2×2列联表中归因风险的统计推断

2×2列联表的统计推断在现代医学,各种临床试验以及现代生物学的研究中通常是感兴趣的问题,其中归因风险的统计推断经常出现在流行病学和健康服务研究中.本文我们针对2×2列联表中归因风险的统计推断给出了一种新的方法,利用2×2列联表中边际和条件概率提出了Score检验,给出了建立在置信区间之上的Score检验的描述.并对Score检验和似然比检验的效果进行评价,依据覆盖概率、左尾错误率及右尾错误率得出结论:Score检验与似然比检验相比更有效.     

右删失情形下平均生存时间的加权Bootstrap推断

在右删失情形下,基于一类合成数据,采用加权Bootstrap方法获得了平均生存时间的加权Bootstrap估计及其加权Bootstrap分布,并就权重是否独立两种情形,证明了此估计的相合性及此分布近似的有效性.基于此,构造了平均生存时间的置信区间.在数值模拟中,取权为Dirichlet(n;1,…,1),并从覆盖概率和区间长度角度,比较了加权Bootstrap和渐近正态逼近产生的置信区间.     

The exact bootstrap method shown on the example of the mean and variance estimation

The bootstrap method is based on resampling of the original randomsample drawn from a population with an unknown distribution. In the article it was shown that because of the progress in computer technology resampling is actually unnecessary if the sample size is not too large. It is possible to automatically generate all possible resamples and calculate all realizations of the required statistic. The obtained distribution can be used in point or interval estimation of population parameters or in testing hypotheses. We should stress that in the exact bootstrap method the entire space of resamples is used and therefore there is no additional bias which results from resampling. The method was used to estimate mean and variance. The comparison of the obtained distributions with the limit distributions confirmed the accuracy of the exact bootstrap method. In order to compare the exact bootstrap method with the basic method (with random sampling) probability that 1,000 resamples would allow for estimating a parameter with a given accuracy was calculated. There is little chance of obtaining the desired accuracy, which is an argument supporting the use of the exact method. Random sampling may be interpreted as discretization of a continuous variable.     

Hybrid bootstrap for mapping quantitative trait loci

The hybrid bootstrap uses resampling ideas to extend the duality approach to interval estimation for a parameter of interest when there are nuisance parameters. The confidence region constructed by the hybrid bootstrap may perform much better than the parametric bootstrap region in situations where the data provide substantial information about the nuisance parameter, but limited information about the parameter of interest. We apply this method to estimate the location of quantitative trait loci (QTL) in interval mapping model. The conditional distribution of quantitative traits, given flanked genetic marker genotypes is often assumed to be the mixture model of two phenotype distributions. The mixing proportions in the model represent the recombination rate between a genetic marker and quantitative trait loci and provides information about the unknown location of the QTL. Since recombination events are unlikely, we will have less information about the location of the QTL than other parameters. This observation makes a hybrid approach to interval estimation for QTL appealing, especially since the necessary distribution theory, which is often a challenge for mixture models, can be handled by bootstrap simulation.     

Likelihood ratio inference for mean residual life of length-biased random variable

In lifetime data analysis, naturally recorded observations are length-biased data if the probability to select an item is proportional to its length. Based on i.i.d. observations of the true distribution, empirical likelihood (EL) procedure is proposed for the inference on mean residual life (MRL) of naturally recorded item. The limit distribution of the EL based log-likelihood ratio is proved to be the chi-square distribution. Under right censorship, since the EL based log-likelihood ratio leads to a scaled chi-square distribution and estimating the scale parameter leads to lower coverage of confidence interval, we propose an algorithm to calculate the likelihood ratio (LR) directly. The corresponding log-likelihood ratio converges to the standard chi-square distribution and the corresponding confidence interval has a better coverage. Simulation studies are used to support the theoretical results.     

Bootstrapping the mode

The problem of constructing bootstrap confidence intervals for the mode of a density is considered. Estimates of the mode are derived from kernel density estimates based on fixed and data-dependent bandwidths. The asymptotic validity of bootstrap techniques to estimate the sampling distribution of the estimates is investigated. In summary, the results are negative in the sense that a straightforward application of a naive bootstrap yields invalid inferences. In particular, the bootstrap fails if resampling is done from the kernel density estimate. On the other hand, if one resamples from a smoother kernel density estimate (which is necessarily different from the one which yields the original estimate of the mode), the bootstrap is consistent. The bootstrap also fails if resampling is done from the empirical distribution, unless the choice of bandwidth is suboptimal. Similar results hold when applying bootstrap techniques to other functionals of a density.     

Inference for the Mean Difference in the Two-Sample Random Censorship Model

Inference for the mean difference in the two-sample random censorship model is an important problem in comparative survival and reliability test studies. This paper develops an adjusted empirical likelihood inference and a martingale-based bootstrap inference for the mean difference. A nonparametric version of Wilks' theorem for the adjusted empirical likelihood is derived, and the corresponding empirical likelihood confidence interval of the mean difference is constructed. Also, it is shown that the martingale-based bootstrap gives a correct first order asymptotic approximation of the corresponding estimator of the mean difference, which ensures that the martingale-based bootstrap confidence interval has asymptotically correct coverage probability. A simulation study is conducted to compare the adjusted empirical likelihood, the martingale-based bootstrap, and Efron's bootstrap in terms of coverage accuracies and average lengths of the confidence intervals. The simulation indicates that the proposed adjusted empirical likelihood and the martingale-based bootstrap confidence procedures are comparable, and both seem to outperform Efron's bootstrap procedure.     

Fast and Accurate Computation of the Exact Confidence Limits for the Common Odds Ratio in Several 2 × 2 Tables

Abstract

Mehta, Patel, and Gray proposed a network algorithm for the exact confidence limits of the common odds ratio in several 2 × 2 tables. Their algorithm was implemented in the StatXact and Egret statistical packages and further discussed by Vollset, Hirji, and Elashoff. The need to evaluate polynomials of potentially very high degrees, however, poses some numerical difficulties. This article presents a method that cuts the degree of polynomials by at least one half. Two other modifications to further stabilize and speed the computation are also proposed.     

Reliability inference of stress–strength model for the truncated proportional hazard rate distribution under progressively Type-II censored samples

This paper considers the reliability inference for the truncated proportional hazard rate stress–strength model based on progressively Type-II censoring scheme. When the stress and strength variables follow the truncated proportional hazard rate distributions, the maximum likelihood estimation and the pivotal quantity estimation of stress–strength reliability are derived. Based on the percentile bootstrap sampling technique, the 95% confidence interval of stress–strength reliability is obtained, as well as the related coverage percentage. Moreover, based on the Fisher Z transformation and the modified generalized pivotal quantity, the 95% modified generalized confidence interval for the stress–strength reliability is obtained. The performance of the proposed method is evaluated by the Monte Carlo simulation. The numerical results show that the pivotal quantity estimators performs better than the maximum likelihood estimators. At last, two real datasets are analyzed by the proposed methodology for illustrative purpose. The results of real example analysis show that our model can be applied to the practical problem, the truncated proportional hazard rate distribution can fit the failure data better than other distributions, and the algorithms in this paper are suitable to handle the small sample data.     

Inference for ordered parameters in multinomial distributions

This paper discusses inference for ordered parameters of multinomial distributions. We first show that the asymptotic distributions of their maximum likelihood estimators (MLEs) are not always normal and the bootstrap distribution estimators of the MLEs can be inconsistent. Then a class of weighted sum estimators (WSEs) of the ordered parameters is proposed. Properties of the WSEs are studied, including their asymptotic normality. Based on those results, large sample inferences for smooth functions of the ordered parameters can be made. Especially, the confidence intervals of the maximum cell probabilities are constructed. Simulation results indicate that this interval estimation performs much better than the bootstrap approaches in the literature. Finally, the above results for ordered parameters of multinomial distributions are extended to more general distribution models. This work was supported by National Natural Science Foundation of China (Grant No. 10371126)     

Accuracy of the bootstrap approximation

Summary The sampling distribution of several commonly occurring statistics are known to be closer to the corresponding bootstrap distribution than the normal distribution, under some conditions on the moments and the smoothness of the population distribution. These conditional approximations are suggestive of the unconditional ones considered in this paper, though one cannot be derived from the other by elementary methods. In this paper, probabilistic bounds are provided for the deviation of the sampling distribution from the bootstrap distribution. The rate of convergence to one, of the probability that the bootstrap approximation outperforms the normal approximation, is obtained. These rates can be applied to obtain theL ^p bounds of Bhattacharya and Qumsiyeh (1989) under weaker conditions. The results apply to studentized versions of functions of multivariate means and thus cover a wide class of common statistics. As a consequence we also obtain approximations to percentiles of studentized means and their appropriate modifications. The results indicate the accuracy of the bootstrap confidence intervals both in terms of the actual coverage probability achieved and also the limits of the confidence interval.Research supported in part by NSA Grant MDA 904-90-H-1001     

带有误差变量混合分布模型的估计

在许多实际同题中，存在一些不可直接观测的变量，对此统计学家们提出了反卷积和混合分布模型来解决这一变量的分布的估计问题。本文对这一问题采用bootstrap模拟方法得出分布函数的估计，并进一步建立该分布函数的非参bootstrap百分位区间，在数值试验中将我们的处理方式与传统的EM算法得到的分布估计和正态逼近区间作比较，数值结果表明用bootstrap模拟方法得到的准确度更好，数值效果更理想。     

基于截尾数据指数Pareto分布应力-强度模型的可靠性

在II型双截尾删失计划下，讨论了当系统被独立的随机施加指数Pareto （EP）压力时的系统可靠性问题.作者给出了系统可靠性参数的不同点估计和区间估计，其中点估计包括一致最小方差无偏估计（UMVUE）和最大似然估计（MLE）；区间估计包括精确置信区间，近似置信区间和bootstrap的区间估计.为了评价不同估计方法效果，作者提供数值模拟结果；最后提供了一个真实数据的分析结果来演示本文提出的方法.     

An algorithm to construct Monte Carlo confidence intervals for an arbitrary function of probability distribution parameters

We derive a new algorithm for calculating an exact confidence interval for a parameter of location or scale family, based on a two-sided hypothesis test on the parameter of interest, using some pivotal quantities. We use this algorithm to calculate approximate confidence intervals for the parameter or a function of the parameter of one-parameter continuous distributions. After appropriate heuristic modifications of the algorithm we use it to obtain approximate confidence intervals for a parameter or a function of parameters for multi-parameter continuous distributions. The advantage of the algorithm is that it is general and gives a fast approximation of an exact confidence interval. Some asymptotic (analytical) results are shown which validate the use of the method under certain regularity conditions. In addition, numerical results of the method compare well with those obtained by other known methods of the literature on the exponential, the normal, the gamma and the Weibull distribution.     

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本文讨论了相互独立且寿命服从单参数 指数分布的两产品在定时截尾寿命试验中的平均寿命比估计, 并给出了该估计量的渐近正态性和置信区间. 如果某一种(或某一类)产品与另一种(或某一类)产品平均寿命比率显著大于1, 那么该种产品就具有耐用优势, 这在投资决策中有重大意义. 通过数值模拟还进一步验证了该比估计量和估计方法的有效性.     

Empirical likelihood based confidence intervals for copulas

Copula as an effective way of modeling dependence has become more or less a standard tool in risk management, and a wide range of applications of copula models appear in the literature of economics, econometrics, insurance, finance, etc. How to estimate and test a copula plays an important role in practice, and both parametric and nonparametric methods have been studied in the literature. In this paper, we focus on interval estimation and propose an empirical likelihood based confidence interval for a copula. A simulation study and a real data analysis are conducted to compare the finite sample behavior of the proposed empirical likelihood method with the bootstrap method based on either the empirical copula estimator or the kernel smoothing copula estimator.     

Bootstrap approximation of nearest neighbor regression function estimates

Let (X, Y) be a random vector in the plane and denote by m(x) = (Y|X = x) the corresponding regression function. We show that the bootstrap approximation for the distribution of a smoothed nearest neighbor estimate of m(x) is valid. Also we compare, by Monte Carlo, confidence intervals which are obtained from both the normal and the bootstrap approximation.     

样本判决系数的样本特性研究

本文用 bootstrap方法估计 R2和 R2的标准误差并构建置信区间 ,用蒙特卡罗方法说明 bootstrap标准误差的精确程度 ,并说明 R2的 95置信水平的置信区间不包含真实值的某些特殊情况在用 R2时不会发生。     

Random weighting method for Cox’s proportional hazards model

Variance of parameter estimate in Cox’s proportional hazards model is based on asymptotic variance. When sample size is small, variance can be estimated by bootstrap method. However, if censoring rate in a survival data set is high, bootstrap method may fail to work properly. This is because bootstrap samples may be even more heavily censored due to repeated sampling of the censored observations. This paper proposes a random weighting method for variance estimation and confidence interval estimation for proportional hazards model. This method, unlike the bootstrap method, does not lead to more severe censoring than the original sample does. Its large sample properties are studied and the consistency and asymptotic normality are proved under mild conditions. Simulation studies show that the random weighting method is not as sensitive to heavy censoring as bootstrap method is and can produce good variance estimates or confidence intervals.     

广义Lorenz 曲线的非参数统计推断

本文讨论了广义Lorenz 曲线的经验似然统计推断. 在简单随机抽样、分层随机抽样和整群随机抽样下, 本文分别定义了广义Lorenz 坐标的pro le 经验似然比统计量, 得出这些经验似然比的极限分布为带系数的自由度为1 的χ² 分布. 对于整个Lorenz 曲线, 基于经验似然方法类似地得出相应的极限过程. 根据所得的经验似然理论, 本文给出了bootstrap 经验似然置信区间构造方法, 并通过数据模拟, 对新给出的广义Lorenz 坐标的bootstrap 经验似然置信区间与渐近正态置信区间以及bootstrap 置信区间等进行了对比研究. 对整个Lorenz 曲线, 基于经验似然方法对其置信域也进行了模拟研究. 最后我们将所推荐的置信区间应用到实例中.