On split sample and randomized confidence intervals for binomial proportions

We study randomized confidence intervals for binomial proportions, comparing coverage, length and the impact of the randomization. It is seen that the recently proposed split sample intervals can be improved upon in various ways. Criticisms of randomized intervals are discussed.     

On Burr XII Distribution Analysis Under Progressive Type-II Hybrid Censored Data

In the current paper, based on progressive type-II hybrid censored samples, the maximum likelihood and Bayes estimates for the two parameter Burr XII distribution are obtained. We propose the use of expectation-maximization (EM) algorithm to compute the maximum likelihood estimates (MLEs) of model parameters. Further, we derive the asymptotic variance-covariance matrix of the MLEs by applying the missing information principle and it can be utilized to construct asymptotic confidence intervals (CIs) for the parameters. The Bayes estimates of the unknown parameters are obtained under the assumption of gamma priors by using Lindley’s approximation and Markov chain Monte Carlo (MCMC) technique. Also, MCMC samples are used to construct the highest posterior density (HPD) credible intervals. Simulation study is conducted to investigate the accuracy of the estimates and compare the performance of CIs obtained. Finally, one real data set is analyzed for illustrative purposes.     

分数填补下两总体分位数差异的经验似然置信区间

设有两个非参数总体,其样本数据不完全,用分数填补法补足缺失数据,得到两总体的"完全"样本数据,在此基础上构造两总体分位数差异的经验似然置信区间.模拟结果显示,分数填补法可以得到更加精确的置信区间.     

Piecewise Linear Approximations for Cure Rate Models and Associated Inferential Issues

Cure rate models offer a convenient way to model time-to-event data by allowing a proportion of individuals in the population to be completely cured so that they never face the event of interest (say, death). The most studied cure rate models can be defined through a competing cause scenario in which the random variables corresponding to the time-to-event for each competing causes are conditionally independent and identically distributed while the actual number of competing causes is a latent discrete random variable. The main interest is then in the estimation of the cured proportion as well as in developing inference about failure times of the susceptibles. The existing literature consists of parametric and non/semi-parametric approaches, while the expectation maximization (EM) algorithm offers an efficient tool for the estimation of the model parameters due to the presence of right censoring in the data. In this paper, we study the cases wherein the number of competing causes is either a binary or Poisson random variable and a piecewise linear function is used for modeling the hazard function of the time-to-event. Exact likelihood inference is then developed based on the EM algorithm and the inverse of the observed information matrix is used for developing asymptotic confidence intervals. The Monte Carlo simulation study demonstrates the accuracy of the proposed non-parametric approach compared to the results attained from the true correct parametric model. The proposed model and the inferential method is finally illustrated with a data set on cutaneous melanoma.     

Exact confidence coefficients of simultaneous confidence intervals for multinomial proportions

Simultaneous confidence intervals for multinomial proportions are useful in many areas of science. Since 1964, approximate simultaneous 1-α confidence intervals have been proposed for multinomial proportions. Although at each point in the parameter space, these confidence sets have asymptotic 1-α coverage probability, the exact confidence coefficients of these simultaneous confidence intervals for a fixed sample size are unknown before.In this paper, we propose a procedure for calculating exact confidence coefficients for simultaneous confidence intervals of multinomial proportions for any fixed sample size. With this methodology, exact confidence coefficients can be clearly derived, and the point at which the infimum of the coverage probability occurs can be clearly identified.     

二项抽样下基于鞍点逼近方法的流行病相对风险置信区间构造

相对风险是流行病学研究中的重要指标之一,它是度量一种暴露因素是否与某病的致病有联系的统计指标.以该指标的数值大小来表明这一暴露因素对某病的发生具有何种影响及影响的大小,体现了暴露与疾病的关联程度.精确地得到相对风险指标的区间估计,对病因推断具有重要意义.但是由于相对风险指标的估计量是两个概率值的估计量的比值,要得到其精确分布一般而言是很困难的,因此已有研究成果大都采用渐近方法估计相对风险的置信区间,这在小样本情况下表现不佳.在二项抽样条件下,对相对风险的点估计、置信区间估计一直被人们所关注.在二项采样下利用鞍点逼近的方法构造相对风险的置信区间,并通过实例与蒙特卡洛模拟,与传统的置信区间构造方法对比,模拟结果显示其优点,尤其是在小样本量条件下估计效果比较好.     

Confidence Intervals and Accuracy Estimation for Heavy-Tailed Generalized Pareto Distributions

The generalized Pareto distribution (GPD) is a two-parameter family of distributions which can be used to model exceedances over a threshold. We compare the empirical coverage of some standard bootstrap and likelihood-based confidence intervals for the parameters and upper p-quantiles of the GPD. Simulation results indicate that none of the bootstrap methods give satisfactory intervals for small sample sizes. By applying a general method of D. N. Lawley, correction factors for likelihood ratio statistics of parameters and quantiles of the GPD have been calculated. Simulations show that for small sample sizes accuracy of confidence intervals can be improved by incorporating the computed correction factors to the likelihood-based confidence intervals. While the modified likelihood method has better empirical coverage probability, the mean length of produced intervals are not longer than corresponding bootstrap confidence intervals. This article also investigates the performance of some bootstrap methods for estimation of accuracy measures of maximum likelihood estimators of parameters and quantiles of the GPD.     

Missing Observations in Spatial Models: A Spectral EM Algorithm

This article presents a model-based thin-plate smoothing method for optimal signal extraction and interpolation of missing data in spatial datasets. The method is based on a spectral EM algorithm where the two steps can be carried out in the frequency domain. In essence, the approach allows both dimensions to be treated separately from each other effectively rendering a likelihood that is easy to evaluate. As a result the algorithm is computationally inexpensive, in terms of both memory size and computing time, while allowing us to obtain an analytic expression for the asymptotic variance of the signal-to-noise ratio with which to construct confidence intervals of the missing data. Some numerical Monte Carlo simulations and a real data example using remotely sensed global aerosol optical thickness data illustrate the results given. Supplemental materials (Matlab computer code and dataset) are available online.     

虚拟解法分析浸入边界法的精度

浸入边界法是对流固耦合系统进行建模和模拟的有效工具,在生物力学领域的应用尤为广泛．该文的工作主要包含两个部分:程序验证和精度分析．前者证明了程序的正确性,后者给出了浸入边界法的精度．两部分工作均使用虚拟解法作为研究工具．在程序验证部分,使用二阶空间离散格式进行数值计算,通过分析各种变量的离散误差,得到的程序计算精度阶是二阶,与理论精度阶一致,证明了数值计算所使用程序的正确性．精度分析部分工作在此基础上展开．引入压强跳跃,在动量方程中加入相应源项,通过分析带有源项的控制方程中各物理量的离散误差,证明浸入边界法只具有一阶精度．同时可以得出以下结论:粗网格无法敏感地捕捉浸入边界的影响;当Euler网格固定时,增加Lagrange标志点的数目并不会改善计算误差．     

Fixed-width confidence interval for covariate-adjusted response-adaptive designs

In this paper, we obtain fixed-width confidence interval for covariate-adjusted response-adaptive designs. Specifically, we consider logistic regression model and the normal regression model for binary and continuous responses, respectively, both in the situations for presence and absence of treatment–covariate interactions. Simulation study and real-data analysis are carried out.     

Subdivision Strategies for Boxes in Branch-and-Bound Nonlinear Solvers and Verification

Most verified solvers for nonlinear interval systems of equations comprise two strategies: a branch-and-bound-type “location” phase for excluding regions that cannot contain a solution, and a “verification” phase for proving that the remaining regions do indeed contain solutions. In the first phase, subdivision is crucial for the efficiency of the solvers. We discuss several ways for subdivision and present robust strategies that are suited for a variety of nonlinear systems. Particular focus is on the choice of subdivision directions, subdivision points and the handling of unbounded intervals. Furthermore we discuss a method to discard parts of a box within subdivision. Numerical evaluations are given based on the nonlinear interval solver SONIC. In the verification phase, further subdivision can increase the strength of the verification tests. In this paper, we address methods for the rigorous implementation of symmetrical subdivision which is needed, e.g., in existence tests based on Borsuk’s theorem.     

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In this paper, we investigate a competing risks model based on exponentiated Weibull distribution under Type-I progressively hybrid censoring scheme. To estimate the unknown parameters and reliability function, the maximum likelihood estimators and asymptotic confidence intervals are derived. Since Bayesian posterior density functions cannot be given in closed forms, we adopt Markov chain Monte Carlo method to calculate approximate Bayes estimators and highest posterior density credible intervals. To illustrate the estimation methods, a simulation study is carried out with numerical results. It is concluded that the maximum likelihood estimation and Bayesian estimation can be used for statistical inference in competing risks model under Type-I progressively hybrid censoring scheme.     

Saddlepoint approximations for tests of dispersion

The nonparametric class of tests for dispersion is widely used for testing the equality of the scale parameters of two populations. This class includes Mood, Siegel–Tukey, Klotz and Moses tests. This paper uses the double saddlepoint approximation to calculate analytical mid-p-values for these tests that are almost exact as the permutation simulation method. The performance of the saddlepoint method is assessed using an extensive simulation study. The speed and accuracy of the saddlepoint method enable us to invert the dispersion tests to calculate (1 ? α)100% confidence intervals for the dispersion parameter.     

Exact Stratified Linear Rank Tests for Ordered Categorical and Binary Data

Abstract

We present an efficient algorithm for generating exact permutational distributions for linear rank statistics defined on stratified 2 × c contingency tables. The algorithm can compute exact p values and confidence intervals for a rich class of nonparametric problems. These include exact p values for stratified two-population Wilcoxon, Logrank, and Van der Waerden tests, exact p values for stratified tests of trend across several binomial populations, exact p values for stratified permutation tests with arbitrary scores, and exact confidence intervals for odds ratios embedded in stratified 2 × c tables. The algorithm uses network-based recursions to generate stratum-specific distributions and then combines them into an overall permutation distribution by convolution. Where only the tail area of a permutation distribution is desired, additional efficiency gains are achieved by backward induction and branch-and-bound processing of the network. The algorithm is especially efficient for highly imbalanced categorical data, a situation where the asymptotic theory is unreliable. The backward induction component of the algorithm can also be used to evaluate the conditional maximum likelihood, and its higher order derivatives, for the logistic regression model with grouped data. We illustrate the techniques with an analysis of two data sets: The leukemia data on survivors of the Hiroshima atomic bomb and data from an animal toxicology experiment provided by the U.S. Food and Drug Administration.     

Two finite difference schemes for time fractional diffusion-wave equation

Time fractional diffusion-wave equations are generalizations of classical diffusion and wave equations which are used in modeling practical phenomena of diffusion and wave in fluid flow, oil strata and others. In this paper we construct two finite difference schemes to solve a class of initial-boundary value time fractional diffusion-wave equations based on its equivalent partial integro-differential equations. Under the weak smoothness conditions, we prove that our two schemes are convergent with first-order accuracy in temporal direction and second-order accuracy in spatial direction. Numerical experiments are carried out to demonstrate the theoretical analysis.     

Analysis of variance with general errors and grouped and non-grouped data: Some iterative algorithms

In this paper we consider some iterative estimation algorithms, which are valid to analyse the variance of data, which may be either non-grouped or grouped with different classification intervals. This situation appears, for instance, when data is collected from different sources and the grouping intervals differ from one source to another. The analysis of variance is carried out by means of general linear models, whose error terms may be general. An initial procedure in the line of the EM, although it does not necessarily agree with it, opens the paper and gives rise to a simplified version where we avoid the double iteration, which implicitly appears in the EM and, also, in the initial procedure mentioned above. The asymptotic stochastic properties of the resulting estimates have been investigated in depth and used to test ANOVA hypothesis.     

Two-sample inference based on one-sample Wilcoxon signed rank statistics

A two-sample test is studied which rejects the null hypothesis of equal population medians when two Wilcoxon distribution free confidence intervals are disjoint. A confidence interval for the difference in population medians is constructed by subtracting the endpoints of two one-sample confidence intervals. Two different ways to select the one-sample intervals are presented. A solution that specifies equal confidence coefficients for the one-sample intervals is recommended. All solutions are shown to have the same asymptotic (Pitman) efficiency as the Mann-Whitney two-sample test.     

Individual confidence intervals for solutions to expected value formulations of stochastic variational inequalities

Stochastic variational inequalities (SVIs) provide a means for modeling various optimization and equilibrium problems where data are subject to uncertainty. Often the SVI cannot be solved directly and requires a numerical approximation. This paper considers the use of a sample average approximation and proposes three methods for computing confidence intervals for components of the true solution. The first two methods use an “indirect approach” that requires initially computing asymptotically exact confidence intervals for the solution to the normal map formulation of the SVI. The third method directly constructs confidence intervals for the true SVI solution; intervals produced with this method meet a minimum specified level of confidence in the same situations for which the first two methods are applicable. We justify the three methods theoretically with weak convergence results, discuss how to implement these methods, and test their performance using three numerical examples.     

Nonparametric inference on the difference of location parameters of correlated variables from fragmentary samples

Summary In this paper, two types of robust estimators and approximate confidence intervals for the difference of location parameters of correlated random variables are proposed and investigated when some observations are missing. It is shown that the suggested estimators are consistent and asymptotically normally distributed. In addition, the proposed approximate confidence intervals are also shown to enjoy some nice asymptotic properties.     

Interval estimation of mean response time for a G/M/1 queueing system: empirical Laplace function approach

Mean response time is an important performance measure for a queueing system. In this paper, we propose a consistent and asymptotically normal (CAN) estimator of the mean response time for a G/M/1 queueing system, which is based on the fixed point of empirical Laplace function. The confidence interval for the mean response time can be constructed by applying the proposed CAN estimator and its estimated variance. And we carried out a simulation study to perform the accuracy of the constructed confidence interval by calculating the coverage percentage and the relative average length of confidence interval. Detailed discussions of all simulation results for three various models of G/M/1‐type system are presented and some valuable conclusions are provided. Copyright © 2006 John Wiley & Sons, Ltd.