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高斯分布扰动下 Gibbs 采样器的收敛性质AI 解读

The exact second eigenvalue of the Markov operator of the Gibbs sampler with random sweep strategy for Gaussian densities is calculated. A comparison lemma yields an upper bound on the second eigenvalue for bounded perturbations of Gaussians which is a significant improvement over previous bounds. For two-block Gibbs sampler algorithms with a perturbation of the form $\chi(g_1(x^{(1)}) + g_2(x^{(2)}))$ the derivative of the second eigenvalue of the algorithm is calculated…

马尔可夫链模拟方法的收敛性AI 解读

The Markov chain simulation method has been successfully used in many problems, including some that arise in Bayesian statistics. We give a self-contained proof of the convergence of this method in general state spaces under conditions that are easy to verify.

固定边际二维列联表的均匀生成与 Diaconis–Efron 条件体积检验AI 解读

Two efficient Monte Carlo algorithms are proposed for uniformlygenerating two-way contingency tables with fixed margins. These permit some improvements on recent work of Diaconis, Efron and Gangolli, especially concerning estimates of the total number of such tables.

灵活风险率估计量的渐近性质AI 解读

Suppose one has a stochastic time-dependent covariate $Z(t)$, and is interested in estimating the hazard relationship $\lambda(t|\overline{Z}(t)) = \omega(Z(t))$, where $\overline{Z}(t)$ denotes the history of $Z(t)$ up to and including time t. In this paper, we consider a model of the form $\exp(s_n(Z(t)))$, where $s_n(Z(t))$ is a spline of finite but arbitrary order, and investigate the behavior of the maximum likelihood estimator of the hazard as the number of knots in…

REML 估计：渐近行为及相关问题AI 解读

The restricted maximum likelihood (REML) estimates of dispersion parameters (variance components) in a general (non-normal) mixed model are defined as solutions of the REML equations. In this paper, we show the REML estimates are consistent if the model is asymptotically identifiable and infinitely informative under the (location) invariant class, and are asymptotically normal (A.N.) if in addition the model is asymptotically nondegenerate. The result does not require…

This paper proposes a consistent nonparametric empirical Bayes estimator of the prior density for directional data. The methodology is to use Fourier analysis on $S^2$ to adapt Euclidean techniques to this non-Euclidean environment. General consistency results are obtained. In addition, a discussion of efficient numerical computation of Fourier transforms on $S^2$ is given, and their applications to the methods suggested in this paper are sketched.

广义核分位数密度估计量的一致相合性AI 解读

Various smoothing methods for quantile density estimation are unified into a generalized kernel smoothing. Based on a stochastic upper bound of the derivatives sequence for a sequence of smoothed Brownian bridges, uniform in-probability consistency of generalized kernel quantile density estimators on any closed subinterval of the open unit interval is derived.

依赖数据密度估计中的带宽选择AI 解读

We address the empirical bandwidth choice problem in cases where the range of dependence may be virtually arbitrarily long. Assuming that the observed data derive from an unknown function of a Gaussian process, it is argued that, unlike more traditional contexts of statistical inference, in density estimation there is no clear role for the classical distinction between short- and long-range dependence. Indeed, the "boundaries" that separate different modes of behaviour for…

密度函数的序贯置信带AI 解读

This paper proposes a fully sequential procedure for constructing a fixed-width confidence band for an unknown density on a finite interval and shows the procedure has the desired coverage probability asymptotically as the width of the band approaches zero. The procedure is based on a result of Bickel and Rosenblatt. Its implementation in the sequential setting cannot be obtained using Anscombe's theorem, because the normalized maximal deviations between the kernel estimate…

偏序偏好的表示AI 解读

This essay considers decision-theoretic foundations for robust Bayesian statistics. We modify the approach of Ramsey, de Finetti, Savage and Anscombe and Aumann in giving axioms for a theory of robust preferences. We establish that preferences which satisfy axioms for robust preferences can be represented by a set of expected utilities. In the presence of two axioms relating to state-independent utility, robust preferences are represented by a set of probability/utility…

贝叶斯推断的局部敏感性诊断AI 解读

We investigate diagnostics for quantifying the effect of small changes to the prior distribution over a k-dimensional parameter space. We show that several previously suggested diagnostics, such as the norm of the Fréchet derivative, diverge at rate $n^{k/2}$ if the base prior is an interior point in the class of priors, under the density ratio topology. Diagnostics based on $\phi$-divergences exhibit similar asymptotic behavior. We show that better asymptotic behavior can…

后验分布的收敛AI 解读

Z.A general (asymptotic) theory of estimation was developed by Ibragimov and Has’minskii under certain conditions on the normalized likelihood ratios. In an earlier work, the present authors studied the limiting behaviour of the posterior distributions under the general setup of Ibragimov and Has’minskii. In particular, they obtained a necessary condition for the convergence of a suitably centered (and normalized) posterior to a constant limit in terms of the limiting…

给定样本均值的后验收敛AI 解读

For various applications one wants to know the asymptotic behavior of $w(\theta | \overline{X})$, the posterior density of a parameter $\theta$ given the mean $\overline{X}$ of the data rather than the full data set. Here we show that $w(\theta | \overline{X})$ is asymptotically normal in an $L^1$ sense, and we identify the mean of the limiting normal and its asymptotic variance. The main results are first proved assuming that $X_1, \dots, X_n, \dots$ are independent and…

极值指数的改进 Pickands 估计量AI 解读

Consider a distribution function that belongs to the weak domain of attraction of an extreme value distribution. The extreme value index $\beta$ will be estimated by mixtures of Pickands estimators, where the weights are generated by a probability measure which satisfies a certain integrability condition. We prove a functional limit theorem for a process of Pickands estimators and asymptotic normality of the refined Pickands estimator. For negative $\beta$ the new estimator…

正则变动指数的渐近有效估计AI 解读

We propose a conditional MLE of the index of regular variation when the functional form of a slowly varying function is assumed known in the tail, and we study its asymptotic properties. We prove asymptotic normality of $P_{\theta}^{k_n}$, a probability measure whose density is the product of the joint conditional density of the $k_n$ largest order statistics from $F_{\theta} (x)$ given $Z_{n-k}$, the $$(n-k)$th order statistic, and a density of $Z_{n-k}$ with parameter $\the…

基于超阈值方法检验极值指数AI 解读

Consider an iid sample $Y_1, \dots, Y_n$ of random variables with common distribution function F, whose upper tail belongs to a neighborhood of the upper tail of a generalized Pareto distribution $H_{\beta}, \beta \epsilon \mathbb{R}$. We investigate the testing problem $\beta = \beta_0$ against a sequence $\beta = \beta_n$ of contiguous 0 n alternatives, based on the point processes $N_n$ of the exceedances among $Y_i$ over a sequence of thresholds $t_n$. It turns out that…

各向同性随机场的局部格点抽样设计AI 解读

For predicting $\int_G v(x)Z(x)dx$, where v is a fixed known function and Z is a stationary random field, a good sampling fesign should have a greater density of observations where v is relatively large in absolute value. Designs using this idea when $G = [0, 1]$ have been studied for some time. For G a region in two dimensions, very little is known about the statistical properties of cubature rules based on designs with varying density. This work proposes a class of designs…

利用格点观测预测随机场积分AI 解读

For a stationary random field Z on $\mathbb{R}^d$, this work studies the asymptotic behavior of predictors of $\int v(x)Z(x)dx$ based on observations on a lattice as the distance between neighbors in the lattice tends to 0. Under a mild condition on the spectral density of Z, an asymptotic expression for the mean-squared error of a predictor of $\int v(x)Z(x)dx$ based on observations on an infinite lattice is derived. For predicting integrals over the unit cube, a simple…

平面标志点数据的形状变化AI 解读

This paper deals with the statistical analysis of matched pairs of shapes of configurations of landmarks in the plane. We provide inference procedures on the complex projective plane for a basic measure of shape change in the plane, on observing that shapes of configurations of $(k + 1)$ landmarks in the plane may be represented as points on $\mathbb{C} P^{k-1}$ and that complex rotations are the only maps on $\mathbb{C} S^{k-1}$ which preserve the usual Hermitian inner…

非参数回归的自动带宽选择与置信区间AI 解读

In the present paper we combine the issues of bandwidth choice and construction of confidence intervals in nonparametric regression. Main emphasis is put on fully data-driven methods. We modify the $\sqrt{n}$-consistent bandwidth selector of Härdle, Hall and Marron such that it is appropriate for heteroscedastic data, and we show how one can optimally choose the bandwidth g of the pilot estimator $\hat{m}_g$. Then we consider classical confidence intervals based on kernel…

短程与长程相关误差下非参数回归的带宽选择AI 解读

We analyse methods based on the block bootstrap and leave-out cross-validation, for choosing the bandwidth in nonparametric regression when errors have an almost arbitrarily long range of dependence. A novel analytical device for modelling the dependence structure of errors is introduced. This allows a concise theoretical description of the way in which the range of dependence affects optimal bandwidth choice. It is shown that, provided block length or leave-out number,…

非参数回归中的可加性检验AI 解读

Additive models are one means of assuaging the curse of dimensionality when nonparametric smoothing methods are used to estimate multivariable regression functions. It is important to have methods for testing the fit of such models, especially in high dimensions where visual assessment of fit becomes difficult. New tests of additivity are proposed in this paper that derive from Fourier series estimators with data-driven smoothing parameters. Other tests related to the…

众数估计中 L^p 误差的最小化AI 解读

We show that, for $L^p$ convergence of the mode of a nonparametric density estimator to the mode of an unknown probability density, finiteness of the pth moment of the underlying distribution is both necessary and sufficient. The basic requirement of existence of finite variance has been overlooked by statisticians, who have earlier considered mean square convergence of nonparametric mode estimators; they have focussed on mean squared error of the asymptotic distribution,…

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