Comparison of ‘Tail Method’ Exact Confidence Limits for the Difference of Binomial Probabilities

Suppose that Y ₁ and Y ₂ are independent and have Binomial(n ₁,p ₁) and Binomial (n ₂,p ₂) distributions respectively. Also suppose that θ=p ₁−p ₂ is the parameter of interest. We consider the problem of finding an exact confidence limit (either upper or lower) for θ. The solution to this problem is very important for statistical practice in the health and life sciences. The ‘tail method’ provides a solution to this problem. This method finds the exact confidence limit by exact inversion of a hypothesis test based on a specified test statistic. Buehler (J. Am. Stat. Assoc. 52, 482–493, 1957) described, for the first time, a finite-sample optimality property of this confidence limit. Consequently, this confidence limit is sometimes called a Buehler confidence limit. An early tail method confidence limit for θ was described by Santner and Snell (J. Am. Stat. Assoc. 75, 386–394, 1980) who used the maximum likelihood estimator of θ as the test statistic. This confidence limit is known to be very inefficient (see e.g. Cytel Software, StatXact, version 6, vol. 2, 2004). The efficiency of the confidence limit resulting from the tail method depends greatly on the test statistic on which it is based. We use the results of Kabaila (Stat. Probab. Lett. 52, 145–154, 2001) and Kabaila and Lloyd (Aust. New Zealand J. Stat. 46, 463–469, 2004, J. Stat. Plan. Inference 136, 3145–3155, 2006) to provide a detailed explanation for the dependence of this efficiency on the test statistic. We consider test statistics that are estimators, Z-statistics and approximate upper confidence limits. This explanation is used to find the situations in which the tail method exact confidence limits based on test statistics that are estimators or Z-statistics are least efficient.     

正态变差系数的经典限

本文推导了正态变差系数的经典精确限.为了满足工程实践的需要,利用Odeh和Owen的计算方法及Brent算法,给出了高精度的可手算的近似限.对不同的置信度γ及样本大小n=1(1)30,40,60,120,样本变差系数ε=0.01(0.01)0.20,计算了正态变差系数的经典精确限表.本文指出,当n≤8,ε≤0.20时,经典精确限C_u略大于Fiducial精确限C_u,F.当n>8.ε≤0.20时.C_u-C_u,F<5×10-6.     

排序法的不合理性及一种修正方法

一、关联系统可靠性精确置信限设一个由 m 个元件(或分系统)组成的关联系统,其可靠性函数为 h(?),(?)=(p_1,p_2,…,P_m).其中 P_i 为第 i 个元件的可靠性,i=(?)设第 i 个元件试验 n_i 次,失效 x_i 次,成功 y_i=n_i—x_i 次,(?),(?)(x_1,x_2,…,x_m),则样本空间为     

Confidence sets and confidence bands for a beta distribution with applications to credit risk management

Incorporating statistical multiple comparisons techniques with credit risk measurement, a new methodology is proposed to construct exact confidence sets and exact confidence bands for a beta distribution. This involves simultaneous inference on the two parameters of the beta distribution, based upon the inversion of Kolmogorov tests. Some monotonicity properties of the distribution function of the beta distribution are established which enable the derivation of an efficient algorithm for the implementation of the procedure. The methodology has important applications to financial risk management. Specifically, the analysis of loss given default (LGD) data are often modeled with a beta distribution. This new approach properly addresses model risk caused by inadequate sample sizes of LGD data, and can be used in conjunction with the standard recommendations provided by regulators to provide enhanced and more informative analyses.     

EM algorithm and its application to testing hypotheses

The conventional method for testing hypotheses is to find an exact or asymptotic distributionof a test statistic But when the model is complex and the sample size is small,difficulty often arises. Thispaper aims to present a method for finding maximum probability with the help of EM algorithm. For any fixedsample size,this method can be used not only to obtain an accurate test but also to check the real level of     

求参数置信限的一种方法

本文介绍一种找参数精确置信限和置信区间的一般方法,关键想法是在样本空间中定义一个序。本文主要考虑单参数统计模型，对序的优良性（即相应的置信限的优良性）作了讨论，指出应以得分函数的大小为依据在样本空间中定义序。还证明了用最大似然估计定义序得到的置信限是一个单调函数的唯一零点，从而通过解方程算出置信限。     

Implementations of the Monte Carlo EM Algorithm

The Monte Carlo EM (MCEM) algorithm is a modification of the EM algorithm where the expectation in the E-step is computed numerically through Monte Carlo simulations. The most exible and generally applicable approach to obtaining a Monte Carlo sample in each iteration of an MCEM algorithm is through Markov chain Monte Carlo (MCMC) routines such as the Gibbs and Metropolis–Hastings samplers. Although MCMC estimation presents a tractable solution to problems where the E-step is not available in closed form, two issues arise when implementing this MCEM routine: (1) how do we minimize the computational cost in obtaining an MCMC sample? and (2) how do we choose the Monte Carlo sample size? We address the first question through an application of importance sampling whereby samples drawn during previous EM iterations are recycled rather than running an MCMC sampler each MCEM iteration. The second question is addressed through an application of regenerative simulation. We obtain approximate independent and identical samples by subsampling the generated MCMC sample during different renewal periods. Standard central limit theorems may thus be used to gauge Monte Carlo error. In particular, we apply an automated rule for increasing the Monte Carlo sample size when the Monte Carlo error overwhelms the EM estimate at any given iteration. We illustrate our MCEM algorithm through analyses of two datasets fit by generalized linear mixed models. As a part of these applications, we demonstrate the improvement in computational cost and efficiency of our routine over alternative MCEM strategies.     

Model Misspecification: Finite Mixture or Homogeneous?

A common problem in statistical modelling is to distinguish between finite mixture distribution and a homogeneous non-mixture distribution. Finite mixture models are widely used in practice and often mixtures of normal densities are indistinguishable from homogenous non-normal densities. This paper illustrates what happens when the EM algorithm for normal mixtures is applied to a distribution that is a homogeneous non-mixture distribution. In particular, a population-based EM algorithm for finite mixtures is introduced and applied directly to density functions instead of sample data. The population-based EM algorithm is used to find finite mixture approximations to common homogeneous distributions. An example regarding the nature of a placebo response in drug treated depressed subjects is used to illustrate ideas.     

A simple algorithm for tighter exact upper confidence bounds with rare attributes in finite universes

When attributes are rare and few or none are observed in the selected sample from a finite universe, sampling statisticians are increasingly being challenged to use whatever methods are available to declare with high probability or confidence that the universe is near or completely attribute-free. This is especially true when the attribute is undesirable. Approximations such as those based on normal theory are frequently inadequate with rare attributes. For simple random sampling without replacement, an appropriate probability distribution for statistical inference is the hypergeometric distribution. But even with the hypergeometric distribution, the investigator is limited from making claims of attribute-free with high confidence unless the sample size is quite large using nonrandomized techniques. For students in statistical theory, this short article seeks to revive the question of the relevance of randomized methods. When comparing methods for construction of confidence bounds in discrete settings, randomization methods are useful in fixing like confidence levels and hence facilitating the comparisons. Under simple random sampling, this article defines and presents a simple algorithm for the construction of exact “randomized” upper confidence bounds which permit one to possibly report tighter bounds than those exact bounds obtained using “nonrandomized” methods. A general theory for exact randomized confidence bounds is presented in Lehmann (1959, p. 81), but Lehmann's development requires more mathematical development than is required in this application. Not only is the development of these “randomized” bounds in this paper elementary, but their desirable properties and their link with the usual nonrandomized bounds are easy to see with the presented approach which leads to the same results as would be obtained using the method of Lehmann.     

双参数指数分布的可靠寿命的广义置信下限

基于双参数指数分布定数截尾数据,利用Weerahanandi给出的广义置信区间的概念,建立了可靠寿命的广义置信下限,并从理论上证明了我们给出的广义置信下限是精确的,即基于广义置信下限的区间估计的覆盖率等于要求的置信水平.广义置信下限需要通过数值方法得到,但是计算方法是简单直接的.在小样本情形下,通过对基于广义置信下限的置信区间与Engelhardt-Bain近似置信区间覆盖率的模拟比较,发现广义置信下限更令人满意.     

Computation of exact confidence limits from discrete data

Suppose that the data have a discrete distribution determined by (∞, ψ) where θ is the scalar parameter of interest and ψ is a nuisance parameter vector. The Buehler 1 - α upper confidence limit for θ is as small as possible, subject to the constraints that (a) its coverage probability is at least 1 - α and (b) it is a nondecreasing function of a pre-specified statisticT. This confidence limit has important biostatistical and reliability applications. The main result of the paper is that for a wide class of models (including binomial and Poisson), parameters of interest 9 and statisticsT (which possess what we call the “logical ordering” property) there is a dramatic increase in the ease with which this upper confidence limit can be computed. This result is illustrated numerically for θ a difference of binomial probabilities. Kabaila & Lloyd (2002) also show that ifT is poorly chosen then an assumption required for the validity of the formula for this confidence limit may not be satisfied. We show that for binomial data this assumption must be satisfied whenT possesses the “logical ordering” property.     

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.     

排序法计算指数寿命型元件串联系统可靠性经典精确最优置信下限

对于样本点是离散的情况,可用对样本点排序的方法确定可靠性置信下限,排序有很多种,有L-P排序、序贯排序、极大似然估计排序、修正L-P排序等。本文提出一种具有直观合理性的新的排序方法,计算指数寿命型元件串联系统可靠性经典精确最优置信下限。     

A Combined Data-Base and Algorithmic Approach to the Pallet-Loading Problem

The current trend in pallet-loading software is for fast, interactive microcomputer packages. Because of the complexity of the problem, it is unlikely that any exact algorithm will guarantee to solve all typical pallet-loading problems within the sort of time range desirable for interactive use. Therefore, the packages tend to be based on heuristic solution methods. This paper describes a way of dealing with those problems which fail to solve within the given time limit. The discussion is in terms of a recent exact algorithm for the pallet-loading problem, but the methodology can be applied in conjunction with any algorithm for which the proportion of problems exceeding the time limit is small. The result is a piece of software which guarantees to solve any typical pallet-loading problem within a few minutes on an IBM-PC.     

可靠度的M-Bayes可信限及其性质

以前作者提出了一种新的参数估计方法-M-Bayes可信限法,并且给出了可靠度的M-Bayes可信下限的估计.给出了另一个可靠度的M-Bayes可信下限的估计,并给出了M-Bayes可信下限的性质一可靠度的两个M-Bayes可信下限与经典置信下限的关系.最后,给出一个例子,从这个例子可以看出本文提出的方法可行且便于应用.     

双边定时截尾样本下广义逆指数分布形状参数的估计

先给出了广义逆指数分布在双边定时截尾样本下形状参数的最大似然估计,并不能得到估计的显式表达式,但证明了参数在(0,+∞)上最大似然估计是唯一存在的.其次提出用EM算法求出形状参数的估计且该估计具有良好的收敛性,还给出了形状参数的EM估计的渐近方差和近似置信区间;最后通过数值模拟,对形状参数的最大似然估计和EM估计的效果进行了比较,说明了用EM算法求形状参数的估计是可行的,并且模拟效果相对比较好.     

Limit theorems for <Emphasis Type="Italic">p</Emphasis>-adic valued asymmetric semistable laws and processes

Limit distributions of scaled sums of p-adic valued i.i.d. are characterized as semistable laws, and a condition to assure the weak convergence of a scaled sum is verified. The limit supremum of the norm of the weakly convergent scaled sum is divergent in fact, and the exact growth rate of the sum is given. It is also shown that, if a scaled sum including a time parameter in the number of the added i.i.d. is considered, the semigroup of the limit distributions corresponds to a p-adic valued Markov process having right continuous sample paths with left limits. These are generalizations of the former results for rotation-symmetric i.i.d., with some necessary modifications.     

某型航空火控系统可靠度置信下限的确定

依据 AMSAA模型利用某型航空火控系统各组成单元的研制阶段试验数据并基于遗传算法确定出单元的时间环境折合系数 ,进而求得各单元 MTBF点估计和置信下限 ,最后利用 L-M法给出该系统可靠度置信下限值 .研究结果表明 ,由此得到的系统可靠性评估结果与依据设计定型后实际使用数据统计得到的可靠性水平相接近 ,应用该方法进行系统可靠性评估能较好地满足工程上的需求 .     

非等定时截尾寿命试验方案指数分布情形平均寿命的置信限

非等定时截尾寿命试验方案是价值高或制造复杂的产品在实际试验中常常采用的方案 ,本文在此方案下给出一种关于指数分布平均寿命的置信限的解决方案。这个置信限在有一定意义下是最优的 ,并且是不保守的。     

Approximate confidence limits of the reliability performances for a cold standby series system

This paper is to investigate the approximate confidence limits of the reliability performances (such as failure rate, reliability function and average life) for a cold standby series system. The Bayesian approximate upper confidence limit of failure rate is obtained firstly, and next Bayesian approximate lower confidence limits for reliability function and average life are presented. The expressions for calculating Bayesian lower confidence limits of the reliability function and average life are also obtained, and an illustrative example is examined numerically by means of the Monte-Carlo simulation. Finally, the accuracy of confidence limits is discussed.