Sequential Order Statistics: Ageing and Stochastic Orderings

Ordered random variables play an important role in statistics, reliability theory, and many applied areas. Sequential order statistics provide a unified approach to a variety of models of ordered random variables. We investigate conditions on the underlying distribution functions on which the sequential order statistics are based, to obtain stochastic comparisons of sequential order statistics given some well known stochastic orderings, such as the usual stochastic, the hazard rate and the likelihood ratio orders, among others. Also, we derive sufficient conditions under which the sequential order statistics are increasing hazard rate, increasing hazard rate average or decreasing hazard rate average. Applications of the main results involving nonhomogeneous pure birth processes are also given.     

Characterizations of the exponential distribution by stochastic ordering properties of the geometric compound

Under the reliability NBU/NWU conditions, the exponential distribution is characterized by stochastic ordering properties which link the geometric compound with minimum order statistics or spacings of order statistics. This somewhat answers a question posed by Kakosyan, Klebanov and Melamed (1984,Characterization of Distributions by the Method of intensively Monotone Operators, Springer, New York). We also show the related results based on the residual life in a renewal process and on record values. Finally, some fundamental properties of the NBUC/NWUC classes of life distributions are investigated.     

Comparisons in the mean residual life order of coherent systems with identically distributed components

The comparisons of the performance of coherent systems (under different stochastic criteria) is an important task in the reliability theory. Several results have been obtained in the literature for the stochastic, hazard rate and likelihood ratio orders. In this paper, we obtain comparison results for the mean residual life order of coherent systems with identically distributed (ID) component lifetimes. These results can be applied not only to the usual case of systems with independent and identically distributed components but also to the case of systems with exchangeable components and to the more general case of just ID components. The results obtained are based on the representation of the system distribution as a distorted distribution of the common components' distribution. Some specific comparison results are given to illustrate the theoretical results. The comparison results for distorted distributions given here can also be applied to other statistical concepts such as order statistics, generalized order statistics or record values. Copyright © 2015 John Wiley & Sons, Ltd.     

Stochastic optimization on Bayesian nets

In this paper we are concerned with stochastic optimization problems in the case when the joint probability distribution, associated with random parameters, can be described by means of a Bayesian net. In such a case we suggest that the structured nature of the probability distribution can be exploited for designing efficient gradient estimation algorithm. Such gradient estimates can be used within the general framework of stochastic gradient (quasi-gradient) solution procedures in order to solve complex non-linear stochastic optimization problems. We describe a gradient estimation algorithm and present a case study related to the reliability of semiconductor manufacturing together with numerical experiments.     

Duane-LR模型下复杂系统的动态可靠性增长评定

基于复杂系统可靠性增长试验的特点，运用Duane可靠性增长模型结合数理统计中的线性回归方法对新批次产品的可靠性参数进行预测。结合产品的少量现场试验数据，利用Bayes方法对系统的可靠性增长试验结果进行评定。文中首先给出了可靠性增长分析的模型，然后运用历次阶段试验中的可靠性增长数据建立动态参数的递推估计模型，在此基础上，运用随机变量函数的分布，给出各阶段可靠性增长试验中可靠性参数的Bayes估计。文中对Weibull、指数和二项分布三种试验结果进行分析，给出计算公式。     

Inequalities for order statistics of samples from distributions with monotone hazard rate

A number of inequalities relating the mathematical expectations of order statistics or functions of order statistics in the case where the sample is taken from a distribution with monotone hazard rate function are presented. These inequalities can be applied in reliability theory and other areas of probability theory and mathematical statistics. In particular, in the paper, they are used to obtain new characterizations of the exponential distribution.     

Discussion of “Birnbaum‐Saunders distribution: A review of models,analysis and applications”

This paper provides the distribution of order statistics from a multivariate Birnbaum‐Saunders (GMBS) distribution which can be used in the reliability and lifetime analyses. Proposing a new generalization of Birnbaum‐Saunders distribution based on the unified skew‐elliptical model, some properties of the order statistics are also studied.     

A stochastic model for the direct and inverse problem of adhesive materials

This work deals with the generation of artificial data based on experimental data for adhesive materials and the application of this data to the inverse and the direct problem. In reality there are only a very limited number of experimental data available. Therefore, the prediction of material behaviour is difficult and a statistical analysis with a stochastic proved thesis is nearly impossible. In order to increase the number of tests a method of stochastic simulation based on time series analysis is applied. With artificial data an arbitrary number of data is available and the process of the parameter identification can be statistically analysed. Additionally, one example is shown, which adapts the analysed material parameter to the direct problem. The stochastic finite element method is used to take into account the distribution and deviation of the fracture strain. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modified information criteria for a uniform approximate equivalence of probability distributions

Summary Information criteria for two-sided uniform ϕ-equivalence, which is a newly introduced strong approximate equivalence of probability distributions, are proposed. The criteria resort to some modified K-L informations defined on suitable approximate main domains and are presented in the form of systems with double inequalities. They present systematic implements to handle many statistical approximation problems and are useful to evaluate related approximation errors quantitatively. Criteria for asymptotic cases are also derived from the presented inequalities. As applications, necessary and sufficient conditions and error evaluations are given for approximate and/or asymptotic equivalences of the probability distributions on sampling with and without replacement from a finite population and on quasi-extreme order statistics from a continuous distribution. The Institute of Statistical Mathematics     

Effect of random system properties on bending response of thermo-mechanically loaded laminated composite plates

In this paper, effect of random variation in system properties on bending response of geometrically linear laminated composite plates subjected to transverse uniform lateral pressure and thermal loading is examined. System parameters such as the lamina material properties, expansion of thermal coefficients, lamina plate thickness and lateral load are modeled as basic random variables. The basic formulation is based on higher order shear deformation theory to model the system behavior of the composite plate. A C⁰ finite element method in conjunction with the first order perturbation technique procedure developed earlier by authors for the plate subjected to lateral loading is employed to obtain the second order response statistics (mean and variance) of the transverse deflection of the plate. Typical numerical results for the second order statistics of the transverse central deflection of geometrically linear composite plates with temperature independent and dependent material properties subjected to uniform temperature and combination of uniform and linearly varying temperature distribution are obtained for various combinations of geometric parameters, uniform lateral pressures, staking sequences and boundary conditions. The performance of the stochastic laminated composite model is demonstrated through comparison of mean transverse central deflection with those results available in literature and standard deviation of the deflection with an independent Monte Carlo simulation.     

Model checking for additive hazards model with multivariate survival data

Multivariate failure time data often arise in biomedical studies due to natural or artificial clustering. With appropriate adjustment for the underlying correlation, the marginal additive hazards model characterizes the hazard difference via a linear link function between the hazard and covariates. We propose a class of graphical and numerical methods to assess the overall fitting adequacy of the marginal additive hazards model. The test statistics are based on the supremum of the stochastic processes derived from the cumulative sum of the martingale-based residuals over time and/or covariates. The distribution of the stochastic process can be approximated through a simulation technique. The proposed tests examine how unusual the observed stochastic process is, compared to a large number of realizations from the approximated process. This class of tests is very general and suitable for various purposes of model fitting evaluation. Simulation studies are conducted to examine the finite sample performance, and the model-checking methods are illustrated with data from an otitis media study.     

Cost–benefit modelling for reliability growth

Decisions during the reliability growth development process of engineering equipment involve trade-offs between cost and risk. However slight, there exists a chance an item of equipment will not function as planned during its specified life. Consequently the producer can incur a financial penalty. To date, reliability growth research has focussed on the development of models to estimate the rate of failure from test data. Such models are used to support decisions about the effectiveness of options to improve reliability. The extension of reliability growth models to incorporate financial costs associated with ‘unreliability’ is much neglected. In this paper, we extend a Bayesian reliability growth model to include cost analysis. The rationale of the stochastic process underpinning the growth model and the cost structures are described. The ways in which this model can be used to support cost–benefit analysis during product development are discussed and illustrated through a simple case.     

Nonlinear and adaptive frame approximation schemes for elliptic PDEs: Theory and numerical experiments

This article is concerned with adaptive numerical frame methods for elliptic operator equations. We show how specific noncanonical frame expansions on domains can be constructed. Moreover, we study the approximation order of best n‐term frame approximation, which serves as the benchmark for the performance of adaptive schemes. We also discuss numerical experiments for second order elliptic boundary value problems in polygonal domains where the discretization is based on recent constructions of boundary adapted wavelet bases on the interval. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Order statistics from mixed exchangeable random variables

Two different exchangeable samples are considered and these two samples are assumed to be independent of each other. From these two samples a new sample is combined and treated as a single set of observations. The distribution of a single order statistic and the joint distribution of two order statistics for a new mixed sample are derived and expressed in terms of joint distribution functions. As a special case the distribution of a single order statistic and the joint distribution of two nonadjacent order statistics from exchangeable random variables are obtained. The results presented in this paper allows widespread applications in modelling of various lifetime data, biomedical sciences, reliability and survival analysis, actuarial sciences etc., where the assumption of independence of data cannot be accepted and the exchangeability is a more realistic assumption.     

Reliability and expectation bounds for coherent systems with exchangeable components

Sharp upper and lower bounds are obtained for the reliability functions and the expectations of lifetimes of coherent systems based on dependent exchangeable absolutely continuous components with a given marginal distribution function, by use of the concept of Samaniego's signature. We first show that the distribution of any coherent system based on exchangeable components with absolutely continuous joint distribution is a convex combination of distributions of order statistics (equivalent to the k-out-of-n systems) with the weights identical with the values of the Samaniego signature of the system. This extends the Samaniego representation valid for the case of independent and identically distributed components. Combining the representation with optimal bounds on linear combinations of distribution functions of order statistics from dependent identically distributed samples, we derive the corresponding reliability and expectation bounds, dependent on the signature of the system and marginal distribution of dependent components. We also present the sequences of exchangeable absolutely continuous joint distributions of components which attain the bounds in limit. As an application, we obtain the reliability bounds for all the coherent systems with three and four exchangeable components, expressed in terms of the parent marginal reliability function and specify the respective expectation bounds for exchangeable exponential components, comparing them with the lifetime expectations of systems with independent and identically distributed exponential components.     

On a characterization of the uniform distribution by generalized order statistics

In this paper a new variant of the Choquet-Deny theorem is obtained and used to prove a characterization of the uniform distribution based on spacings of generalized order statistics. This result extends two recent characterizations of the uniform distribution.     

Malliavin calculus and asymptotic expansion for martingales

Summary. We present an asymptotic expansion of the distribution of a random variable which admits a stochastic expansion around a continuous martingale. The emphasis is put on the use of the Malliavin calculus; the uniform nondegeneracy of the Malliavin covariance under certain truncation plays an essential role as the Cramér condition did in the case of independent observations. Applications to statistics are presented. Received: 5 September 1995 / In revised form: 20 October 1996     

广义次序统计量间隔的多维随机排序

本文研究了附加于广义次序统计量底分布以及参数的条件, 使得人们在多维似然比序和多维通常随机序意义下对广义次序统计量的间隔向量进行比较, 同时也给出了文中主要结果的应用.     

Rates of Convergence for Bivariate Extremes

Under a second order regular variation condition, rates of convergence of the distribution of bivariate extreme order statistics to its limit distribution are given both in the total variation metric and in the uniform metric.     

A note on uniform asymptotic normality of intermediate order statistics

It is proved that under fairly general von Mises-type conditions on the underlying distribution, the intermediate order statistics, properly standardized, converge uniformly over all Borel sets to the standard normal distribution. This closes the gap between central order statistics and extremes, where uniform convergence under mild conditions is well-known.