Stochastic Comparisons with Applications PDF Download

Author: Jee S. Kim
Publisher:
ISBN:
Category :
Languages : en
Pages : 19

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Book Description
This paper surveys recent advances in stochastic comparisons of order statistics along with reliability applications. Section 2 presents inequalities for linear combinations of order statistics from restricted families. Comparisons of linear combinations of order statistics from distributions F and G are obtained for/G F convex and for/G F starshaped. These results yield conservative upper and lower tolerance limits. For G exponential and F IFR or IFRA, stochastic comparisons are presented for the total time on test, used in life testing. Section 3 presents stochastic comparisons of order statistics from underlying heterogeneous distributions. Given two sets of independent components (possibly unlike), majorization conditions are given which insure that any k-out-of-n system constructed from components in the first set will have reliability at least as great as that of a corresponding system constructed from components in the second set. Since the ordered failure times of the components represent order statistics from heterogeneous distributions, the order statistics from one set of underlying distributions (F1, ..., Fn) can be compared stochastically with those from another set (F1, ..., Fn).

Author: Subhash C. Kochar
Publisher: Springer Nature
ISBN: 303112104X
Category : Mathematics
Languages : en
Pages : 280

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Book Description
This book emphasizes the use of stochastic orders as motivational tools for developing new statistical procedures. Stochastic orders have found useful applications in many disciplines, including reliability theory, survival analysis, risk theory, finance, nonparametric methods, economics and actuarial science. Written by a statistician, this volume clarifies the connection between stochastic orders and nonparametric methods. The importance of order statistics and spacings is well recognized. Classically, they mainly focus on the case when the observations are independent and identically distributed, however, several new developments have extended the comparison of order statistics to the case of non-identically distributed or non-independent observations. In addition to giving a detailed discussion of various topics in the general area of stochastic orders, a substantial part of the book is devoted to recent research on stochastic comparisons of order statistics and spacings, including a long chapter on dependence among them. The book will be useful for graduate students and researchers in statistics, economics, actuarial science and other related disciplines. In particular, with close to 300 references, it will be a valuable resource for reliability theorists, applied probabilists and statisticians. Readers are expected to have taken a first-year graduate level course in mathematical statistics or in applied probability.

Author: Haijun Li
Publisher: Springer Science & Business Media
ISBN: 1461468922
Category : Mathematics
Languages : en
Pages : 459

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Book Description
Stochastic Orders in Reliability and Risk Management is composed of 19 contributions on the theory of stochastic orders, stochastic comparison of order statistics, stochastic orders in reliability and risk analysis, and applications. These review/exploratory chapters present recent and current research on stochastic orders reported at the International Workshop on Stochastic Orders in Reliability and Risk Management, or SORR2011, which took place in the City Hotel, Xiamen, China, from June 27 to June 29, 2011. The conference’s talks and invited contributions also represent the celebration of Professor Moshe Shaked, who has made comprehensive, fundamental contributions to the theory of stochastic orders and its applications in reliability, queueing modeling, operations research, economics and risk analysis. This volume is in honor of Professor Moshe Shaked. The work presented in this volume represents active research on stochastic orders and multivariate dependence, and exemplifies close collaborations between scholars working in different fields. The Xiamen Workshop and this volume seek to revive the community workshop tradition on stochastic orders and dependence and strengthen research collaboration, while honoring the work of a distinguished scholar.

Author: N Balakrishnan
Publisher: Birkhäuser
ISBN: 9780817672553
Category : Mathematics
Languages : en
Pages :

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Book Description
This is the first book specifically devoted to the topic of stochastic orderings for order statistics. Combining classical and recent results, the work is a comprehensive overview of ordering properties for order statistics and related concepts. Special emphasis is placed on the connections between order statistics and coherent systems, applying results to reliability and survival studies. The work is an excellent reference for advanced undergraduates, graduate students, and researchers in statistics, mathematics, computer science, engineering, and medicine. The book may serve as a supplementary textbook for courses on ordering for order statistics and ordering coherent systems.

Author: F. Proschan
Publisher:
ISBN:
Category :
Languages : en
Pages : 19

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Book Description
In this paper, the authors stochastically compare whole vectors of order statistics. This stochastic vector comparison yields as special cases a number of previous results such as the Pledger-Proschan results and additional results stochastically comparing partial or complete sums of order statistics. It is shown that some of these results concerning stochastic comparison of sums remain valid if one does not order the sample, but instead uses the ordering specified by the constants of proportionality of the hazard functions.

Author: Moshe Shaked
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 580

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Book Description
Stochastic orders and inequalities are being used at an accelerated rate in many diverse areas of probability and statistics. This book provides the first unified, systematic, and accessible treatment of stochasticorders, addressing the growing importance of these orders with the presentation of numerous results that illustrate their usefulness and applicability. Ten insightful chapters emphasize the applications by specialists in probability and statistics, economics, operations research, and reliability theory. Applications include multivariate variability, epidemics, comparisons of risk and risk aversion, scheduling, and systems reliability theory.

Author: Asit P. Basu
Publisher: World Scientific
ISBN: 9789810233600
Category : Technology & Engineering
Languages : en
Pages : 512

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Book Description
This volume presents recent results in reliability theory by leading experts in the world. It will prove valuable for researchers, and users of reliability theory. It consists of refereed invited papers on a broad spectrum of topics in reliability. The subjects covered include Bayesian reliability, Bayesian reliability modeling, confounding in a series system, DF tests, Edgeworth approximation to reliability, estimation under random censoring, fault tree reduction for reliability, inference about changes in hazard rates, information theory and reliability, mixture experiment, mixture of Weibull distributions, queuing network approach in reliability theory, reliability estimation, reliability modeling, repairable systems, residual life function, software spare allocation systems, stochastic comparisons, stress-strength models, system-based component test plans, and TTT-transform.

Author: Maochao Xu
Publisher:
ISBN:
Category : Statistics
Languages : en
Pages : 132

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Book Description
The statistics literature has mostly focused on the case when the data available is in the form of a random sample. In many cases, the observations are not identically distributed. Such samples are called heterogeneous samples. The study of heterogeneous samples is of great interest in many areas, such as statistics, econometrics, reliability engineering, operation research and risk analysis. Stochastic orders between probability distributions is a widely studied concept. There are several kinds of stochastic orders that are used to compare different aspects of probability distributions like location, variability, skewness, dependence, etc. In this dissertation, most of the work is devoted to investigating the properties of statistics based on heterogeneous samples with the aid of stochastic orders. We will see the effect of the change in the stochastic properties of various functions of observations as their parameters change. The theory of majorization will be used for this purpose. First, order statistics from heterogeneous samples will be investigated. Order statistics appear everywhere in statistics and related areas. The k-out-of-n systems are building blocks of a coherent system. The lifetime of such a system is the same as that of the (n-k+1)th order statistic in a sample size of n. Stochastic comparisons between order statistics have been studied extensively in the literature in case the parent observations are independent and identically distributed. However, in practice this assumption is often violated as different components in a system may not have the same distribution. Comparatively less work has been done in the case of heterogeneous random variables, mainly because of the reason that their distribution theory is very complicated. Some open problems in the literature have been solved in the dissertation. Some new problems associated with order statistics have been investigated in the thesis. Next, stochastic properties of spacings based on heterogeneous observations are studied. Spacings are of great interest in many areas of statistics, in particular, in the characterizations of distributions, goodness-of-fit tests, life testing and reliability models. In particular, the stochastic properties of the sample range are investigated in detail. Applications in reliability theory are highlighted. The relative dependence between extreme order statistics will be investigated in Chapter 4. In particular, the open problem discussed in Dolati, et al. (2008) is solved in this Chapter. In the last Chapter, convolutions of random variables from heterogeneous samples will be investigated. Convolutions have been widely used in many areas to model many practical situations. For example, in reliability theory, it arises as the lifetime of a redundant standby system; in queuing theory, it is used to model the total service time by an agent in a system; in insurance, it is used to model total claims on a number of policies in the individual risk model. I will compare the dispersion and skewness properties of convolutions of different heterogeneous samples. The tail behavior of convolutions are investigated as well. The work in this dissertation has significant applications in many diverse areas of applied probability and statistics. For example, statistics based on order statistics and spacings from heterogeneous samples arise in studying the robust properties of statistical procedures; the work on order statistics will also provide a better estimation of lifetime of a coherent system in reliability engineering; convolution results will be of great interest in insurance and actuarial science for evaluating risks.

Author: N. Balakrishnan
Publisher: Springer Science & Business Media
ISBN: 0817644229
Category : Mathematics
Languages : en
Pages : 439

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Book Description
S. Panchapakesan has made significant contributions to ranking and selection and has published in many other areas of statistics, including order statistics, reliability theory, stochastic inequalities, and inference. Written in his honor, the twenty invited articles in this volume reflect recent advances in these areas and form a tribute to Panchapakesan’s influence and impact on these areas. Featuring theory, methods, applications, and extensive bibliographies with special emphasis on recent literature, this comprehensive reference work will serve researchers, practitioners, and graduate students in the statistical and applied mathematics communities.

Author: Karl Mosler
Publisher: Springer Science & Business Media
ISBN: 3642499724
Category : Mathematics
Languages : en
Pages : 385

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Book Description
A bibliography on stochastic orderings. Was there a real need for it? In a time of reference databases as the MathSci or the Science Citation Index or the Social Science Citation Index the answer seems to be negative. The reason we think that this bibliog raphy might be of some use stems from the frustration that we, as workers in the field, have often experienced by finding similar results being discovered and proved over and over in different journals of different disciplines with different levels of mathematical so phistication and accuracy and most of the times without cross references. Of course it would be very unfair to blame an economist, say, for not knowing a result in mathematical physics, or vice versa, especially when the problems and the languages are so far apart that it is often difficult to recognize the analogies even after further scrutiny. We hope that collecting the references on this topic, regardless of the area of application, will be of some help, at least to pinpoint the problem. We use the term stochastic ordering in a broad sense to denote any ordering relation on a space of probability measures. Questions that can be related to the idea of stochastic orderings are as old as probability itself. Think for instance of the problem of comparing two gambles in order to decide which one is more favorable.