On Bayesian group sequential sampling procedures

A general group sequential statistical decision model is investigated. Using a Bayesian approach and Bayesian dynamic programming, we examine structural properties of the cost functions and of optimal sampling procedures. In particular, we consider variable-sample-size-sequential probability ratio tests and show that the so-called onion-skins conjecture is false.     

指数分布场合恒加试验缺失数据的Bayes统计分析

主要讨论了当寿命分布是指数分布时恒定应力加速寿命试验中常见几类数据类型(完全样本,分组样本,删失样本)的Bayes统计分析方法.运用G ibbs抽样迭代算法成功解决了Bayes分析中极为复杂的后验边际分布的计算问题,得到了满足顺序约束的参数的Bayes估计.模拟结果表明各场合下此方法均较好.     

Bayesian priors in sequential binomial design

The status of sequential analysis in Bayesian inference is revisited. The information on the experimental design, including the stopping rule, is one part of the evidence, prior to the sampling. Consequently this information must be incorporated in the prior distribution. This approach allows to relax the likelihood principle when appropriate. It is illustrated in the case of successive Binomial trials. Using Jeffreys' rule, a prior based on the Fisher information and conditional on the design characteristics is derived. The corrected Jeffreys prior, which involves a new distribution called Beta-J, extends the classical Jeffreys priors for the Binomial and Pascal sampling models to more general stopping rules. As an illustration, we show that the correction induced on the posterior is proportional to the bias induced by the stopping rule on the maximum likelihood estimator. To cite this article: P. Bunouf, B. Lecoutre, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

A sequential test for the failure rate of an exponential distribution with censored data

This paper provides an optimal sequential decision procedure for deciding between two composite hypotheses about the unknown failure rate of an exponential distribution, using censored data. The procedure has two components, a stopping time and a decision function. The optimal stopping time minimizes the expected total loss due to a wrong decision plus cost of observing the process. The optimal decision function is easily characterized once a stopping time has been specified. The main result determines the continuation region for the optimal decision procedure     

概率方法在纺织产品质量检验中的应用

本文用统计分析方法分析了针织厂原产品质量检验方案 ,指出了其不合理性 ;从该厂的实际出发 ,针对三种不同批量建立了新的检验方案。实际证明 ,新方案既可以确保产品出厂质量 (限定在AQL值内 ) ,又可以对生产质量水平起到监督的作用     

An application of the sampling theorem

Let B_σ,p,1 <-p<-∞, be the set of all functions from L₈(R) which can be continued to entire functions of exponential type <-σ. The well known Shannon sampling theorem and its generalization [1] state that every f∈B_σ,p, 1<p<∞, can be represented as

Group sequential comparison of two binomial proportions under ranked set sampling design

Ranked set sampling (RSS) is a technique for incorporating auxiliary (concomitant) information into estimation and testing procedures right at the design stage. In this paper, we propose group sequential testing procedures for comparing two treatments with binary outcomes under an RSS scheme with perfect ranking. We compare the power, the average sample sizes and type I errors of the proposed tests to those of the group sequential tests based on simple random sampling schemes. We illustrate the usefulness of the methodology by using data from a clinical trial on leukemia.     

An application of the sampling theorem

Let B_σ,p,1 <-p<-∞, be the set of all functions from L₈(R) which can be continued to entire functions of exponential type <-σ. The well known Shannon sampling theorem and its generalization [1] state that every f∈B_σ,p, 1 $$f(x) = \mathop \Sigma \limits_{j \in z} f(j\pi /\sigma )\tfrac{{sin\sigma (x - j\pi /\sigma )}}{{\sigma (x - j\pi /\sigma )}}, \sigma > 0$$ It is well known that the Shannon sampling theorem plays an important role in many applied fields [1,2,7]. In this paper we give an application of sampling theorem to approximation theory.     

A new sequential space-filling sampling strategy for elementary effects-based screening method

To predict or control the response of a complicated numerical model which involves a large number of input variables but is mainly affected by only a part of variables, it is necessary to screening those active variables. This paper proposes a new space-filling sampling strategy, which is used to screening the parameters based on the Morris’ elementary effect method. The beginning points of sampling trajectories are selected by using the maximin principle of Latin Hypercube Sampling method. The remaining points of trajectories are determined by using the one-factor-at-a-time design. Being different from other sampling strategies to determine the sequence of factors randomly in one-factor-at-a-time design, the proposed method formulates the sequence of factors by a deterministic algorithm, which sequentially maximizes the Euclidean distance among sampling trajectories. A new efficient algorithm is proposed to transform the distance maximization problem to a coordinate sorting problem, which saves computational cost much. After the elementary effects are computed using the sampling points, a detailed criterion is presented to select the active factors. Two mathematic examples and an engineering problem are used to validate the proposed sampling method, which demonstrates the priority in computational efficiency, space-filling performance, and screening efficiency.     

An estimation method for the Neyman chi-square divergence with application to test of hypotheses

We propose a new definition of the Neyman chi-square divergence between distributions. Based on convexity properties and duality, this version of the χ² is well suited both for the classical applications of the χ² for the analysis of contingency tables and for the statistical tests in parametric models, for which it is advocated to be robust against outliers.We present two applications in testing. In the first one, we deal with goodness-of-fit tests for finite and infinite numbers of linear constraints; in the second one, we apply χ²-methodology to parametric testing against contamination.     

Design of progressively censored group sampling plans for Weibull distributions: An optimization problem

Optimization algorithms provides efficient solutions to many statistical problems. Essentially, the design of sampling plans for lot acceptance purposes is an optimization problem with several constraints, usually related to the quality levels required by the producer and the consumer. An optimal acceptance sampling plan is developed in this paper for the Weibull distribution with unknown scale parameter. The proposed plan combines grouping of items, sudden death testing in each group and progressive group removals, and its decision criterion is based on the uniformly most powerful life test. A mixed integer programming problem is first solved for determining the minimum number of failures required and the corresponding acceptance constant. The optimal number of groups is then obtained by minimizing a balanced estimation of the expected test cost. Excellent approximately optimal solutions are also provided in closed-forms. The sampling plan is considerably flexible and allows to save experimental time and cost. In general, our methodology achieves solutions that are quite robust to small variations in the Weibull shape parameter. A numerical example about a manufacturing process of gyroscopes is included for illustration.     

An application of a general sampling theorem

An extension of the sampling theorem to multi-band signals is discussed and an application to the compression of speech outlined.     

Bayesian graduation of mortality rates: An application to reserve evaluation

This paper presents Bayesian graduation models of mortality rates, using Markov chain Monte Carlo (MCMC) techniques. Graduated annual death probabilities are estimated through the predictive distribution of the number of deaths, which is assumed to follow a Poisson process, considering that all individuals in the same age class die independently and with the same probability. The resulting mortality tables are formulated through dynamic Bayesian models. Calculation of adequate reserve levels is exemplified, via MCMC, making use of the value at risk concept, demonstrating the importance of using “true” observed mortality figures for the population exposed to risk in determining the survival coverage rate.     

An application of least cost acceptance sampling schemes

Summary A technique is described for determining most economical single sample inspection schemes in the situation where costs associated with inspection and failure to detect defective items can be estimated. A computational procedure is outlined, and a comparison made between the schemes determined by this method, and those previously in use. A least-cost decision rule is also derived for sequential sampling schemes.

---

Zusammenfassung Ein Verfahren zum Aufstellen wirtschaftlichster einfacher Stichprobenverfahren wird für den Fall beschrieben, in dem die Kosten für das Überprüfen und das Nichtentdecken fehlerhafter Stücke geschätzt werden können. Es wird ein numerisches Verfahren aufgezeigt und ein Vergleich zwischen dem hier beschriebenen Verfahren und den früher verwendeten durchgeführt. Eine Entscheidungsregel für geringste Kosten wird ebenso für sequentielle Stichprobenpläne hergeleitet.

---

---

Now with the Operations Research Section of B.O.A.C., Comet House, London Airport, Hounslow, Middx., England.

---

Vorgel. v.:M. Sasieni     

Application of the sequential parametric convex approximation method to the design of robust trusses

We study an algorithm recently proposed, which is called sequential parametric approximation method, that finds the solution of a differentiable nonconvex optimization problem by solving a sequence of differentiable convex approximations from the original one. We show as well the global convergence of this method under weaker assumptions than those made in the literature. The optimization method is applied to the design of robust truss structures. The optimal structure of the model considered minimizes the total amount of material under mechanical equilibrium, displacements and stress constraints. Finally, Robust designs are found by considering load perturbations.     

An innovative unification process for probabilistic hesitant fuzzy elements and its application to decision making

Fuzzy Optimization and Decision Making - The probabilistic hesitant fuzzy element (PHFE) is a worthwhile extension of hesitant fuzzy element (HFE) which is a means of allowing the decision makers...     

A sequential parametric convex approximation method with applications to nonconvex truss topology design problems

We describe a general scheme for solving nonconvex optimization problems, where in each iteration the nonconvex feasible set is approximated by an inner convex approximation. The latter is defined using an upper bound on the nonconvex constraint functions. Under appropriate conditions, a monotone convergence to a KKT point is established. The scheme is applied to truss topology design (TTD) problems, where the nonconvex constraints are associated with bounds on displacements and stresses. It is shown that the approximate convex problem solved at each inner iteration can be cast as a conic quadratic programming problem, hence large scale TTD problems can be efficiently solved by the proposed method.     

An optimization-based approach for the design of Bayesian networks

Bayesian networks model conditional dependencies among the domain variables, and provide a way to deduce their interrelationships as well as a method for the classification of new instances. One of the most challenging problems in using Bayesian networks, in the absence of a domain expert who can dictate the model, is inducing the structure of the network from a large, multivariate data set. We propose a new methodology for the design of the structure of a Bayesian network based on concepts of graph theory and nonlinear integer optimization techniques.     

An application of Lipschitzian global optimization to product design

The issue of finding feasible mixture designs is formulated and solved as a Lipschitzian global optimization problem. The solution algorithm is based on a simplicial partition strategy. Implementation aspects and extension possibilities are treated in some detail, providing also numerical examples.