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多元正则变动条件下的极端风险区域估计AI 解读

When considering d possibly dependent random variables, one is often interested in extreme risk regions, with very small probability p. We consider risk regions of the form {z ∈ ℝd : f(z) ≤ β}, where f is the joint density and β a small number. Estimation of such an extreme risk region is difficult since it contains hardly any or no data. Using extreme value theory, we construct a natural estimator of an extreme risk region and prove a refined form of consistency, given a…

迭代滤波AI 解读

Inference for partially observed Markov process models has been a longstanding methodological challenge with many scientific and engineering applications. Iterated filtering algorithms maximize the likelihood function for partially observed Markov process models by solving a recursive sequence of filtering problems. We present new theoretical results pertaining to the convergence of iterated filtering algorithms implemented via sequential Monte Carlo filters. This theory…

基于尖峰与厚尾先验的变量选择：上界化—最小化方法AI 解读

We develop a method to carry out MAP estimation for a class of Bayesian regression models in which coefficients are assigned with Gaussian-based spike and slab priors. The objective function in the corresponding optimization problem has a Lagrangian form in that regression coefficients are regularized by a mixture of squared l2 and l0 norms. A tight approximation to the l0 norm using majorization–minimization techniques is derived, and a coordinate descent algorithm in…

具有非参数连接函数的单指标与多指标函数型回归AI 解读

Fully nonparametric methods for regression from functional data have poor accuracy from a statistical viewpoint, reflecting the fact that their convergence rates are slower than nonparametric rates for the estimation of high-dimensional functions. This difficulty has led to an emphasis on the so-called functional linear model, which is much more flexible than common linear models in finite dimension, but nevertheless imposes structural constraints on the relationship between…

检验跳跃具有有限活动性还是无限活动性AI 解读

We propose statistical tests to discriminate between the finite and infinite activity of jumps in a semimartingale discretely observed at high frequency. The two statistics allow for a symmetric treatment of the problem: we can either take the null hypothesis to be finite activity, or infinite activity. When implemented on high-frequency stock returns, both tests point toward the presence of infinite-activity jumps in the data.

单指标模型的估计函数方法（EFM）AI 解读

Single-index models are natural extensions of linear models and circumvent the so-called curse of dimensionality. They are becoming increasingly popular in many scientific fields including biostatistics, medicine, economics and financial econometrics. Estimating and testing the model index coefficients β is one of the most important objectives in the statistical analysis. However, the commonly used assumption on the index coefficients, ‖β‖ = 1, represents a nonregular…

多元凸回归函数的非参数最小二乘估计AI 解读

This paper deals with the consistency of the nonparametric least squares estimator of a convex regression function when the predictor is multidimensional. We characterize and discuss the computation of such an estimator via the solution of certain quadratic and linear programs. Mild sufficient conditions for the consistency of this estimator and its subdifferentials in fixed and stochastic design regression settings are provided.

Fisher 信息的一致矩界及其时间序列应用AI 解读

In this paper, a uniform (over some parameter space) moment bound for the inverse of Fisher’s information matrix is established. This result is then applied to develop moment bounds for the normalized least squares estimate in (nonlinear) stochastic regression models. The usefulness of these results is illustrated using time series models. In particular, an asymptotic expression for the mean squared prediction error of the least squares predictor in autoregressive moving…

核密度估计的带宽选择：Oracle 不等式与自适应极小极大最优性AI 解读

We address the problem of density estimation with "$\mathbb{L}_{s}$-loss by selection of kernel estimators. We develop a selection procedure and derive corresponding $\mathbb{L}_{s}$-risk oracle inequalities. It is shown that the proposed selection rule leads to the estimator being minimax adaptive over a scale of the anisotropic Nikol’skii classes. The main technical tools used in our derivations are uniform bounds on the $\mathbb{L}_{s}$-norms of empirical processes…

随机设计回归中的变点与自助法AI 解读

In this paper we study the consistency of different bootstrap procedures for constructing confidence intervals (CIs) for the unique jump discontinuity (change-point) in an otherwise smooth regression function in a stochastic design setting. This problem exhibits nonstandard asymptotics, and we argue that the standard bootstrap procedures in regression fail to provide valid confidence intervals for the change-point. We propose a version of smoothed bootstrap, illustrate its…

稀疏情形下若干多重检验程序的渐近贝叶斯最优性AI 解读

Within a Bayesian decision theoretic framework we investigate some asymptotic optimality properties of a large class of multiple testing rules. A parametric setup is considered, in which observations come from a normal scale mixture model and the total loss is assumed to be the sum of losses for individual tests. Our model can be used for testing point null hypotheses, as well as to distinguish large signals from a multitude of very small effects. A rule is defined to be…

TFT自助法：频域重抽样与时域时间序列复制AI 解读

A new time series bootstrap scheme, the time frequency toggle (TFT)-bootstrap, is proposed. Its basic idea is to bootstrap the Fourier coefficients of the observed time series, and then to back-transform them to obtain a bootstrap sample in the time domain. Related previous proposals, such as the “surrogate data” approach, resampled only the phase of the Fourier coefficients and thus had only limited validity. By contrast, we show that the appropriate resampling of phase and…

长程依赖下 U 过程的渐近性质AI 解读

Let (Xi)i≥1 be a stationary mean-zero Gaussian process with covariances $\rho(k)=\mathbb {E}(X_{1}X_{k+1})$ satisfying ρ(0) = 1 and ρ(k) = k−DL(k), where D is in (0, 1), and L is slowly varying at infinity. Consider the U-process {Un(r), r ∈ I} defined as $$U_n(r) =\frac{1}{n(n−1)} \sum_{1≤i≠j≤n}1_{\{G(X_i, X_j)≤r\}},$$ where I is an interval included in ℝ, and G is a symmetric function. In this paper, we provide central and noncentral limit theorems for Un. They are used to…

随机矩阵相干性的极限分布及协方差检验与压缩感知应用AI 解读

Testing covariance structure is of significant interest in many areas of statistical analysis and construction of compressed sensing matrices is an important problem in signal processing. Motivated by these applications, we study in this paper the limiting laws of the coherence of an n × p random matrix in the high-dimensional setting where p can be much larger than n. Both the law of large numbers and the limiting distribution are derived. We then consider testing the…

函数型线性回归与白噪声逆问题的渐近等价性AI 解读

We consider the statistical experiment of functional linear regression (FLR). Furthermore, we introduce a white noise model where one observes an Itô process, which contains the covariance operator of the corresponding FLR model in its construction. We prove asymptotic equivalence of FLR and this white noise model in LeCam’s sense under known design distribution. Moreover, we show equivalence of FLR and an empirical version of the white noise model for finite sample sizes.…

高阶扰乱数字网对光滑被积函数达到最优均方根误差收敛率AI 解读

We study a random sampling technique to approximate integrals ∫[0, 1]s f(x) dx by averaging the function at some sampling points. We focus on cases where the integrand is smooth, which is a problem which occurs in statistics. The convergence rate of the approximation error depends on the smoothness of the function f and the sampling technique. For instance, Monte Carlo (MC) sampling yields a convergence of the root mean square error (RMSE) of order N−1/2 (where N is the…

广义 lasso 的解路径AI 解读

We present a path algorithm for the generalized lasso problem. This problem penalizes the ℓ1 norm of a matrix D times the coefficient vector, and has a wide range of applications, dictated by the choice of D. Our algorithm is based on solving the dual of the generalized lasso, which greatly facilitates computation of the path. For D = I (the usual lasso), we draw a connection between our approach and the well-known LARS algorithm. For an arbitrary D, we derive an unbiased…

高维矩阵降秩估计量的最优选择AI 解读

We introduce a new criterion, the Rank Selection Criterion (RSC), for selecting the optimal reduced rank estimator of the coefficient matrix in multivariate response regression models. The corresponding RSC estimator minimizes the Frobenius norm of the fit plus a regularization term proportional to the number of parameters in the reduced rank model. The rank of the RSC estimator provides a consistent estimator of the rank of the coefficient matrix; in general, the rank of…

局部最优设计中的 de la Garza 现象札记AI 解读

The celebrated de la Garza phenomenon states that for a polynomial regression model of degree p−1 any optimal design can be based on at most p design points. In a remarkable paper, Yang [Ann. Statist. 38 (2010) 2499–2524] showed that this phenomenon exists in many locally optimal design problems for nonlinear models. In the present note, we present a different view point on these findings using results about moment theory and Chebyshev systems. In particular, we show that…

高维数据中基于阈值化的稀疏线性判别分析AI 解读

In many social, economical, biological and medical studies, one objective is to classify a subject into one of several classes based on a set of variables observed from the subject. Because the probability distribution of the variables is usually unknown, the rule of classification is constructed using a training sample. The well-known linear discriminant analysis (LDA) works well for the situation where the number of variables used for classification is much smaller than…

The delta method is a popular and elementary tool for deriving limiting distributions of transformed statistics, while applications of asymptotic distributions do not allow one to obtain desirable accuracy of approximation for tail probabilities. The large and moderate deviation theory can achieve this goal. Motivated by the delta method in weak convergence, a general delta method in large deviations is proposed. The new method can be widely applied to driving the moderate…

个体化治疗规则的性能保证AI 解读

Because many illnesses show heterogeneous response to treatment, there is increasing interest in individualizing treatment to patients [11]. An individualized treatment rule is a decision rule that recommends treatment according to patient characteristics. We consider the use of clinical trial data in the construction of an individualized treatment rule leading to highest mean response. This is a difficult computational problem because the objective function is the…

偏序下的多元分位数AI 解读

This paper focuses on generalizing quantiles from the ordering point of view. We propose the concept of partial quantiles, which are based on a given partial order. We establish that partial quantiles are equivariant under order-preserving transformations of the data, robust to outliers, characterize the probability distribution if the partial order is sufficiently rich, generalize the concept of efficient frontier, and can measure dispersion from the partial order…

平面形状平均测地线的内蕴推断与基于叶片生长的树木区分AI 解读

For planar landmark based shapes, taking into account the non-Euclidean geometry of the shape space, a statistical test for a common mean first geodesic principal component (GPC) is devised which rests on one of two asymptotic scenarios. For both scenarios, strong consistency and central limit theorems are established, along with an algorithm for the computation of a Ziezold mean geodesic. In application, this allows to verify the geodesic hypothesis for leaf growth of…