General Asymptotic Confidence Bands Based on Kernel-type Function Estimators

We establish uniform and non-uniform asymptotic simultaneous confidence bands for functionals of the distribution based on kernel-type estimators, which include the Nadaraya-Watson kernel estimators of regression functions and the Akaike-Parzen-Rosenblatt kernel density estimators. Our theorems, based upon functional limit laws derived by modern empirical process theory, allow data-driven local bandwidths for these statistics. This revised version was published online in June 2006 with corrections to the Cover Date.     

Empirical Likelihood Confidence Bands in Density Estimation

Abstract

Empirical likelihood methods are developed for constructing confidence bands in problems of nonparametric density estimation. These techniques have an advantage over more conventional methods in that the shape of the bands is determined solely by the data. We show how to construct an empirical likelihood functional, rather than a function, and contour it to produce the confidence bands. Analogs of Wilks's theorem are established in this infinite-parameter setting and may be used to select the appropriate contour. An alternative calibration, based on the bootstrap, is also suggested. Large-sample theory is developed to show that the bands have asymptotically correct coverage, and a numerical example is presented to demonstrate the technique. Comparisons are made with the use of bootstrap replications to choose both the shape and size of the bands.     

Simultaneous Posterior Probability Statements From Monte Carlo Output

This article considers the problem of making simultaneous probability statements in multivariate inferential problems based on samples from a posterior distribution. The calculation of simultaneous credible bands is reviewed and—as an alternative—contour probabilities are proposed. These are defined as 1 minus the content of the highest posterior density region which just covers a certain point of interest. We discuss a Monte Carlo method to estimate contour probabilities and distinguish whether or not the functional form of the posterior density is available. In the latter case, an approach based on Rao-Blackwellization is proposed. We highlight that this new estimate has an important invariance property. We illustrate the performance of the different methods in three applications.     

Confidence Bands for Isotonic Median Curves Using Sign Tests

Suppose that one observes independent random variables (X₁, Y₁), (X₂, Y₂), …, (X_n, Y_n) in R² with unknown distributions, except that Median(Y_i | X_i = M(x) for some unknown isotonic function M. We describe an explicit algorithm for the computation of confidence bands for the median function M whose running time is of order O(n²). The bands rely on multiscale sign tests and are shown to have desirable asymptotic properties.     

全概率公式的推广

首先给出了普通事件在普通条件和Fuzzy条件下的Fuzzy条件概率及Fuzzy事件在普通条件和Fuzzy条件下的Fuzzy条件概率公式,并通过对普通事件的全概率公式进行推广,得到普通事件和Fuzzy事件分别在普通划分和Fuzzy划分下的全概率公式     

Sequential Confidence Bands for Quantile Densities Under Truncated and Censored Data

In this paper an asymptotic distribution is obtained for the maximal deviation between the kernel quantile density estimator and the quantile density when the data are subject to random left truncation and right censorship. Based on this result we propose a fully sequential procedure for construct ing a fixed-width confidence band for the quantile density on a finite interval and show that the procedure has the desired coverage probability asymptotically as the width of the band approaches zero.     

Uniform Comparison of Tails of (Non-Symmetric) Probability Measures and Their Symmetrized Counterparts with Applications

Let (, ) be a separable Banach space and let be a class of probability measures on , and let denote the symmetrization of . We provide two sufficient conditions (one in terms of certain quantiles and the other in terms of certain moments of relative to μ and , ) for the “uniform comparison” of the μ and measure of the complements of the closed balls of centered at zero, for every . As a corollary to these “tail comparison inequalities,” we show that three classical results (the Lévy-type Inequalities, the Kwapień-Contraction Inequality, and a part of the It?–Nisio Theorem) that are valid for the symmetric (but not for the general non-symmetric) independent -valued random vectors do indeed hold for the independent random vectors whose laws belong to any which satisfies one of the two noted conditions and which is closed under convolution. We further point out that these three results (respectively, the tail comparison inequalities) are valid for the centered log-concave, as well as, for the strictly α-stable (or the more general strictly (r, α) -semistable) α ≠ 1 random vectors (respectively, probability measures). We also present several examples which we believe form a valuable part of the paper.      

全概率公式的推广及应用

给出两种全概率公式的推广形式,从而弱化了全概率公式中事件列是互不相容的条件,最后给出相关的应用.     

随机矩阵乘积的尾概率

在一定条件下,本文给出了一列正随机矩阵乘积的尾概率估计,它以指数的速度消失;然后,在一维的情形,基于更新过程的残差等待时间的拉普拉斯变换,建立了极限常数的两种不同形式表达式之间的联系.     

全概率公式的推广及其应用

全概率公式及其思想在概率统计与随机过程中具有重要作用.给出了公式在条件概率下的推广形式与具体应用.同时,得到了独立性条件下的特殊形式.     

基于置信分布的多个对数正态总体的公共均值的置信区间的构建（英文）

在实际应用中需要拟合正的偏态数据时,对数正态分布是通常的选择.当通过多重比较确定了多个对数正态分布总体的均值相同时,如何能够利用更多的信息,同时使用这些对数正态分布总体的信息来构建公共均值的置信区间成为了众多学者颇为关注的问题.本篇文章提出了一种新的基于置信分布的方法来构建多个对数正态总体公共均值的置信区间,该方法通过对相关量的样本方差进行加权来提高效率.进而对文中提出的基于置信分布的置信区间的构建方法进行了蒙特卡洛模拟研究,模拟结果表明,我们提出的构建方法可以得到很好的覆盖概率和较短的区间宽度.文章的最后用三个实际数据来验证了文中所提出方法的表现.     

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The log-normal distribution is a common choice for modeling positively skewed data arising from many practical applications.This article introduces a new method of constructing confidence interval for a common mean shared by several log-normal populations through confidence distributions, which combines all information from independent sources. We develop a non-trivial weighting approach by taking account of the sample variances of related quantities to enhance efficiency. Combined confidence distributions are used to construct confidence intervals for the common mean and a simplified version of one existing method is also proposed. We conduct simulation studies to evaluate the performance of the proposed methods in comparison with existing methods. Our simulation results show that the weighting approach yields shorter interval length than the non-weighting approach. The newly proposed confidence intervals perform very well in terms of empirical coverage probability and average interval length. Finally, applications of the proposed methodology is illustrated through three real data examples.     

Strong Convergence for Weighted Sums of Negatively Associated Arrays

Let {Xni} be an array of rowwise negatively associated random variables and Tnk=k∑i=1 i^a Xni for a ≥ -1, Snk =∑｜i｜≤k Ф（i/nη）1/nη Xni for η∈（0,1],where Ф is some function. The author studies necessary and sufficient conditions of ∞∑n=1 AnP（max 1≤k≤n｜Tnk｜〉εBn）〈∞ and ∞∑n=1 CnP（max 0≤k≤mn｜Snk｜〉εDn）〈∞ for all ε 〉 0, where An, Bn, Cn and Dn are some positive constants, mn ∈ N with mn /nη →∞. The results of Lanzinger and Stadtmfiller in 2003 are extended from the i.i.d, case to the case of the negatively associated, not necessarily identically distributed random variables. Also, the result of Pruss in 2003 on independent variables reduces to a special case of the present paper; furthermore, the necessity part of his result is complemented.     

Simulation-Based Simultaneous Confidence Bands in Multiple Linear Regression With Predictor Variables Constrained in Intervals

This article presents a method for the construction of a simultaneous confidence band for the normal-error multiple linear regression model. The confidence bands considered have their width proportional to the standard error of the estimated regression function, and the predictor variables are allowed to be constrained in intervals. Past articles in this area gave exact bands only for the simple regression model. When there is more than one predictor variable, only conservative bands are proposed in the statistics literature. This article advances this methodology by providing simulation-based confidence bands for regression models with any number of predictor variables. Additionally, a criterion is proposed to assess the sensitivity of a simultaneous confidence band. This criterion is defined to be the probability that a false linear regression model is excluded from the band at least at one point and hence this false linear regression model is correctly declared as a false model by the band. Finally, the article considers and compares several computational algorithms for obtaining the confidence band.     

Confidence sets centered at C p -estimators

Suppose X _n is an observation, or average of observations, on a discretized signal _n that is measured at n time points. The random vector X _n has a N( _n , ² _n I) distribution, the mean and variance being unknown. Under squared error loss, the unbiased estimator X _n of _n can be improved by variable-selection. Consider the candidate estimator _n (A) whose i-th component equals the i-th component of X _n whenever i/(n+1) lies in A and vanishes otherwise. Allow the set A to range over a large collection of possibilities. A C _p -estimator is a candidate estimator that minimizes estimated quadratic loss over A. This paper constructs confidence sets that are centered at a C _p -estimator, have correct asymptotic coverage probabiligy for _n , and are geometrically smaller than or equal to the competing confidence balls centered at X _n . The asymptotics are locally uniform in the parameters ( _n , ² _n ). The results illustrate an approach to inference after variable-selection.     

Confidence intervals for quantile estimation using Jackknife techniques

We consider the inference on quantiles, Q _y (β), with jackknife techniques, in finite populations of a variable, Y, using the quantile information on an auxiliary variable, X. Jackknife techniques are applied to estimate quantiles and the behaviour of these estimators is analyzed. Their properties are studied for simple random sampling. We also examine the confidence intervals obtained with jackknife variances.     

模糊事件贝叶斯公式

讨论了模糊事件的概率及其基本性质 ,并通过对经典贝叶斯公式的推广 ,提出了模糊事件的贝叶斯公式 .