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一类 Matérn 型高斯随机场的固定域渐近理论AI 解读

Stein [Statist. Sci. 4 (1989) 432–433] proposed the Matérn-type Gaussian random fields as a very flexible class of models for computer experiments. This article considers a subclass of these models that are exactly once mean square differentiable. In particular, the likelihood function is determined in closed form, and under mild conditions the sieve maximum likelihood estimators for the parameters of the covariance function are shown to be weakly consistent with respect to…

线性泛函的自适应估计AI 解读

Adaptive estimation of linear functionals over a collection of parameter spaces is considered. A between-class modulus of continuity, a geometric quantity, is shown to be instrumental in characterizing the degree of adaptability over two parameter spaces in the same way that the usual modulus of continuity captures the minimax difficulty of estimation over a single parameter space. A general construction of optimally adaptive estimators based on an ordered modulus of…

含离群值非参数回归中的交叉验证AI 解读

A popular data-driven method for choosing the bandwidth in standard kernel regression is cross-validation. Even when there are outliers in the data, robust kernel regression can be used to estimate the unknown regression curve [Robust and Nonlinear Time Series Analysis. Lecture Notes in Statist. (1984) 26 163–184]. However, under these circumstances standard cross-validation is no longer a satisfactory bandwidth selector because it is unduly influenced by extreme prediction…

现时状态数据下部分线性变换模型的惩罚对数似然估计AI 解读

We consider partly linear transformation models applied to current status data. The unknown quantities are the transformation function, a linear regression parameter and a nonparametric regression effect. It is shown that the penalized MLE for the regression parameter is asymptotically normal and efficient and converges at the parametric rate, although the penalized MLE for the transformation function and nonparametric regression effect are only n1/3 consistent. Inference…

Grenander 估计量的 Lk 误差渐近正态性AI 解读

We investigate the limit behavior of the \(L_k\)-distance between a decreasing density \(f\) and its nonparametric maximum likelihood estimator \(\hat{f}_n\) for \(k \geq 1\). Due to the inconsistency of \(\hat{f}_n\) at zero, the case \(k = 2.5\) turns out to be a kind of transition point. We extend asymptotic normality of the \(L_1\)-distance to the \(L_k\)-distance for \(1 \leq k < 2.5\), and obtain the analogous limiting result for a modification of the \(L_k\)-distance…

回归误差密度的估计AI 解读

Estimation of the density of regression errors is a fundamental issue in regression analysis and it is typically explored via a parametric approach. This article uses a nonparametric approach with the mean integrated squared error (MISE) criterion. It solves a long-standing problem, formulated two decades ago by Mark Pinsker, about estimation of a nonparametric error density in a nonparametric regression setting with the accuracy of an oracle that knows the underlying…

非必为高斯噪声下的小波阈值法：功能表现AI 解读

For signals belonging to balls in smoothness classes and noise with enough moments, the asymptotic behavior of the minimax quadratic risk among soft-threshold estimates is investigated. In turn, these results, combined with a median filtering method, lead to asymptotics for denoising heavy tails via wavelet thresholding. Some further comparisons of wavelet thresholding and of kernel estimators are also briefly discussed.

重复测量与生存时间联合分析中极大似然估计的渐近结果AI 解读

Maximum likelihood estimation has been extensively used in the joint analysis of repeated measurements and survival time. However, there is a lack of theoretical justification of the asymptotic properties for the maximum likelihood estimators. This paper intends to fill this gap. Specifically, we prove the consistency of the maximum likelihood estimators and derive their asymptotic distributions. The maximum likelihood estimators are shown to be semiparametrically efficient.

右删失现患队列中长度偏倚生存函数无条件非参数最大似然估计的渐近行为AI 解读

Right censored survival data collected on a cohort of prevalent cases with constant incidence are length-biased, and may be used to estimate the length-biased (i.e., prevalent-case) survival function. When the incidence rate is constant, so-called stationarity of the incidence, it is more efficient to use this structure for unconditional statistical inference than to carry out an analysis by conditioning on the observed truncation times. It is well known that, due to the…

离散分布混合密度的非参数估计AI 解读

By a mixture density is meant a density of the form πμ(⋅)=∫πθ(⋅)×μ(dθ), where (πθ)θ∈Θ is a family of probability densities and μ is a probability measure on Θ. We consider the problem of identifying the unknown part of this model, the mixing distribution μ, from a finite sample of independent observations from πμ. Assuming that the mixing distribution has a density function, we wish to estimate this density within appropriate function classes. A general approach is proposed…

多元正态混合分布的地形结构AI 解读

Multivariate normal mixtures provide a flexible method of fitting high-dimensional data. It is shown that their topography, in the sense of their key features as a density, can be analyzed rigorously in lower dimensions by use of a ridgeline manifold that contains all critical points, as well as the ridges of the density. A plot of the elevations on the ridgeline shows the key features of the mixed density. In addition, by use of the ridgeline, we uncover a function that…

随机变量函数之和的估计：实例与信息界AI 解读

This paper concerns the estimation of sums of functions of observable and unobservable variables. Lower bounds for the asymptotic variance and a convolution theorem are derived in general finite- and infinite-dimensional models. An explicit relationship is established between efficient influence functions for the estimation of sums of variables and the estimation of their means. Certain “plug-in” estimators are proved to be asymptotically efficient in finite-dimensional…

递归蒙特卡洛滤波：算法与理论分析AI 解读

Recursive Monte Carlo filters, also called particle filters, are a powerful tool to perform computations in general state space models. We discuss and compare the accept–reject version with the more common sampling importance resampling version of the algorithm. In particular, we show how auxiliary variable methods and stratification can be used in the accept–reject version, and we compare different resampling techniques. In a second part, we show laws of large numbers and a…

分类数据中的可忽略性AI 解读

We study the problem of ignorability in likelihood-based inference from incomplete categorical data. Two versions of the coarsened at random assumption (car) are distinguished, their compatibility with the parameter distinctness assumption is investigated and several conditions for ignorability that do not require an extra parameter distinctness assumption are established. It is shown that car assumptions have quite different implications depending on whether the underlying…

相依序列样本分位数的 Bahadur 表示AI 解读

We establish the Bahadur representation of sample quantiles for linear and some widely used nonlinear processes. Local fluctuations of empirical processes are discussed. Applications to the trimmed and Winsorized means are given. Our results extend previous ones by establishing sharper bounds under milder conditions and thus provide new insight into the theory of empirical processes for dependent random variables.

分数单整的精确局部 Whittle 估计AI 解读

An exact form of the local Whittle likelihood is studied with the intent of developing a general-purpose estimation procedure for the memory parameter (d) that does not rely on tapering or differencing prefilters. The resulting exact local Whittle estimator is shown to be consistent and to have the same $N(0,\frac{1}{4})$ limit distribution for all values of d if the optimization covers an interval of width less than $\frac{9}{2}$ and the initial value of the process is…

谱密度具有未知极点的平稳过程半参数估计AI 解读

We consider the estimation of the location of the pole and memory parameter, λ0 and α, respectively, of covariance stationary linear processes whose spectral density function f(λ) satisfies f(λ)∼C|λ−λ0|−α in a neighborhood of λ0. We define a consistent estimator of λ0 and derive its limit distribution Zλ0. As in related optimization problems, when the true parameter value can lie on the boundary of the parameter space, we show that Zλ0 is distributed as a normal random…

平稳与非平稳分数阶时间序列推断的效率改进AI 解读

We consider a time series model involving a fractional stochastic component, whose integration order can lie in the stationary/invertible or nonstationary regions and be unknown, and an additive deterministic component consisting of a generalized polynomial. The model can thus incorporate competing descriptions of trending behavior. The stationary input to the stochastic component has parametric autocorrelation, but innovation with distribution of unknown form. The model is…

一类新的广义贝叶斯极小极大岭回归估计量AI 解读

Let y=Aβ+ɛ, where y is an N×1 vector of observations, β is a p×1 vector of unknown regression coefficients, A is an N×p design matrix and ɛ is a spherically symmetric error term with unknown scale parameter σ. We consider estimation of β under general quadratic loss functions, and, in particular, extend the work of Strawderman [J. Amer. Statist. Assoc. 73 (1978) 623–627] and Casella [Ann. Statist. 8 (1980) 1036–1056, J. Amer. Statist. Assoc. 80 (1985) 753–758] by finding…

小波阈值的经验贝叶斯选择AI 解读

This paper explores a class of empirical Bayes methods for level-dependent threshold selection in wavelet shrinkage. The prior considered for each wavelet coefficient is a mixture of an atom of probability at zero and a heavy-tailed density. The mixing weight, or sparsity parameter, for each level of the transform is chosen by marginal maximum likelihood. If estimation is carried out using the posterior median, this is a random thresholding procedure; the estimation can also…

方向与射影形状AI 解读

This paper deals with projective shape analysis, which is a study of finite configurations of points modulo projective transformations. The topic has various applications in machine vision. We introduce a convenient projective shape space, as well as an appropriate coordinate system for this shape space. For generic configurations of k points in m dimensions, the resulting projective shape space is identified as a product of k−m−2 copies of axial spaces ℝPm. This…

高维球面与形状空间上的统计分析AI 解读

We consider the statistical analysis of data on high-dimensional spheres and shape spaces. The work is of particular relevance to applications where high-dimensional data are available—a commonly encountered situation in many disciplines. First the uniform measure on the infinite-dimensional sphere is reviewed, together with connections with Wiener measure. We then discuss densities of Gaussian measures with respect to Wiener measure. Some nonuniform distributions on…

等值线回归：充分降维的一般方法AI 解读

We propose a novel approach to sufficient dimension reduction in regression, based on estimating contour directions of small variation in the response. These directions span the orthogonal complement of the minimal space relevant for the regression and can be extracted according to two measures of variation in the response, leading to simple and general contour regression (SCR and GCR) methodology. In comparison with existing sufficient dimension reduction techniques, this…

Boosting is one of the most significant advances in machine learning for classification and regression. In its original and computationally flexible version, boosting seeks to minimize empirically a loss function in a greedy fashion. The resulting estimator takes an additive function form and is built iteratively by applying a base estimator (or learner) to updated samples depending on the previous iterations. An unusual regularization technique, early stopping, is employed…