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有限样本非参数推断与大样本效率AI 解读

Given a sample $X_1,\dots, X_n$ from a distribution $F$, the problem of constructing nonparametric confidence intervals for the mean $\mu(F)$ is considered. Unlike bootstrap procedures or those based on normal approximations, we insist on any procedure being truly nonparametric in the sense that the probability that the confidence interval contains $\mu(F)$ based on a sample of size $n$ from $F$ be at least $1 - \alpha$ for all $F$ and all $n$. Bahadur and Savage proved it…

响应曲面中光滑断裂线的追踪估计AI 解读

We suggest a sequential, or “tracking,”algorithm for estimating a smooth fault line in a response surface. The method starts with an approximation to a point on the line, and from there the line is tracked as it meanders through the plane. The technique differs from recent approaches in that it does not require a large part of the plane to be searched for evidence of a fault line. This offers potential computational savings, and produces a method that is invariant under…

光滑高斯随机场模型中结构化相关矩阵的估计AI 解读

This article considers the estimation of structured correlation matrices in infinitely differentiable Gaussian random field models.The problem is essentially motivated by the stochastic modeling of smooth deterministic responses in computer experiments.In particular, the log-likelihood function is determined explicitly in closed-form and the sieve maximum likelihood estimators are shown to be strongly consistent under mild conditions.

张量积空间 ANOVA 模型AI 解读

To deal with the curse of dimensionality in high-dimensional nonparametric problems, we consider using tensor product space ANOVA models, which extend the popular additive models and are able to capture interactions of any order. The multivariate function is given an ANOVA decomposition, that is, it is expressed as a constant plus the sum of functions of one variable (main effects), plus the sum of functions of two variables (two-factor interactions)and so on. We assume the…

正态均值的经验贝叶斯检验AI 解读

We exhibit an empirical Bayes test $\delta_n^*$ for the normal mean testing problem using a linear error loss. Under the condition that the critical point of a Bayes test is within some known compact interval, $\delta_n^*$ is shown to be asymptotically optimal and its associated regret Bayes risk converges to zero at a rate $O(n^{-1}(\ln n)^{1.5})$, where $n$ is the number of past experiences available when the current component decision problem is considered. Under the same…

多元数据的经验几何：反卷积方法AI 解读

Let $\{Y_j : j = 1,\ldots,n\}$ be independent observations in $\mathbb{R}^m, m \geq 1$ with common distribution $Q$. Suppose that $Y_j = X_j + \xi_j, j = 1,\ldots,n$, where $\{X_j, \xi_j, j = 1,\ldots,n\}$ are independent, $X_ j, j = 1,\ldots,n$ have common distribution $P$ and $\xi_ j, j = 1, \ldots,n$ have common distribution $\mu$, so that $Q = P * \mu$. The problem is to recover hidden geometric structure of the support of $P$ based on the independent observations $Y_j$.…

若干非参数问题的贝叶斯视角AI 解读

We study the Bayesian approach to nonparametric function estimation problems such as nonparametric regression and signal estimation. We consider the asymptotic properties of Bayes procedures for conjugate (= Gaussian) priors. We show that so long as the prior puts nonzero measure on the very large parameter set of interest then the Bayes estimators are not satisfactory. More specifically, we show that these estimators do not achieve the correct minimax rate over norm bounded…

后验分布的收敛速率AI 解读

We consider the asymptotic behavior of posterior distributions and Bayes estimators for infinite-dimensional statistical models. We give general results on the rate of convergence of the posterior measure. These are applied to several examples, including priors on finite sieves, log-spline models, Dirichlet processes and interval censoring.

曲线估计的尺度空间视角AI 解读

Scale space theory from computer vision leads to an interesting and novel approach to nonparametric curve estimation. The family of smooth curve estimates indexed by the smoothing parameter can be represented as a surface called the scale space surface. The smoothing parameter here plays the same role as that played by the scale of resolution in a visual system. In this paper, we study in detail various features of that surface from a statistical viewpoint. Weak convergence…

层析成像问题中的密度有效估计AI 解读

The aim of tomography is to reconstruct a multidimensional function from observations of its integrals over hyperplanes. We consider the model that corresponds to the case of positron emission tomography. We have $n$ i.i.d.observations from a probability density proportional to $Rf$, where $Rf$ stands for the Radon transform of the density $f$.We assume that $f$ is an $N$-dimensional density such that its Fourier transform is exponentially decreasing. We find an estimator of…

广义线性模型中的置信带AI 解读

Generalized linear models (GLM) include many useful models. This paper studies simultaneous confidence regions for the mean response function in these models. The coverage probabilities of these regions are related to tail probabilities of maxima of Gaussian random fields, asymptotically, and hence, the so-called tube formula is applicable without any modification. However, in the generalized linear models, the errors are often nonadditive and non-Gaussian and may be…

加性逻辑回归：提升法的统计视角（含讨论与作者回应）AI 解读

Boosting is one of the most important recent developments in classification methodology. Boosting works by sequentially applying a classification algorithm to reweighted versions of the training data and then taking a weighted majority vote of the sequence of classifiers thus produced. For many classification algorithms, this simple strategy results in dramatic improvements in performance. We show that this seemingly mysterious phenomenon can be understood in terms of…

如何绘制 Hill 图AI 解读

An abundance of high quality data sets requiring heavy tailed models necessitates reliable methods of estimating the shape parameter governing the degree of tail heaviness. The Hill estimator is a popular method for doing this but its practical use is encumbered by several difficulties. We show that an alternative method of plotting Hill estimator values is more revealing than the standard method unless the underlying data comes from a Pareto distribution.

多元 M 估计量的密度AI 解读

When a unique $M$-estimate exists, its density is obtained as a corollary to a more general theorem which asserts that under mild conditions the intensity function of the point process of solutions of the estimating equations exists and is given by the density of the estimating function standardized by multiplying it by the inverse of its derivative. We apply the results to give a result for Huber’s proposal 2 applied to regression and scale estimates. We also give a…

拟合优度检验的全局功效函数AI 解读

It is shown that the global power function of any nonparametric test is flat on balls of alternatives except for alternatives coming from a finite dimensional subspace. The present benchmark is here the upper one-sided (or two-sided) envelope power function. Every choice of a test fixes a priori a finite dimensional region with high power. It turns out that also the level points are far away from the corresponding Neyman–Pearson test level points except for a finite number…

一致随机序约束下某类分布一致估计量的渐近分布理论AI 解读

We identify the asymptotic behavior of the estimators proposed by Rojo and Samaniego and Mukerjee of a distribution $F$ assumed to be uniformly stochastically smaller than a known baseline distribution $G$.We show that these estimators are $\sqrt{n}$-convergent to a limit distribution with mean squared error smaller than or equal to the mean squared error of the empirical survival function. By examining the mean squared error of the limit distribution, we are able to…

密度估计中的策略混合AI 解读

General results on adaptive density estimation are obtained with respect to any countable collection of estimation strategies under Kullback-Leibler and squared $L_2$ losses. It is shown that without knowing which strategy works best for the underlying density, a single strategy can be constructed by mixing the proposed ones to be adaptive in terms of statistical risks. A consequence is that under some mild conditions, an asymptotically minimax-rate adaptive estimator exists…

曲线点值估计的自适应置信区间AI 解读

We present a procedure associated with nonlinear wavelet methods that provides adaptive confidence intervals around $f (x_0)$, in either a white noise model or a regression setting. A suitable modification in the truncation rule for wavelets allows construction of confidence intervals that achieve optimal coverage accuracy up to a logarithmic factor. The procedure does not require knowledge of the regularity of the unknown function $f$; it is also efficient for functions…

功效差距趋零与渐近相对效率AI 解读

The shortcoming of a test is the difference between the maximal attainable power and the power of the test under consideration. Vanishing shortcoming, when the number of observations tends to infinity, is therefore an optimality property of a test. Other familiar optimality criteria are based on the asymptotic relative efficiency of the test. The relations between these optimality criteria are investigated. It turns out that vanishing shortcoming is seemingly slightly…

预检验程序的检验尺寸与功效AI 解读

A pretest procedure consists of a preliminary test on a nuisance parameter, investigating whether it equals a given value or not, followed by the main testing problem on the parameter of interest. In case of acceptance of the preliminary test, the main test is applied in the restricted family with the given value of the nuisance parameter, while otherwise the test is performed in the complete family, including the nuisance parameter. For an appropriate class of tests,…

半参数偏倚抽样模型中最大似然估计的大样本理论AI 解读

Vardi [Ann.Statist.13 178-203 (1985)] introduced an $s$-sample biased sampling model with known selection weight functions, gave a condition under which the common underlying probability distribution $G$ is uniquely estimable and developed simple procedure for computing the nonparametric maximum likelihood estimator (NPMLE) $\mathbb{G}_n$ of $G$. Gill, Vardi and Wellner thoroughly described the large sample properties of Vardi’s NPMLE, giving results on uniform consistency,…

更新理论中的非参数估计（二）：更新型方程的解AI 解读

Nonparametric estimators of solutions of renewal-type equations are proposed in terms of the empirical renewal function. Using the approach of Grübel and Pitts [Ann .Stat .21 1431 –1451 (1993)], who studied asymptotic properties of the empirical renewal function, a number of properties of our estimators are established, including strong consistency, asymptotic normality and efficiency, and asymptotic validity of bootstrap confidence bounds. The results are illustrated by…

成分数未知的混合模型贝叶斯分析：可逆跳跃方法的替代方案AI 解读

Richardson and Green present a method of performing a Bayesian analysis of data from a finite mixture distribution with an unknown number of components. Their method is a Markov Chain Monte Carlo (MCMC) approach, which makes use of the “reversible jump” methodology described by Green. We describe an alternative MCMC method which views the parameters of the model as a (marked) point process, extending methods suggested by Ripley to create a Markov birth-death process with an…

通过线性函数校准检验回归均值的单调性AI 解读

A new approach to testing for monotonicity of a regression mean, not requiring computation of a curve estimator or a bandwidth, is suggested. It is based on the notion of “running gradients ” over short intervals, although from some viewpoints it may be regarded as an analogue for monotonicity testing of the dip/excess mass approach for testing modality hypotheses about densities. Like the latter methods, the new technique does not suffer difficulties caused by almost-flat…