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单阶段抽样设计的函数型中心极限定理AI 解读

For a joint model-based and design-based inference, we establish functional central limit theorems for the Horvitz–Thompson empirical process and the Hájek empirical process centered by their finite population mean as well as by their super-population mean in a survey sampling framework. The results apply to single-stage unequal probability sampling designs and essentially only require conditions on higher order correlations. We apply our main results to a Hadamard…

随机梯度估计量的渐近与有限样本性质AI 解读

Stochastic gradient descent procedures have gained popularity for parameter estimation from large data sets. However, their statistical properties are not well understood, in theory. And in practice, avoiding numerical instability requires careful tuning of key parameters. Here, we introduce implicit stochastic gradient descent procedures, which involve parameter updates that are implicitly defined. Intuitively, implicit updates shrink standard stochastic gradient descent…

离散观测标量扩散的非参数贝叶斯后验收缩率AI 解读

We consider nonparametric Bayesian inference in a reflected diffusion model $dX_{t}=b(X_{t})\,dt+\sigma(X_{t})\,dW_{t}$, with discretely sampled observations $X_{0},X_{\Delta},\ldots,X_{n\Delta}$. We analyse the nonlinear inverse problem corresponding to the “low frequency sampling” regime where $\Delta>0$ is fixed and $n\to\infty$. A general theorem is proved that gives conditions for prior distributions $\Pi$ on the diffusion coefficient $\sigma$ and the drift function $b$…

Hellinger 邻域上的稳健模型判别设计AI 解读

To aid in the discrimination between two, possibly nonlinear, regression models, we study the construction of experimental designs. Considering that each of these two models might be only approximately specified, robust “maximin” designs are proposed. The rough idea is as follows. We impose neighbourhood structures on each regression response, to describe the uncertainty in the specifications of the true underlying models. We determine the least favourable—in terms of…

白 Wishart 矩阵极端特征值的稀有事件分析AI 解读

In this paper, we consider the extreme behavior of the extremal eigenvalues of white Wishart matrices, which plays an important role in multivariate analysis. In particular, we focus on the case when the dimension of the feature $p$ is much larger than or comparable to the number of observations $n$, a common situation in modern data analysis. We provide asymptotic approximations and bounds for the tail probabilities of the extremal eigenvalues. Moreover, we construct…

相关观测下最优设计的一种新方法AI 解读

This paper presents a new and efficient method for the construction of optimal designs for regression models with dependent error processes. In contrast to most of the work in this field, which starts with a model for a finite number of observations and considers the asymptotic properties of estimators and designs as the sample size converges to infinity, our approach is based on a continuous time model. We use results from stochastic analysis to identify the best linear…

波动率的非参数变点分析AI 解读

In this work, we develop change-point methods for statistics of high-frequency data. The main interest is in the volatility of an Itô semimartingale, the latter being discretely observed over a fixed time horizon. We construct a minimax-optimal test to discriminate continuous paths from paths with volatility jumps, and it is shown that the test can be embedded into a more general theory to infer the smoothness of volatilities. In a high-frequency setting, we prove weak…

有无稀疏性时反卷积密度线性泛函估计的极小极大理论AI 解读

The present paper considers the problem of estimating a linear functional $\Phi=\int_{-\infty}^{\infty}\varphi(x)f(x)\,dx$ of an unknown deconvolution density $f$ on the basis of $n$ i.i.d. observations, $Y_{1},\ldots,Y_{n}$ of $Y=\theta+\xi$, where $\xi$ has a known pdf $g$, and $f$ is the pdf of $\theta$. The objective of the present paper is to develop the a general minimax theory of estimating $\Phi$, and to relate this problem to estimation of functionals $\Phi_{n}=n^{-1}\sum_{i=1}^{n}\varphi(\theta_{i})$…

干扰模型中普遍最优圆周设计的判别AI 解读

Many applications of block designs exhibit neighbor and edge effects. A popular remedy is to use the circular design coupled with the interference model. The search for optimal or efficient designs has been intensively studied in recent years. The circular neighbor balanced designs at distances 1 and 2 (CNBD2), including orthogonal array of type I ($\mathrm{OA}_{I}$) of strength $2$, are the two major designs proposed in literature for the purpose of estimating the direct…

随机曲面非参数协方差算子的可分离性检验AI 解读

The assumption of separability of the covariance operator for a random image or hypersurface can be of substantial use in applications, especially in situations where the accurate estimation of the full covariance structure is unfeasible, either for computational reasons, or due to a small sample size. However, inferential tools to verify this assumption are somewhat lacking in high-dimensional or functional data analysis settings, where this assumption is most relevant. We…

大型噪声矩阵中子矩阵定位的计算与统计边界AI 解读

We study in this paper computational and statistical boundaries for submatrix localization. Given one observation of (one or multiple nonoverlapping) signal submatrix (of magnitude $\lambda$ and size $k_{m}\times k_{n}$) embedded in a large noise matrix (of size $m\times n$), the goal is to optimal identify the support of the signal submatrix computationally and statistically. Two transition thresholds for the signal-to-noise ratio $\lambda/\sigma$ are established in terms…

贝叶斯变点检测的最优性AI 解读

By introducing suitable loss random variables of detection, we obtain optimal tests in terms of the stopping time or alarm time for Bayesian change-point detection not only for a general prior distribution of change-points but also for observations being a Markov process. Moreover, the optimal (minimal) average detection delay is proved to be equal to $1$ for any (possibly large) average run length to false alarm if the number of possible change-points is finite.

操作时间与样本内密度预测AI 解读

In this paper, we consider a new structural model for in-sample density forecasting. In-sample density forecasting is to estimate a structured density on a region where data are observed and then reuse the estimated structured density on some region where data are not observed. Our structural assumption is that the density is a product of one-dimensional functions with one function sitting on the scale of a transformed space of observations. The transformation involves…

In this paper, we propose a test for deciding whether the jump activity index of a discretely observed Itô semimartingale of pure-jump type (i.e., one without a diffusion) varies over a fixed interval of time. The asymptotic setting is based on observations within a fixed time interval with mesh of the observation grid shrinking to zero. The test is derived for semimartingales whose “spot” jump compensator around zero is like that of a stable process, but importantly the…

半鞅检测与拟合优度检验AI 解读

In quantitative finance, we often fit a parametric semimartingale model to asset prices. To ensure our model is correct, we must then perform goodness-of-fit tests. In this paper, we give a new goodness-of-fit test for volatility-like processes, which is easily applied to a variety of semimartingale models. In each case, we reduce the problem to the detection of a semimartingale observed under noise. In this setting, we then describe a wavelet-thresholding test, which…

惩罚模型选择中的弱信号识别与推断AI 解读

Weak signal identification and inference are very important in the area of penalized model selection, yet they are underdeveloped and not well studied. Existing inference procedures for penalized estimators are mainly focused on strong signals. In this paper, we propose an identification procedure for weak signals in finite samples, and provide a transition phase in-between noise and strong signal strengths. We also introduce a new two-step inferential method to construct…

高维重复测量数据的协方差结构检验AI 解读

In regression analysis with repeated measurements, such as longitudinal data and panel data, structured covariance matrices characterized by a small number of parameters have been widely used and play an important role in parameter estimation and statistical inference. To assess the adequacy of a specified covariance structure, one often adopts the classical likelihood-ratio test when the dimension of the repeated measurements ($p$) is smaller than the sample size ($n$).…

可信集的覆盖率与局部半径速率AI 解读

In the mildly ill-posed inverse signal-in-white-noise model, we construct confidence sets as credible balls with respect to the empirical Bayes posterior resulting from a certain two-level hierarchical prior. The quality of the posterior is characterized by the contraction rate which we allow to be local, that is, depending on the parameter. The issue of optimality of the constructed confidence sets is addressed via a trade-off between its “size” (the local radial rate) and…

函数型数据的非线性充分降维AI 解读

We propose a general theory and the estimation procedures for nonlinear sufficient dimension reduction where both the predictor and the response may be random functions. The relation between the response and predictor can be arbitrary and the sets of observed time points can vary from subject to subject. The functional and nonlinear nature of the problem leads to construction of two functional spaces: the first representing the functional data, assumed to be a Hilbert space,…

针对单调旋转对称备择的高维球面均匀性检验AI 解读

We consider the problem of testing uniformity on high-dimensional unit spheres. We are primarily interested in nonnull issues. We show that rotationally symmetric alternatives lead to two Local Asymptotic Normality (LAN) structures. The first one is for fixed modal location $\mathbf{{\theta}}$ and allows to derive locally asymptotically most powerful tests under specified $\mathbf{{\theta}}$. The second one, that addresses the Fisher–von Mises–Langevin (FvML) case, relates…

核方法的随机压缩：快速且最优的非参数回归AI 解读

Kernel ridge regression (KRR) is a standard method for performing nonparametric regression over reproducing kernel Hilbert spaces. Given $n$ samples, the time and space complexity of computing the KRR estimate scale as $\mathcal{O}(n^{3})$ and $\mathcal{O}(n^{2})$, respectively, and so is prohibitive in many cases. We propose approximations of KRR based on $m$-dimensional randomized sketches of the kernel matrix, and study how small the projection dimension $m$ can be chosen…

马尔可夫网络直接稀疏变化学习的支持一致性AI 解读

We study the problem of learning sparse structure changes between two Markov networks $P$ and $Q$. Rather than fitting two Markov networks separately to two sets of data and figuring out their differences, a recent work proposed to learn changes directly via estimating the ratio between two Markov network models. In this paper, we give sufficient conditions for successful change detection with respect to the sample size $n_{p},n_{q}$, the dimension of data $m$ and the number…

高维尖峰协方差经验特征结构的渐近理论AI 解读

We derive the asymptotic distributions of the spiked eigenvalues and eigenvectors under a generalized and unified asymptotic regime, which takes into account the magnitude of spiked eigenvalues, sample size, and dimensionality. This regime allows high dimensionality and diverging eigenvalues and provides new insights into the roles that the leading eigenvalues, sample size, and dimensionality play in principal component analysis. Our results are a natural extension of those…

网络向量自回归AI 解读

We consider here a large-scale social network with a continuous response observed for each node at equally spaced time points. The responses from different nodes constitute an ultra-high dimensional vector, whose time series dynamic is to be investigated. In addition, the network structure is also taken into consideration, for which we propose a network vector autoregressive (NAR) model. The NAR model assumes each node’s response at a given time point as a linear combination…