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降维后的统计推断AI 解读

The methodologies of sufficient dimension reduction have undergone extensive developments in the past three decades. However, there has been a lack of systematic and rigorous development of post dimension reduction inference, which has seriously hindered its applications. The current common practice is to treat the estimated sufficient predictors as the true predictors and use them as the starting point of the downstream statistical inference. However, this naive inference…

条件独立性检验的困难与广义协方差度量AI 解读

It is a common saying that testing for conditional independence, that is, testing whether whether two random vectors $X$ and $Y$ are independent, given $Z$, is a hard statistical problem if $Z$ is a continuous random variable (or vector). In this paper, we prove that conditional independence is indeed a particularly difficult hypothesis to test for. Valid statistical tests are required to have a size that is smaller than a pre-defined significance level, and different tests…

支撑边界恢复中复合泊松先验的非参数贝叶斯分析AI 解读

Given data from a Poisson point process with intensity $(x,y)\mapston\mathbf{1}(f(x)\leq y)$, frequentist properties for the Bayesian reconstruction of the support boundary function $f$ are derived. We mainly study compound Poisson process priors with fixed intensity proving that the posterior contracts with nearly optimal rate for monotone support boundaries and adapts to Hölder smooth boundaries. We then derive a limiting shape result for a compound Poisson process prior…

连接恒定步长随机梯度下降与马尔可夫链AI 解读

We consider the minimization of a strongly convex objective function given access to unbiased estimates of its gradient through stochastic gradient descent (SGD) with constant step size. While the detailed analysis was only performed for quadratic functions, we provide an explicit asymptotic expansion of the moments of the averaged SGD iterates that outlines the dependence on initial conditions, the effect of noise and the step size, as well as the lack of convergence in the…

不依赖特定分布与相关结构的高维均值双样本检验AI 解读

We propose a two-sample test for high-dimensional means that requires neither distributional nor correlational assumptions, besides some weak conditions on the moments and tail properties of the elements in the random vectors. This two-sample test based on a nontrivial extension of the one-sample central limit theorem (Ann. Probab. 45 (2017) 2309–2352) provides a practically useful procedure with rigorous theoretical guarantees on its size and power assessment. In…

GRID：高维非参数回归的变量选择与结构发现方法AI 解读

We consider nonparametric regression in high dimensions where only a relatively small subset of a large number of variables are relevant and may have nonlinear effects on the response. We develop methods for variable selection, structure discovery and estimation of the true low-dimensional regression function, allowing any degree of interactions among the relevant variables that need not be specified a-priori. The proposed method, called the GRID, combines empirical…

高维尖峰模型中的假设检验AI 解读

We consider the five classes of multivariate statistical problems identified by James (Ann. Math. Stat. 35 (1964) 475–501), which together cover much of classical multivariate analysis, plus a simpler limiting case, symmetric matrix denoising. Each of James’ problems involves the eigenvalues of $E^{-1}H$ where $H$ and $E$ are proportional to high-dimensional Wishart matrices. Under the null hypothesis, both Wisharts are central with identity covariance. Under the…

A major challenge for building statistical models in the big data era is that the available data volume far exceeds the computational capability. A common approach for solving this problem is to employ a subsampled dataset that can be handled by available computational resources. We propose a general subsampling scheme for large-scale multiclass logistic regression and examine the variance of the resulting estimator. We show that asymptotically, the proposed method always…

利用稳健调整剖面得分进行双样本汇总数据孟德尔随机化推断AI 解读

Mendelian randomization (MR) is a method of exploiting genetic variation to unbiasedly estimate a causal effect in presence of unmeasured confounding. MR is being widely used in epidemiology and other related areas of population science. In this paper, we study statistical inference in the increasingly popular two-sample summary-data MR design. We show a linear model for the observed associations approximately holds in a wide variety of settings when all the genetic variants…

基于分区的级数估计量的大样本性质AI 解读

We present large sample results for partitioning-based least squares nonparametric regression, a popular method for approximating conditional expectation functions in statistics, econometrics and machine learning. First, we obtain a general characterization of their leading asymptotic bias. Second, we establish integrated mean squared error approximations for the point estimator and propose feasible tuning parameter selection. Third, we develop pointwise inference methods…

L4–L2 范数等价条件下的稳健协方差估计AI 解读

Let $X$ be a centered random vector taking values in $\mathbb{R}^{d}$ and let $\Sigma=\mathbb{E}(X\otimes X)$ be its covariance matrix. We show that if $X$ satisfies an $L_{4}-L_{2}$ norm equivalence (sometimes referred to as the bounded kurtosis assumption), there is a covariance estimator $\hat{\Sigma}$ that exhibits almost the same performance one would expect had $X$ been a Gaussian vector. The procedure also improves the current state-of-the-art regarding high…

噪声图像边界的稳健且速率最优的 Gibbs 后验推断AI 解读

Detection of an image boundary when the pixel intensities are measured with noise is an important problem in image segmentation. From a statistical point of view, a challenge is that likelihood-based methods require modeling the pixel intensities inside and outside the image boundary, even though these distributions are typically not of interest. Since misspecification of the pixel intensity distributions can negatively affect inference on the image boundary, it would be…

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…

含干扰泛函的调查加权估计方程推断AI 解读

This paper provides a rigorous treatment on design-based estimating equation inference using complex survey data in the presence of nuisance functionals. The proposed design-based framework covers parameters from inequality measures and measures on performance evaluation in economic, business and financial studies but the scope of the paper is more broad. Unlike nuisance parameters in other settings where profile analysis is commonly used, the nuisance functionals are…

纯空间模型的自适应推断AI 解读

In a general class of semiparametric pure spatial models (having no explanatory variables) allowing nonlinearity in the parameter and the weight matrix, we propose adaptive tests and estimates which are asymptotically efficient in the presence of unknown, nonparametric distributional form. Feasibility of adaptive estimation is verified and its efficiency improvement over Gaussian pseudo maximum likelihood is shown to be either less than, or more than, for models with…

共同冲击面板模型中异质系数的有效估计AI 解读

This paper investigates the estimation and inference issues of heterogeneous coefficients in panel data models with common shocks. We propose a novel two-step method to estimate the heterogeneous coefficients. We establish the asymptotic theory of our estimators, including consistency, asymptotic representation, and limiting distribution. Our two-step method can effectively address the limitations of the existing methods, such as the common correlated effects method proposed…

按特征排序的投资组合：估计与推断AI 解读

Portfolio sorting is ubiquitous in the empirical finance literature, where it has been widely used to identify pricing anomalies. Despite its popularity, little attention has been paid to the statistical properties of the procedure. We develop a general framework for portfolio sorting by casting it as a nonparametric estimator. We present valid asymptotic inference methods and a valid mean square error expansion of the estimator leading to an optimal choice for the number of…

强依赖下的分位相关图AI 解读

We develop the limit theory of the quantilogram and cross-quantilogram under long memory. We establish the sub-root-n central limit theorems for quantilograms that depend on nuisance parameters. We propose a moving block bootstrap (MBB) procedure for inference and establish its consistency, thereby enabling a consistent confidence interval construction for the quantilograms. The newly developed reduction principles for the quantilograms serve as the main technical devices…

识别协整面板中的潜在分组模式AI 解读

We consider a panel cointegration model with latent group structures that allows for heterogeneous long-run relationships across groups. We extend Su, Shi, and Phillips (2016, Econometrica 84(6), 2215–2264) classifier-Lasso (C-Lasso) method to the nonstationary panels and allow for the presence of endogeneity in both the stationary and nonstationary regressors in the model. In addition, we allow the dimension of the stationary regressors to diverge with the sample size. We…

大型似不相关回归系统：惩罚准极大似然估计AI 解读

In this article, using a shrinkage estimator, we propose a penalized quasi-maximum likelihood estimator (PQMLE) to estimate a large system of equations in seemingly unrelated regression models, where the number of equations is large relative to the sample size. We develop the asymptotic properties of the PQMLE for both the error covariance matrix and model coefficients. In particular, we derive the asymptotic distribution of the coefficient estimator and the convergence rate…

汇率可预测性与动态贝叶斯学习AI 解读

Summary We consider how an investor in the foreign exchange market can exploit predictive information by means of flexible Bayesian inference. Using a variety of vector autoregressive models, the investor is able, each period, to learn about important data features. The developed methodology synthesizes a wide array of established approaches for modeling exchange rate dynamics. In a thorough investigation of monthly exchange rate predictability for 10 countries, we find that…