Sequential Fixed-Width Confidence Interval for the Product of Two Means

For the product of two population means, the problem of constructing a fixed-width confidence interval with preassigned coverage probability is considered. It is shown that the optimal sample sizes which minimize the total sample size and at the same time guarantee a fixed-width confidence interval of desired coverage depend on the unknown parameters. In order to overcome this, a fully sequential procedure consisting of a sampling scheme and a stopping rule are proposed. It is then shown that the sequential confidence interval is asymptotically consistent and the stopping rule is asymptotically efficient, as the width goes to zero. Furthermore, a second order result for the difference between the expected stopping time and the (total) optimal fixed sample size is established. The theoretical results are supported by appropriate simulations.     

Comparison of some sequential procedures with related optimal stopping rules

Summary For the problem of estimating the mean of independent, identically distributed random variables, with loss equal to a linear combination of squared error and sample size, certain sequential procedures have been shown to be asymptotically optimal when compared with the best fixed sample size rule. In this paper it is shown that these procedures are asymptotically suboptimal when compared with a closely related optimal stopping rule.A preliminary version of this paper was presented at the International Meeting on Sample Survey Analysis and on Sequential Analysis, Jerusalem, Israel, June 1982     

On bounded maximum width sequential confidence ellipsoids based on generalized U-statistics

For a vector of (estimable) functionals of several independent distributions, sequential confidence ellipsoids (of bounded maximum width) based on a class of generalized U-statistics are studied. A stopping rule along with a procedure for choosing the component sample sizes at each stage is developed, so that the proposed confidence ellipsoid has a confidence coefficient asymptotically (as the prescribed maximum width shrinks to zero) equal to a preassigned 1 - α (0 < α < 1), and the expected total sample size is minimized for the procedure. Asymptotic efficiency of the procedure is also studied. The case of von Mises' functionals is treated briefly at the end.     

A stopping rule for symbolic dynamic filtering

One of the key issues in symbolic dynamic filtering (SDF) is how to obtain a lower bound on the length of symbol blocks for computing the state probability vectors of probabilistic finite-state automata (PFSA). Having specified an absolute error bound at a confidence level, this short work formulates a stopping rule by making use of Markov chain Monte Carlo (MCMC) computations.     

A stopping rule for the Robbins-Monro method

A stopping rule for the multidimensional Robbins-Monro stochastic approximation method is developed in this paper. Both moving average and stationary -mixing type of correlated noise processes are treated. Sequentially determined confidence ellipsoids are constructed to fulfill the goal for the determination of the stopping rule. The limit behavior of the algorithm is investigated. It is shown that the stopped Robbins-Monro process is asymptotically normal. Such asymptotic normality is established by means of weak convergence methods.Communicated by Y. C. Ho     

Confident estimation for density of a biological population based on line transect sampling

Line transect sampling is a very useful method in survey of wildlife population. Confident interval estimation for density D of a biological population is proposed based on a sequential design. The survey area is occupied by the population whose size is unknown. A stopping rule is proposed by a kernel-based estimator of density function of the perpendicular data at a distance. With this stopping rule, we construct several confidence intervals for D by difference procedures. Some bias reduction techniques are used to modify the confidence intervals. These intervals provide the desired coverage probability as the bandwidth in the stopping rule approaches zero. A simulation study is also given to illustrate the performance of this proposed sequential kernel procedure.     

A Practical Sequential Stopping Rule for High-Dimensional Markov Chain Monte Carlo

A current challenge for many Bayesian analyses is determining when to terminate high-dimensional Markov chain Monte Carlo simulations. To this end, we propose using an automated sequential stopping procedure that terminates the simulation when the computational uncertainty is small relative to the posterior uncertainty. Further, we show this stopping rule is equivalent to stopping when the effective sample size is sufficiently large. Such a stopping rule has previously been shown to work well in settings with posteriors of moderate dimension. In this article, we illustrate its utility in high-dimensional simulations while overcoming some current computational issues. As examples, we consider two complex Bayesian analyses on spatially and temporally correlated datasets. The first involves a dynamic space-time model on weather station data and the second a spatial variable selection model on fMRI brain imaging data. Our results show the sequential stopping rule is easy to implement, provides uncertainty estimates, and performs well in high-dimensional settings. Supplementary materials for this article are available online.     

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.     

On fixed-width confidence intervals associated with maximum likelihood estimation

A sequential procedure is proposed to determine the sample size for a fixed-width confidence interval for an unknown parameter with its maximum likelihood estimator as the center of the interval. It is established that the sequential procedure is asymptotically consistent and efficient.     

Tukey两步同时置信区间

利用两阶段抽样,构造出Tukey两步同时置信区间,它同时满足预先给定的可靠度和精度的要求.且利用数值计算的方法给出了第一阶段最优抽样量.     

Determination of sample sizes for setting β-content tolerance limits controlling both tails of the normal distribution

The problem of determining sample size for two-sided-β-content tolerance intervals which control both tails of the normal distribution is investigated. The tolerance limits are defined to assure that the tail proportions do not exceed specified values p₁ and p₂. We determine the minimum sample size so that, at a given confidence level, the tail proportions will not be too small, i.e., so that the tolerance interval will not be overly conservative. Tables of the sample size n and the corresponding factors are provided.     

A Response-Driven Adaptive Design Based on the Klein Urn

The Klein design is a response-driven random rule to allocate experimental subjects between two treatments. It aims to allocate more subjects to the treatment that is performing better, and therefore it is useful when ethical issues are of prime interest. It behaves asymptotically as the drop-the-loser rule, which is known to have a high degree of compromise between ethics and inferential properties. Besides, the Klein design has a powerful stochastic structure, which permits to obtain exact values, for each sample size n, for its main operating characteristics, such as variability of allocations, expected failure rate and power, selection bias or accidental bias. These properties of the Klein design are thoroughly studied and we obtain exact and asymptotic results.     

序贯抽样在计算机辅助电话调查中的应用

样本量的确定是抽样设计中的关键问题,传统方法利用总体方差和调查费用的有关信息来确定样本量可能产生两种结果,一种是样本量过低,无法保证希望的估计精度要求;一种是样本量过高,导致调查经费的浪费。计算机辅助电话调查中即时的数据运算和管理功能为序贯抽样的应用奠定了基础。利用前期抽取样本的计算结果,可以规定进一步需抽取的样本量,最终样本量是对真正期望样本量的一个最佳近似,它比传统方法更能保证以最少的费用满足预先设定的精度要求。     

Asymptotic Optimality of Certain Multihypothesis Sequential Tests: Non‐i.i.d. Case

It is known that certain combinations of one‐sided sequential probability ratio tests are asymptotically optimal (relative to the expected sample size) for problems involving a finite number of possible distributions when probabilities of errors tend to zero and observations are independent and identically distributed according to one of the underlying distributions. The objective of this paper is to show that two specific constructions of sequential tests asymptotically minimize not only the expected time of observation but also any positive moment of the stopping time distribution under fairly general conditions for a finite number of simple hypotheses. This result appears to be true for general statistical models which include correlated and non‐homogeneous processes observed either in discrete or continuous time. For statistical problems with nuisance parameters, we consider invariant sequential tests and show that the same result is valid for this case. Finally, we apply general results to the solution of several particular problems such as a multi‐sample slippage problem for correlated Gaussian processes and for statistical models with nuisance parameters. This revised version was published online in June 2006 with corrections to the Cover Date.     

Asymptotic optimality in sequential interval estimation

The problem of setting a fixed width confidence interval for the mean of a normal distribution with unknown variance is considered. Several procedures are reviewed. An asymptotic lower bound for the expected sample size of any sequential sampling plan with the specified confidence coefficient is obtained as the width of the interval decreases to 0.     

Assessing process capability based on the lower confidence bound of Cpk for asymmetric tolerances

It has been proved that process capability indices provide very efficient measures of the capability of processes from many different perspectives. At the present time, the C_pk index is used more than any other index for measuring process capability. However, most existing research works for capability testing have focused on processes with symmetric tolerances, but not for asymmetric tolerances. A lower confidence bound estimates the minimum process capability, conveying critical information regarding product quality, which is essential to quality assurance. The sample size determination, which provides the sample sizes necessary to achieve a desired lower confidence bound, is directly related to the cost of the data collection plan. This paper provides explicit formulas with efficient algorithms to obtain the lower confidence bounds and sample sizes required for specified precision of the estimation on C_pk for processes with asymmetric tolerances. A Matlab computer program using a binary search method is developed. For the practitioners to use in their in-plant applications, we tabulate lower confidence bounds for some commonly used capability requirement and the sampling accuracy of C_pk for sample sizes determination. A realistic example of forging process is presented to illustrate the applicability of the proposed method.     

Frequentist model averaging for linear mixed-effects models

Linear mixed-effects models are a powerful tool for the analysis of longitudinal data. The aim of this paper is to study model averaging for linear mixed-effects models. The asymptotic distribution of the frequentist model average estimator is derived, and a confidence interval procedure with an actual coverage probability that tends to the nominal level in large samples is developed. The two confidence intervals based on the model averaging and based on the full model are shown to be asymptotically equivalent. A simulation study shows good finite sample performance of the model average estimators.     

Constructing the best linear combination of diagnostic markers via sequential sampling

We study the best linear combination of markers in terms of the area under the receiver operating characteristic curve, since no single marker is perfect for classification purposes. The sequential fixed-width confidence interval estimate method is applied. We show that the proposed procedure is efficient in terms of the total sample size, with an optimal ratio of cases to controls, and is asymptotically consistent. The performance of our method is illustrated by synthesized data and a real example.     

Stopping Tests for Markov Chain Monte-Carlo Methods

Markov Chain Monte-Carlo methods produce a random sample of a given distribution by simulating a Markov chain for which the desired distribution is a reversible measure. In order to generate a sample of size n, we propose to run n independent copies of the chain all starting from the same initial state. If n is large enough, the cutoff phenomenon yields a natural stopping rule. Indeed, the access to equilibrium can be detected using empirical estimates for the expectation of a state function. The method is illustrated by the generation of random samples of stable sets on an undirected graph.     

On sequential estimation for branching processes with immigration

Consider a Galton–Watson process with immigration. The limiting distributions of the nonsequential estimators of the offspring mean have been proved to be drastically different for the critical case and subcritical and supercritical cases. A sequential estimator, proposed by Sriram et al. (Ann. Statist. 19 (1991) 2232), was shown to be asymptotically normal for both the subcritical and critical cases. Based on a certain stopping rule, we construct a class of two-stage estimators for the offspring mean. These estimators are shown to be asymptotically normal for all the three cases. This gives, without assuming any prior knowledge, a unified estimation and inference procedure for the offspring mean.