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多线性形式最大值的尾概率及其应用AI 解读

Let Z be a k­way array consisting of independent standard normal variables. For column vectors h1, …, hk, define a multilinear form of degree k by $(h_1 \otimes \cdots \otimes h_k)' \vec(Z)$. We derive formulas for upper tail probabilities of the maximum of a multilinear form with respect to the hi’s under the condition that the hi’s are unit vectors, and of its standardized statistic obtained by dividing by the norm of Z. We also give formulas for the maximum of a symmetric…

主成分分析中最大特征值的分布AI 解读

Let x(1) denote the square of the largest singular value of an n × p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x(1) is the largest principal component variance of the covariance matrix $X'X$, or the largest eigenvalue of a p­variate Wishart distribution on n degrees of freedom with identity covariance. Consider the limit of large p and n with $n/p = \gamma \ge 1$. When centered by $\mu_p = (\sqrt{n-1} + \sqrt{p})^2$ and scaled…

极值理论中的极小极大风险界AI 解读

Asymptotic minimax risk bounds for estimators of a positive extreme value index under zero-one loss are investigated in the classical i.i.d. setup. To this end, we prove the weak convergence of suitable local experiments with Pareto distributions as center of localization to a white noise model, which was previously studied in the context of nonparametric local density estimation and regression. From this result we derive upper and lower bounds on the asymptotic minimax risk…

约束M估计与回降型M估计的正则性和唯一性AI 解读

Constrained M-estimates of multivariate location and scatter are found by finding the global minimum of an objective function subject to a constraint. They are related to redescending M-estimates of multivariate location and scatter since any stationary point of the objective function corresponds to such an M-estimate. Unfortunately, even for the population form of the estimator, that is, the constrained M-functional, the objective function may have multiple stationary…

非椭圆回归变量下稳健回归的最大偏差曲线AI 解读

Maximum bias curves for some regression estimates were previously derived assuming that (i) the intercept term is known and/or (ii) the regressors have an elliptical distribution. We present a single method to obtain the maximum bias curves for a large class of regression estimates. Our results are derived under very mild conditions and, in particular, do not require the restrictive assumptions (i) and (ii) above. Using these results it is shown that the maximum bias curves…

位置参数的最优稳健 M 估计AI 解读

We find optimal robust estimates for the location parameter of n independent measurements from a common distribution F that belongs to a contamination neighborhood of a normal distribution. We follow an asymptotic minimax approach similar to Huber's but work with full neighborhoods of the central parametric model including nonsymmetric distributions. Our optimal estimates minimize monotone functions of the estimate's asymptotic variance and bias, which include asymptotic…

定性假设的多尺度检验AI 解读

Suppose that one observes a process Y on the unit interval, where dY(t) =n1/2 f(t)dt +dW (t) with an unknown function parameter f, given scale parameter n <=1 ("sample size") and standard Brownian motion W. We propose two classes of tests of qualitative nonparametric hypotheses about f such as monotonicity or concavity. These tests are asymptotically optimal and adaptive in a certain sense. They are constructed via a new class of multiscale statistics and an extension of…

凸模型、MLS与模型误设AI 解读

We analyze the asymptotic behavior of maximum likelihood estimators (MLE) in convex dominated models when the true distribution generating the independent data does not necessarily belong to the model. Inspired by the Hellinger distance and its properties, we introduce a family of divergences (contrast functions) which allow a unified treatment of well- and misspecified convex models. Convergence and rates of convergence of the MLE with respect to our divergences are…

非线性结构型测量误差模型的半参数估计AI 解读

In the nonlinear structural errors-in-variables model, we propose a consistent estimator of the unknown parameter using a modified least squares criterion. We give an upper bound of its rate of convergence which is strongly related to the regularity of the regression function and is generally slower than the parametric rate of convergence n-1/2. Nevertheless, the rate is of order n-1/2 for some particular analytic regression functions. For instance, when the regression…

广义似然比统计量与 Wilks 现象AI 解读

Likelihood ratio theory has had tremendous success in parametric inference, due to the fundamental theory of Wilks. Yet, there is no general applicable approach for nonparametric inferences based on function estimation. Maximum likelihood ratio test statistics in general may not exist in nonparametric function estimation setting. Even if they exist, they are hard to find and can not be optimal as shown in this paper. We introduce the generalized likelihood statistics to…

摘要暂缺

局部平稳过程的似然近似AI 解读

A new approximation to the Gaussian likelihood of a multivariate locally stationary process is introduced. It is based on an approximation of the inverse of the covariance matrix of such processes. The new quasi likelihood is a generalization of the classical Whittle likelihood for stationary processes. Several approximation results are proved for the likelihood function. For parametric models, asymptotic normality and efficiency of the resulting estimator are derived for…

自适应优化与 D 最优实验设计AI 解读

We consider the situation where one has to maximize a function $\eta(\theta, \mathbf{x})$ with respect to $\mathbf{x} \epsilon \mathbb{R}^q$, when $\theta$ is unknown and estimated by least squares through observations $y_k = \mathbf{f}^{\top(\mathbf{x}_k)\theta + \varepsilon_k$, with $\varepsilon_k$ some random error. Classical applications are regulation and extremum control problems. The approach we adopt corresponds to maximizing the sum of the current estimated objective and a penalization for poor estimation: $…

多项式回归的受约束 D 与 D₁ 最优设计AI 解读

In the common polynomial regression model of degree m we consider the problem of determining the $D$- and $D_1$-optimal designs subject to certain constraints for the $D_1$-efficiencies in the models of degree $m - j, m - j + 1,\dots, m + k(m > j \geq 0, k \geq 0 \text{given})$.We present a complete solution of these problems, which on the one hand allow a fast computation of the constrained optimal designs and, on the other hand, give an answer to the question of the…

Motivated by an application in computerized adaptive tests, we consider the following sequential design problem.There are $J$ jobs to be processed according to a predetermined order. A single machine is available to process these $J$ jobs. Each job under processing evolves stochastically as a Markov chain and earns rewards as it is processed, not otherwise. The Markov chain has transition probabilities parameterized by an unknown parameter $\theta$. The objective is to…

指数族中序贯检验与固定样本量检验错误概率的渐近近似AI 解读

Asymptotic approximations for the error probabilities of sequential tests of composite hypotheses in multiparameter exponential families are developed herein for a general class of test statistics, including generalized likelihood ratio statistics and other functions of the sufficient statistics. These results not only generalize previous approximations for Type I error probabilities of sequential generalized likelihood ratio tests, but also pro- vide a unified treatment of…

回归模型选择的渐近极小极大遗憾规则与维数惩罚AI 解读

This paper addresses the topic of model selection in regression.We emphasize the case of two models, testing which model provides a better prediction based on $n$ observations. Within a family of selection rules, based on maximizing a penalized log-likelihood under a normal model, we search for asymptotically minimax rules over a class $\mathscr{G}$ of possible joint distributions of the explanatory and response variables. For the class $\mathscr{G}$ of multivariate normal…

BIC 马尔可夫阶数估计量的一致性AI 解读

The Bayesian Information Criterion (BIC) estimates the order of a Markov chain (with finite alphabet $A$) from observation of a sample path $x_1, x_2,\dots, x_n$, as that value $k = \hat{k}$ that minimizes the sum of the negative logarithm of the $k$th order maximum likelihood and the penalty term $\frac{|A|^k(|A|-1)}{2}\log n$. We show that $\hat{k}$ equals the correct order of the chain, eventually almost surely as $n \rightarrow \infty$, thereby strengthening earlier…

大规模伯努利数据的平滑样条 ANOVA 模型与随机化 GACVAI 解读

We propose the randomized Generalized Approximate Cross Validation (ranGACV) method for choosing multiple smoothing parameters in penalized likelihood estimates for Bernoulli data. The method is intended for application with penalized likelihood smoothing spline ANOVA models. In addition we propose a class of approximate numerical methods for solving the penalized likelihood variational problem which, in conjunction with the ranGACV method allows the application of smoothing…

乘法级联的统计估计AI 解读

The probability distribution of the cascade generators in a random multiplicative cascade represents a hidden parameter which is reflected in the fine scale limiting behavior of the scaling exponents (sample moments) of a single sample cascade realization as a.s.constants.We identify a large class of cascade generators uniquely determined by these scaling exponents. For this class we provide both asymptotic consistency and confidence intervals for two different estimators of…

具有近似频率学有效性的贝叶斯预测AI 解读

We characterize priors which asymptotically match the posterior coverage probability of a Bayesian prediction region with the corresponding frequentist coverage probability. This is done considering both posterior quantiles and highest predictive density regions with reference to a future observation. The resulting priors are shown to be invariant under reparameterization. The role of Jeffreys’ prior in this regard is also investigated. It is further shown that, for any…

非线性回归与神经网络训练的局部贪心逼近AI 解读

A criterion for local estimation and approximation in nonlinear regres- sion and neural network training is introduced and motivated. $N$th-order greedy approximation for the regression (or target) function based on the criterion is shown to converge at rate $O(1/N^{1/2})$ in the nonsampling case.

非参数回归中降低偏差的数据锐化方法AI 解读

We consider methods for kernel regression when the explanatory and/or response variables are adjusted prior to substitution into a conven- tional estimator.This “data-sharpening” procedure is designed to preserve the advantages of relatively simple, low-order techniques, for example, their robustness against design sparsity problems, yet attain the sorts of bias reductions that are commonly associated only with high-order methods.We consider Nadaraya–Watson and local-linear…

最大似然估计的序贯置信域AI 解读

The goal of this paper is to develop a general framework for constructing sequential fixed size confidence regions based on maximum likelihood estimates. Asymptotic properties of the sequential procedure for setting up the confidence regions are analyzed under very broad assumptions on the underlying parametric model. It is shown that the proposed sequential procedure is asymptotically optimal in the sense that it approximates the optimal fixed-sample size procedure. It is…