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随机设计非参数回归的渐近等价理论AI 解读

This paper establishes the global asymptotic equivalence between the nonparametric regression with random design and the white noise under sharp smoothness conditions on an unknown regression or drift function. The asymptotic equivalence is established by constructing explicit equivalence mappings between the nonparametric regression and the white-noise experiments, which provide synthetic observations and synthetic asymptotic solutions from any one of the two experiments…

Poisson 强度估计与正漂移扩散估计的渐近等价性AI 解读

We consider a diffusion model of small variance type with positive drift density varying in a nonparametric set. We investigate Gaussian and Poisson approximations to this model in the sense of asymptotic equivalence of experiments. It is shown that observation of the diffusion process until its first hitting time of level one is a natural model for the purpose of inference on the drift density. The diffusion model can be discretized by the collection of level crossing times…

超越混合条件的弱依赖与非参数回归渐近理论AI 解读

We consider a new concept of weak dependence, introduced by Doukhan and Louhichi [Stochastic Process. Appl. 84 (1999) 313–342], which is more general than the classical frameworks of mixing or associated sequences. The new notion is broad enough to include many interesting examples such as very general Bernoulli shifts, Markovian models or time series bootstrap processes with discrete innovations. Under such a weak dependence assumption, we investigate nonparametric…

一类稳健且具有完全渐近效率的回归估计量AI 解读

This paper introduces a new class of robust estimators for the linear regression model. They are weighted least squares estimators, with weights adaptively computed using the empirical distribution of the residuals of an initial robust estimator. It is shown that under certain general conditions the asymptotic breakdown points of the proposed estimators are not less than that of the initial estimator, and the finite sample breakdown point can be at most $1/n$ less. For the…

稳健回归估计的自助法AI 解读

We introduce a new computer-intensive method to estimate the distribution of robust regression estimates. The basic idea behind our method is to bootstrap a reweighted representation of the estimates. To obtain a bootstrap method that is asymptotically correct, we include the auxiliary scale estimate in our reweighted representation of the estimates. Our method is computationally simple because for each bootstrap sample we only have to solve a linear system of equations. The…

正性、单调性与凸性的非参数检验AI 解读

We consider the problem of estimating the distance from an unknown signal, observed in a white-noise model, to convex cones of positive/monotone/convex functions. We show that, when the unknown function belongs to a Hölder class, the risk of estimating the $L_r$-distance, $1 \leq r < \infty$, from the signal to a cone is essentially the same (up to a logarithmic factor) as that of estimating the signal itself. The same risk bounds hold for the test of positivity,…

含类别预测变量的回归充分降维AI 解读

In this article, we describe how the theory of sufficient dimension reduction, and a well-known inference method for it (sliced inverse regression), can be extended to regression analyses involving both quantitative and categorical predictor variables. As statistics faces an increasing need for effective analysis strategies for high-dimensional data, the results we present significantly widen the applicative scope of sufficient dimension reduction and open the way for a new…

回归条件均值的降维AI 解读

In many situations regression analysis is mostly concerned with inferring about the conditional mean of the response given the predictors, and less concerned with the other aspects of the conditional distribution. In this paper we develop dimension reduction methods that incorporate this consideration. We introduce the notion of the Central Mean Subspace (CMS), a natural inferential object for dimension reduction when the mean function is of interest. We study properties of…

依赖性对密度估计随机精度度量的影响AI 解读

In kernel density estimation, those data values that make a nondegenerate contribution to the estimator (computed at a given point) tend to be spaced well apart. This property has the effect of suppressing many of the conventional consequences of long-range dependence, for example, slower rates of convergence, which might otherwise be revealed by a traditional loss-or risk-based assessment of performance. From that viewpoint, dependence has to be very long-range indeed…

保序回归的风险界AI 解读

Nonasymptotic risk bounds are provided for maximum likelihood-type isotonic estimators of an unknown nondecreasing regression function, with general average loss at design points. These bounds are optimal up to scale constants, and they imply uniform $n^{-1/3}$-consistency of the $\ell_p$ risk for unknown regression functions of uniformly bounded variation, under mild assumptions on the joint probability distribution of the data, with possibly dependent observations.

各向异性回归中的随机速率：附讨论与作者回应AI 解读

In the context of minimax theory, we propose a new kind of risk, normalized by a random variable, measurable with respect to the data. We present a notion of optimality and a method to construct optimal procedures accordingly. We apply this general setup to the problem of selecting significant variables in Gaussian white noise. In particular, we show that our method essentially improves the accuracy of estimation, in the sense of giving explicit improved confidence sets in $L_2$…

响应延迟下临床试验自适应设计的渐近性质AI 解读

For adaptive clinical trials using a generalized Friedman’s urn design, we derive the limiting distribution of the urn composition under staggered entry and delayed response. The stochastic delay mechanism is assumed to depend on both the treatment assigned and the patient’s response. A very general setup is employed with $K$ treatments and $L$ responses. When $L = K =2$, one example of a generalized Friedman’s urn design is the randomized play-the-winner rule. An…

可谐过程的谱分析AI 解读

Spectral estimation of nonstationary but harmonizable processes is considered. Given a single realization of the process, periodogram-like and consistent estimators are proposed for spectral mass estimation when the spectral support of the process consists of lines. Such a process can arise in signals of a moving source from array data or multipath signals with Doppler stretch from a single receiver. Such processes also include periodically correlated (or cyclostationary)…

多重假设检验与第一类错误的期望次数AI 解读

The performance of multiple test procedures with respect to error control is an old issue. Assuming that all hypotheses are true we investigate the behavior of the expected number of type I errors (ENE) as a characteristic of certain multiple tests controlling the familywise error rate (FWER) or the false discovery rate (FDR) at a prespecified level. We derive explicit formulas for the distribution of the number of false rejections as well as for the ENE for single-step,…

单纯形约束模型中遗传距离及其他量的估计偏差AI 解读

The genetic distance between two loci on a chromosome is defined as the mean number of crossovers between the loci. The parameters of the crossover distribution are constrained by the parameters of the distribution of chiasmata. Ott (1996) derived the maximum likelihood estimator (MLE) of the parameters of the crossover distribution and the MLE of the mean. We demonstrate that the MLE of the mean is pointwise less than or equal to the empirical mean number of crossovers. It…

分类数据的边际模型AI 解读

Statistical models defined by imposing restrictions on marginal distributions of contingency tables have received considerable attention recently. This paper introduces a general definition of marginal log-linear parameters and describes conditions for a marginal log-linear parameter to be a smooth parameterization of the distribution and to be variation independent. Statistical models defined by imposing affine restrictions on the marginal log-linear parameters are…

含协变量多臂老虎机问题的非参数估计与随机分配AI 解读

We study a multi-armed bandit problem in a setting where covariates are available. We take a nonparametric approach to estimate the functional relationship between the response (reward) and the covariates. The estimated relationships and appropriate randomization are used to select a good arm to play for a greater expected reward. Randomization helps balance the tendency to trust the currently most promising arm with further exploration of other arms. It is shown that, with…

Cox 比例风险模型与脆弱性模型的变量选择AI 解读

A class of variable selection procedures for parametric models via nonconcave penalized likelihood was proposed in Fan and Li (2001a). It has been shown there that the resulting procedures perform as well as if the subset of significant variables were known in advance. Such a property is called an oracle property. The proposed procedures were illustrated in the context of linear regression, robust linear regression and generalized linear models. In this paper, the nonconcave…

弱基假设及其对提升回归与分类的含义AI 解读

When studying the training error and the prediction error for boosting, it is often assumed that the hypotheses returned by the base learner are weakly accurate, or are able to beat a random guesser by a certain amount of difference. It has been an open question how much this difference can be, whether it will eventually disappear in the boosting process or be bounded by a positive amount. This question is crucial for the behavior of both the training error and the…

逐步多重检验程序中错误发现率的若干结果AI 解读

The concept of false discovery rate (FDR) has been receiving increasing attention by researchers in multiple hypotheses testing. This paper produces some theoretical results on the FDR in the context of stepwise multiple testing procedures with dependent test statistics. It was recently shown by Benjamini and Yekutieli that the Benjamini–Hochberg step-up procedure controls the FDR when the test statistics are positively dependent in a certain sense. This paper strengthens…

二项比例置信区间及其渐近展开AI 解读

We address the classic problem of interval estimation of a binomial proportion. The Wald interval $\hat{p}\pm z_{\alpha/2} n^{-1/2} (\hat{p} (1 - \hat{p}))^{1/2}$ is currently in near universal use. We first show that the coverage properties of the Wald interval are persistently poor and defy virtually all conventional wisdom. We then proceed to a theoretical comparison of the standard interval and four additional alternative intervals by asymptotic expansions of their…

AMP 链图马尔可夫模型的分离与完备性AI 解读

Pearl ’s well-known $d$-separation criterion for an acyclic directed graph (ADG) is a pathwise separation criterion that can be used to efficiently identify all valid conditional independence relations in the Markov model determined by the graph. This paper introduces $p$-separation, a pathwise separation criterion that efficiently identifies all valid conditional independences under the Andersson–Madigan–Perlman (AMP) alternative Markov property for chain graphs ( =…

无限参数线性回归中的自适应预测与估计AI 解读

The problem of adaptive prediction and estimation in the stochastic linear regression model with infinitely many parameters is considered.We suggest a prediction method that is sharp asymptotically minimax adaptive over ellipsoids in $\ell_2$. The method consists in an application of blockwise Stein’s rule with “weakly” geometrically increasing blocks to the penalized least squares fits of the first $N$ coefficients. To prove the results we develop oracle inequalities for a…

线性泛函的精确自适应估计AI 解读

We consider estimation of a linear functional $T(f)$ where $f$ is an unknown function observed in Gaussian white noise.We find asymptotically sharp adaptive estimators on various scales of smoothness classes in multidimensional situations. The results allow evaluating explicitly the effect of dimension and treating general scales of classes. Furthermore, we establish a connection between sharp adaptation and optimal recovery. Namely, we propose a scheme that reduces the…