\dots,X_{n}$ in a separable Hilbert space $\mathbb{H}$ with unknown covariance operator $\Sigma $. The complexity of the problem is characterized by its effective rank $\mathbf{r}(\Sigma):=\frac{\operatorname{tr}(\Sigma)}{\|\Sigma \|}$, where $\mathrm{tr}(\Sigma)$ denotes the trace of $\Sigma $ and $\|\Sigma\|$ denotes its operator norm. We develop a method of bias reduction in the…

模型选择后具有一致有效性的置信区间AI 解读

We suggest general methods to construct asymptotically uniformly valid confidence intervals post-model-selection. The constructions are based on principles recently proposed by Berk et al. (Ann. Statist. 41 (2013) 802–837). In particular, the candidate models used can be misspecified, the target of inference is model-specific, and coverage is guaranteed for any data-driven model selection procedure. After developing a general theory, we apply our methods to practically…

通过样本分割一致选择变点数量AI 解读

In multiple change-point analysis, one of the major challenges is to estimate the number of change-points. Most existing approaches attempt to minimize a Schwarz information criterion which balances a term quantifying model fit with a penalization term accounting for model complexity that increases with the number of change-points and limits overfitting. However, different penalization terms are required to adapt to different contexts of multiple change-point problems and…

This paper proposes a numerical bootstrap method that is consistent in many cases where the standard bootstrap is known to fail and where the $m$-out-of-$n$ bootstrap and subsampling have been the most commonly used inference approaches. We provide asymptotic analysis under both fixed and drifting parameter sequences, and we compare the approximation error of the numerical bootstrap with that of the $m$-out-of-$n$ bootstrap and subsampling. Finally, we discuss applications…

典范与弯曲指数族随机图模型的集中性和一致性结果AI 解读

Statistical inference for exponential-family models of random graphs with dependent edges is challenging. We stress the importance of additional structure and show that additional structure facilitates statistical inference. A simple example of a random graph with additional structure is a random graph with neighborhoods and local dependence within neighborhoods. We develop the first concentration and consistency results for maximum likelihood and $M$-estimators of a wide…

多臂老虎机问题：一种有效率的非参数解法AI 解读

Lai and Robbins (Adv. in Appl. Math. 6 (1985) 4–22) and Lai (Ann. Statist. 15 (1987) 1091–1114) provided efficient parametric solutions to the multi-armed bandit problem, showing that arm allocation via upper confidence bounds (UCB) achieves minimum regret. These bounds are constructed from the Kullback–Leibler information of the reward distributions, estimated from specified parametric families. In recent years, there has been renewed interest in the multi-armed bandit…

弱可识别性下的主成分方向检验AI 解读

We consider the problem of testing, on the basis of a $p$-variate Gaussian random sample, the null hypothesis $\mathcal{H}_{0}:\boldsymbol{\theta}_{1}=\boldsymbol{\theta}_{1}^{0}$ against the alternative $\mathcal{H}_{1}:\boldsymbol{\theta}_{1}\neq \boldsymbol{\theta}_{1}^{0}$, where $\boldsymbol{\theta}_{1}$ is the “first” eigenvector of the underlying covariance matrix and $\boldsymbol{\theta}_{1}^{0}$ is a fixed unit $p$-vector. In the classical setup where eigenvalues $\lambda_{1}>\lambda_{2}\geq \cdots \geq \lambda_{p}$…

稀疏高维回归：可扩展精确算法与相变AI 解读

We present a novel binary convex reformulation of the sparse regression problem that constitutes a new duality perspective. We devise a new cutting plane method and provide evidence that it can solve to provable optimality the sparse regression problem for sample sizes $n$ and number of regressors $p$ in the 100,000s, that is, two orders of magnitude better than the current state of the art, in seconds. The ability to solve the problem for very high dimensions allows us to…

非标准条件下基于 M 估计量的 Bootstrap 置信域AI 解读

Suppose that a confidence region is desired for a subvector $\theta $ of a multidimensional parameter $\xi =(\theta ,\psi )$, based on an M-estimator $\hat{\xi }_{n}=(\hat{\theta }_{n},\hat{\psi }_{n})$ calculated from a random sample of size $n$. Under nonstandard conditions $\hat{\xi }_{n}$ often converges at a nonregular rate $r_{n}$, in which case consistent estimation of the distribution of $r_{n}(\hat{\theta }_{n}-\theta )$, a pivot commonly chosen for confidence…

随机梯度下降中模型参数的统计推断AI 解读

The stochastic gradient descent (SGD) algorithm has been widely used in statistical estimation for large-scale data due to its computational and memory efficiency. While most existing works focus on the convergence of the objective function or the error of the obtained solution, we investigate the problem of statistical inference of true model parameters based on SGD when the population loss function is strongly convex and satisfies certain smoothness conditions. Our main…

多层网络一致社区检测的谱方法与矩阵分解方法AI 解读

We consider the problem of estimating a consensus community structure by combining information from multiple layers of a multi-layer network using methods based on the spectral clustering or a low-rank matrix factorization. As a general theme, these “intermediate fusion” methods involve obtaining a low column rank matrix by optimizing an objective function and then using the columns of the matrix for clustering. However, the theoretical properties of these methods remain…

一元全变差去噪与趋势滤波的自适应风险界AI 解读

We study trend filtering, a relatively recent method for univariate nonparametric regression. For a given integer $r\geq1$, the $r$th order trend filtering estimator is defined as the minimizer of the sum of squared errors when we constrain (or penalize) the sum of the absolute $r$th order discrete derivatives of the fitted function at the design points. For $r=1$, the estimator reduces to total variation regularization which has received much attention in the statistics and…

加权随机区块模型中社群估计的最优速率AI 解读

Community identification in a network is an important problem in fields such as social science, neuroscience and genetics. Over the past decade, stochastic block models (SBMs) have emerged as a popular statistical framework for this problem. However, SBMs have an important limitation in that they are suited only for networks with unweighted edges; in various scientific applications, disregarding the edge weights may result in a loss of valuable information. We study a…

基于包络模型的稀疏偏最小二乘AI 解读

Sparse partial least squares (SPLS) is widely used in applied sciences as a method that performs dimension reduction and variable selection simultaneously in linear regression. Several implementations of SPLS have been derived, among which the SPLS proposed in Chun and Keleş (J. R. Stat. Soc. Ser. B. Stat. Methodol. 72 (2010) 3–25) is very popular and highly cited. However, for all of these implementations, the theoretical properties of SPLS are largely unknown. In this…

In sequential causal inference, two types of causal effects are of practical interest, namely, the causal effect of the treatment regime (called the sequential causal effect) and the blip effect of treatment on the potential outcome after the last treatment. The well-known $G$-formula expresses these causal effects in terms of the standard parameters. In this article, we obtain a new $G$-formula that expresses these causal effects in terms of the point observable effects of…

模型辅助变量聚类：极小极大最优恢复与算法AI 解读

The problem of variable clustering is that of estimating groups of similar components of a $p$-dimensional vector $X=(X_{1},\ldots ,X_{p})$ from $n$ independent copies of $X$. There exists a large number of algorithms that return data-dependent groups of variables, but their interpretation is limited to the algorithm that produced them. An alternative is model-based clustering, in which one begins by defining population level clusters relative to a model that embeds notions…

稀疏切片逆回归：最优收敛率与自适应估计AI 解读

Sliced inverse regression (SIR) is an innovative and effective method for sufficient dimension reduction and data visualization. Recently, an impressive range of penalized SIR methods has been proposed to estimate the central subspace in a sparse fashion. Nonetheless, few of them considered the sparse sufficient dimension reduction from a decision-theoretic point of view. To address this issue, we in this paper establish the minimax rates of convergence for estimating the…