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多元回归的投影估计及其在函数型 ANOVA 模型中的应用AI 解读

A general theory on rates of convergence of the least-squares projection estimate in multiple regression is developed. The theory is applied to the functional ANOVA model, where the multivariate regression function is modeled as a specified sum of a constant term, main effects (functions of one variable) and selected interaction terms (functions of two or more variables). The least-squares projection is onto an approximating space constructed from arbitrary linear spaces of…

局部平稳过程的自适应协方差估计AI 解读

It is shown that the covariance operator of a locally stationary process has approximate eigenvectors that are local cosine functions. We model locally stationary processes with pseudo-differential operators that are time-varying convolutions. An adaptive covariance estimation is calculated by searching first for a "best" local cosine basis which approximates the covariance by a band or a diagonal matrix. The estimation is obtained from regularized versions of the diagonal…

一般变换模型中具有根号n收敛率的非参数估计AI 解读

We propose simple estimators for the transformation function $\Lambda$ and the distribution function F of the error for the model $$\Lambda (Y) = \alpha + \mathbf{X} \mathbf{\beta} + \varepsilon.$$ It is proved that these estimators are consistent and can achieve the unusual $n^{-1/2}$ rate of convergence on any finite interval under some regularity conditions. We show that our estimators are more attractive than another class of estimators proposed by Horowitz. Interesting…

双重删失数据下的回归 M 估计AI 解读

The M-estimators are proposed for the linear regression model with random design when the response observations are doubly censored. The proposed estimators are constructed as some functional of a Campbell-type estimator $\hat{F}_n$ for a bivariate distribution function based on data which are doubly censored in one coordinate. We establish strong uniform consistency and asymptotic normality of $\hat{F}_n$ and derive the asymptotic normality of the proposed regression…

非参数函数估计中的超效率AI 解读

Fixed parameter asymptotic statements are often used in the context of nonparametric curve estimation problems (e.g., nonparametric density or regression estimation). In this context several forms of superefficiency can occur. In contrast to what can happen in regular parametric problems, here every parameter point (e.g., unknown density or regression function) can be a point of superefficiency. We begin with an example which shows how fixed parameter asymptotic statements…

分段凸函数估计：初步估计量AI 解读

Given noisy data, function estimation is considered when the unknown function is known a priori to be either convex or concave on each of a small number of regions where the function. When the number of regions is unknown, the model selection problem is to determine the number of convexity change points. For kernel estimates in Gaussian noise, the number of false change points is evaluated as a function of the smoothing parameter. To insure that the number of false convexity…

论筛法与惩罚法AI 解读

We develop a general theory which provides a unified treatment for the asymptotic normality and efficiency of the maximum likelihood estimates (MLE's) in parametric, semiparametric and nonparametric models. We find that the asymptotic behavior of substitution estimates for estimating smooth functionals are essentially governed by two indices: the degree of smoothness of the functional and the local size of the underlying parameter space. We show that when the local size of…

非参数置信区间研究AI 解读

An inequality is given for the expected length of a confidence interval given that a particular distribution generated the data and assuming that the confidence interval has a given coverage probability over a family of distributions. As a corollary, attempts to adapt to the regularity of the true density within derivative smoothness classes cannot improve the rate of convergence of the length of the confidence interval over minimax fixed-length intervals and still maintain…

The problem of optimal adaptive estimation of a function at a given point from noisy data is considered. Two procedures are proved to be asymptotically optimal for different settings. First we study the problem of bandwidth selection for nonparametric pointwise kernel estimation with a given kernel. We propose a bandwidth selection procedure and prove its optimality in the asymptotic sense. Moreover, this optimality is stated not only among kernel estimators with a variable…

互信息、度量熵与累积相对熵风险AI 解读

Assume ${P_{\theta}: \theta \epsilon \Theta}$ is a set of probability distributions with a common dominating measure on a complete separable metric space Y. A state $\theta^* \epsilon \Theta$ is chosen by Nature. A statistician obtains n independent observations $Y_1, \dots, Y_n$ from Y distributed according to $P_{\theta^*}$. For each time t between 1 and n, based on the observations $Y_1, \dots, Y_{t-1}$, the statistician produces an estimated distribution $\hat{P}_t$ for $P_{\theta^*}$…

稳态模拟中的多重比较程序AI 解读

Suppose that there are $k \geq 2$ different systems (i.e., stochastic processes), where each system has an unknown steady-state mean performance and unknown asymptotic variance. We allow for the asymptotic variances to be unequal and for the distributions of the k systems to be different. We consider the problem of running independent, single-stage simulations to make multiple comparisons of the steady-state means of the different systems. We derive asymptotically valid (as…

多元分布的变换经验过程与修正 Kolmogorov–Smirnov 检验AI 解读

A general way of constructing classes of goodness-of-fit tests for multivariate samples is presented. These tests are based on a random signed measure that plays the same role as the empirical process in the construction of the classical Kolmogorov-Smirnov tests. The resulting tests are consistent against any fixed alternative, and, for each sequence of contiguous alternatives, a test in each class can be chosen so as to optimize the discrimination of those alternatives.

光滑或分片光滑锥备择下卡方混合分布的权重AI 解读

We study the problem of testing a simple null hypothesis about the multivariate normal mean vector against smooth or piecewise smooth cone alternatives. We show that the mixture weights of the $\bar{\chi}^2$ distribution of the likelihood ratio test can be characterized as mixed volumes of the cone and its dual. The weights can be calculated by integration involving the second fundamental form on the boundary of the cone. We illustrate our technique by examples involving a…

生物等效性问题的无偏检验AI 解读

It is shown that the standard two one-sided tests procedure for bioequivalence is a biased test. Better tests exist. In this paper, an unbiased $\alpha$-level test and other tests which are uniformly more powerful than the two one-sided tests procedure are constructed. Its power can be noticeably larger than that of the $\alpha$-level two one-sided tests procedure.

傅里叶回归中基于整数格点的 D 最优设计AI 解读

A theory of optimum orthogonal fractions is developed for Fourier regression models using integer lattice designs. These provide alternatives to simple grids (product designs) in the case when specified main effects and interaction terms are required to be analyzed. The challenge is to obtain sample sizes which are polynomial in the dimension rather than exponential. This is achieved for certain models with special algorithms based on both algebraic generation and more…

支撑集估计的插入式方法AI 解读

We suggest a new approach, based on the use of density estimators, for the problem of estimating the (compact) support of a multivariate density. This subject (motivated in terms of pattern analysis by Grenander) has interesting connections with detection and clustering. A natural class of density-based estimators is defined. Universal consistency results and convergence rates are established for these estimators, with respect to the usual measure-based metric $d_{\mu}$ betwe…

粒度平滑AI 解读

A new method for smoothing a multivariate data set is introduced that is based on a simple geometric operation. This method is applied to the problem of estimating level sets of a density and minimum volume sets with given probability content. Building on existing techniques, the resulting estimator combines excellent theoretical and computational properties: It converges with the minimax rates (up to log factors) in most cases where these rates are known and, at the same…

删失同方差线性回归的估计：经验过程方法AI 解读

Pertaining to the estimating equations proposed by Tsiatis, based on the linear rank test, we show the existence of local confounding between the baseline hazard function and the covariates. Due to the local confounding, an estimating equation in Tsiatis' family with a larger time point of truncation could contain less information about the regression parameter than the estimating equation with a smaller time point of truncation. This phenomenon further indicates significant…

非渐近通用平滑参数、核复杂度与 Yatracos 类AI 解读

We introduce a method to select a smoothing factor for kernel density estimation such that, for all densities in all dimensions, the $L_1$ error of the corresponding kernel estimate is not larger than three times the error of the estimate with the optimal smoothing factor plus a constant times $\sqrt{\log n/n}$, where n is the sample size, and the constant depends only on the complexity of the kernel used in the estimate. The result is nonasymptotic, that is, the bound is…

估计收敛率、Kolmogorov熵与回归中的降维原则AI 解读

$L_1$-optimal minimum distance estimators are provided for a projection pursuit regression type function with smooth functional components that are either additive or multiplicative, in the presence of or without interactions. The obtained rates of convergence of the estimate to the true parameter depend on Kolmogorov's entropy of the assumed model and confirm Stone's heuristic dimensionality reduction principle. Rates of convergence are also obtained for the error in…

符号不变分布下位置参数的同时估计AI 解读

Estimation of $p, p \geq 3$, location parameters of a distribution of a p-dimensional random vector $\mathsf{X}$ is considered under quadratic loss. Explicit estimators which are better than the best invariant one are given for a sign-invariantly distributed random vector $\mathsf{X}$. The results depend only on the second and the third moments of $|| \mathsf{X} - \theta ||$. The generalizations to concave loss functions and to n observations are also considered.…

多元正态均值的精确置信集AI 解读

A class of confidence sets with constant coverage probability for the mean of a p-variate normal distribution is proposed through a pseudo-empirical-Bayes construction. When the dimension is greater than 2, by combining analytical results with some exact numerical calculations the proposed sets are proved to have a uniformly smaller volume than the usual confidence region. Sufficient conditions for the connectedness of the proposed confidence sets are also derived. In…

双指标置换统计量中心极限定理的误差界AI 解读

An error bound in the normal approximation to the distribution of the double-indexed permutation statistics is derived. The derivation is based on Stein's method and on an extension of a combinatorial method of Bolthausen. The result can be applied to obtain the convergence rate of order $n^{-1/2}$ for some rank-related statistics, such as Kendall's tau, Spearman's rho and the Mann-Whitney-Wilcoxon statistic. Its applications to graph-related nonparametric statistics of…

样本均值分布的两种渐近展开之间的关系及应用AI 解读

Although the cumulative distribution function may be differentiated to obtain the corresponding density function, whether or not a similar relationship exists between their asymptotic expansions remains a question. We provide a rigorous argument to prove that Lugannani and Rice's asymptotic expansion for the cumulative distribution function of the mean of a sample of i.i.d. observations may be differentiated to obtain Daniels's asymptotic expansion for the corresponding…