Statistical Theory Of Reliability Research Articles

Some results on the odds rate order along with its statistical testing

Recently, Lando, et al. [2022. Properties of increasing odds rate distributions with a statistical application. Journal of Statistical Planning and Inference 221,313–-325 ] define the odds rate order and show that it is stronger than the usual stochastic order, but weaker than the hazard rate order. The purpose of this paper is to investigate the other properties of this new stochastic order. Some characterization and preservation results and its relationship with other well-known stochastic orders are given. In addition, two relative ordering based on the odds rate functions are also defined and their relation with the odds rate order is investigated. Applications in statistical theory of reliability and a statistical test for the odds rate order are also included.

Refinement bounds for the expected number of renewal epochs over a finite interval

ABSTRACT Primary quantities of interest in renewal theory are the expected number of renewals in ( 0 , t ] , known as renewal function (denoted by U ( t ) ), and the expected number of renewals in ( t , t + h ] , given by the difference U ( t + h ) − U ( t ) .One of the most known bounds for the expected number of renewals over a finite interval is the one delivered by Barlow and Proschan [Statistical theory of reliability and life testing. MD: Silver Spring; 1981. Reprint Edition. To begin with]. The essential quantity of this bound is the difference U ( t + h ) − U ( t ) − U ( h ) . In this paper, initially, we offer bounds for the conditional tails of the backward and forward recurrence times, and we give a renewal-type equation for the difference U ( t + h ) − U ( t ) − U ( h ) . Employing those results, we refine the provided bound by Barlow and Proschan [Statistical theory of reliability and life testing. MD: Silver Spring; 1981. Reprint Edition. To begin with]. The results are illustrated with numerical examples.

Approximations for success run probabilities in Bernoulli trials

Concise and convenient bounds are obtained for the probability mass and cumulative distribution functions associated with the first success run of length k k in a sequence of n n Bernoulli trials. Results are compared to an approximation obtained by the Stein–Chen method as well as to bounds obtained from statistical reliability theory. These approximation formulas are used to obtain precise estimates of the expectation value associated with the occurrence of at least one success run of length k k within N N concurrent sequences of Bernoulli trials.

Applications of TP2 Functions in Theory of Stochastic Orders: A Review of some Useful Results

In the literature on Statistical Reliability Theory and Stochastic Orders, several results based on theory of TP2/RR2 functions have been extensively used in establishing various properties. In this paper, we provide a review of some useful results in this direction and highlight connections between them.

ОЦЕНКА ДОСТОВЕРНОСТИ ПРЕОБРАЗОВАНИЯ ИНФОРМАЦИИ В ИНФОРМАЦИОННЫХ СИСТЕМАХ

Objective: To find the method of data processing validation in the process of data systems engineering (IS), that provides for reliability of information at different stages of data processing (keying, reading different types of transfer support, storing, channeling, processing, presenting results to the user, etc.). Methods: The methods of statistical theory of reliability and the theory of information coding were applied, as the reliability of control options (by a human-operator, by means of hardware device, via program, etc.) significantly differs which leads to efficiency changes of one and the same method, depending on its concrete implementation. Measures on reducing a number of occurring distortions were taken in addition to error control methods. Commonly used parameters of the given methods prevent from the unified position assessment of the simultaneous application effect of different control methods and measures, aimed at reducing the number of errors, not related to control, taking into account their mutual influence, reliability and implementation structure. Results: The task of using different methods of error control (protective coding, checking sums application, duplication of operations with results comparison, data visual inspection, acceptable range check when entering some data, etc.) was solved, as well as the methods not related to control (the selection of input values from a pre-arranged list, errorfree performance of the staff incentive, etc.). In the process of IS engineering it became possible to assess the effect from application of a separate method and a group of different methods of reliability improvement at different stages of data processing. Practical importance: The considered approach makes it possible to assess detectivity of separate and the complex of control methods at the stages of data conversion in the process of IS engineering, as well as to consider from unified positions the impact of measures, not related to control, on information integrity.

The application of statistical reliability theory in the context of Intelligent Environments: a tutorial review

Intelligent Environments often require the integration of multi-modal sensing and actuating technologies with high performance real-time computation, including artificial intelligence systems for analysis, learning patterns and reasoning. Such systems may be complex, and involve multiple components. However, in order to make them affordable, Intelligent Environments sometimes require many of their individual components to be low-cost. Nevertheless, in many applications—including safety-critical systems, and systems monitoring the health and well-being of vulnerable individuals, it is essential that these Intelligent Environment systems are reliable, which the issue of affordability must not compromise. If such environments are to find real application and deployment in these types of domain, it is necessary to be able to obtain accurate predictions of how probable any potential failure of the system is in any given timeframe, and of statistical parameters regarding the expected time to the first, or between successive, failures. Such quantities must be kept within what are deemed to be acceptable tolerances if the Intelligent Environment is to be suitable for applications in these critical areas, without requiring excessively high levels of human monitoring and/or intervention. In this paper, an introductory overview of statistical reliability theory is presented. The applicability of this to the context of Intelligent Environments—particularly those involving safety critical or other sensitive issues—is discussed, along with how such reliability modelling can be used to influence the design, implementation and application of an Intelligent Environment.

Some elementary ideas concerning the complexity of system structure

A system of iid Bernoulli components is the starting point in the statistical theory of reliability. This simplification allows for the development of a rich, though elementary, theory for the structure of the system. Two representations play a prominent role in the study of structural reliability. One is the minimal path set representation and the other is the signature representation. By combining the two representations with the Gibbs measure for the state of components, one obtains terms that can be interpreted as the complexity of the system structure. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010

Competing auctions with endogenous quantities

Two sellers decide on their discrete supply of a homogenous good. There is a finite number of buyers with unit demand and privately known valuations. In the first model, there is a centralized market place where a uniform auction takes place. In the second, there are two distinct auction sites, each with one seller, and buyers decide where to bid. Using the theory of potential games, we show that in the one-site auction model there is always an equilibrium in pure-strategies. In contrast, if the distribution of buyers values has an increasing failure rate, and if the marginal cost of production is relatively low, there is no pure-strategy equilibrium where both sellers make positive profits in the competing sites model. We also identify conditions under which an equilibrium with a unique active site exists. We deal with the finite and discrete models by using several results about order statistics developed by Richard Barlow and Frank Proschan [R. Barlow, F. Proschan, Mathematical Theory of Reliability, Wiley, New York, 1965; R. Barlow, F. Proschan, Inequalities for linear combinations of order statistics from restricted families, Ann. Math. Statist. 37 (1966) 1593–1601; R. Barlow, F. Proschan, Statistical Theory of Reliability and Life Testing, McArdle Press, Silver Spring, 1975].

Calculations of System Aging through the Stochastic Entropy

The present research discusses four ‘physical’ models of system and calculates thereliability function during system’s aging and maturity on the basis of the system structure.

Rent seeking and the value of time

This paper focuses on the question of how rivals’ rent-seeking expenditures and investment expenditures are affected by the temporal dimension of those cash flows as well as the timing of the cash flow of monopoly rents. The paper applies methods from statistical reliability theory to derive five propositions establishing the conditions that must be satisfied if the rivals apportion their rent seeking and investment expenditures to maximize their certainty equivalents of the monopoly rent. The propositions explicate the responses of the rivals to changes in economic parameters characterizing the rent-seeking contest such as a change in the duration of the monopoly rent cash flow or a change in the number of rivals.

Comparing two reliability upper bounds for multistate systems

The path-cut reliability bound due to Esary and Proschan [J. Am. Stat. Assoc. 65 (1970) 329] and the minimax reliability bound due to Barlow and Proschan [Statistical Theory of Reliability and Life Testing: Probability Models, 1981] for binary systems have been generalized to multistate systems by Block and Savits [J. Appl. Probab. 19 (1982) 391]. Some comparison results concerning the two multistate lower bounds for various types of multistate systems are given by Meng [Probab. Eng. Inform. Sci. 16 (2002) 485]. In this note we compare the two multistate upper bounds and present results which generalize some previous ones obtained by Maymin [J. Stat. Plan. Inference 16 (1987) 337] for binary systems. Examples are given to illustrate our results.

Monotone Log-Odds Rate Distributions in Reliability Analysis

Monotone failure rate models [Barlow Richard, E., Marshall, A. W., Proschan, Frank. (1963). Properties of probability distributions with monotone failure rate. Annals of Mathematical Statistics 34:375–389, and Barlow Richard, E., Proschan, Frank. (1965). Mathematical Theory of Reliability. New York: John Wiley & Sons, Barlow Richard, E., Proschan, Frank. (1966a). Tolerance and confidence limits for classes of distributions based on failure rate. Annals of Mathematical Statistics 37(6):1593–1601, Barlow Richard, E., Proschan, Frank. (1966b). Inequalities for linear combinations of order statistics from restricted families. Annals of Mathematical Statistics 37(6):1574–1592, Barlow Richard, E., Proschan, Frank. (1975). Statistical Theory of Reliability and Life Testing. New York: Holt, Rinehart and Winston, Inc.] have become one of the most important models of failure time for reliability practitioners to consider and use. The above authors also developed models and bounds for monotone increasing failure rates (IFR) and for monotone decreasing failure rates (DFR). The IFR models and bounds appear to be especially useful for describing and bounding the hazard of aging. This article extends a new model for time to failure based onthe log odds rate [Zimmer William, J., Wang Yao, Pathak, P. K. (1998). Log-odds rate and monotone log-odds rate distributions. Journal of Quality Technology 30(4):376–385.] which is comparable to the model based on the failure rate. It is shown that in the case of increasing log odds rate (ILOR) in terms of log time (ln t), the model is less stringent than the IFR model for aging. The characterization of distributions of failure time by log odds rate is also derived. It is shown that the logistic distribution has the property of constant log odds rate over time and that the log logistic distribution has the property of constant log odds rate with respect to ln t. Some other properties of ILOR distributions are presented and bounds based on the relationship to the log logistic distribution are provided for distributions which are ILOR with respect to ln t. Motivational examples are provided. The ILOR bounds are compared to the more stringent bounds based on the IFR model. Bounds on system reliability are also provided for certain systems.

The total time on test transform and the excess wealth stochastic orders of distributions

For nonnegative random variables X and Y we write X ≤TTT Y if ∫0 F -1(p)(1-F(x))dx ≤ ∫0 G -1(p)(1-G(x))dx all p ∈ (0,1), where F and G denote the distribution functions of X and Y respectively. The purpose of this article is to study some properties of this new stochastic order. New properties of the excess wealth (or right-spread) order, and of other related stochastic orders, are also obtained. Applications in the statistical theory of reliability and in economics are included.

The total time on test transform and the excess wealth stochastic orders of distributions

For nonnegative random variables X and Y we write X ≤TTTY if ∫0F-1(p)(1-F(x))dx ≤ ∫0G-1(p)(1-G(x))dx all p ∈ (0,1), where F and G denote the distribution functions of X and Y respectively. The purpose of this article is to study some properties of this new stochastic order. New properties of the excess wealth (or right-spread) order, and of other related stochastic orders, are also obtained. Applications in the statistical theory of reliability and in economics are included.

Global Availability and Reliability Assessment of the GPS and Galileo Global Navigation Satellite Systems

The performance of a Global Navigation Satellite System can be quantified by availability, accuracy, and reliability. The deployment of Galileo by the European Union will provide double the number of navigation satellites currently available to users and offer improvements in accuracy and reliability. This paper briefly describes the design of the proposed Galileo system in comparison with the existing Global Positioning System (GPS). Availability and accuracy are then defined and statistical reliability theory is reviewed. Worldwide availability, accuracy, and reliability estimates for GPS, Galileo, and the combined GPS/Galileo system are obtained by software simulation. The simulation results are presented using Horizontal and Vertical Dilution of Precision as measures of availability and accuracy and maximum Horizontal Position Error due to one undetected blunder as a measure of reliability. The validity of these quantities as measures of accuracy and reliability is discussed and compared with other me...

Domains of attraction of asymptotic reliability functions

Theorems about domains of attraction of limit distribution of extreme statistics of independent identically distributed random variables are well known and included in [R.E. Barlow, F. Proshan, Statistical Theory of Reliability of Life Testing, Probability Models, Holt. Reinehart and Winston, New York, 1975; H. Chernoff, H. Teicher, Trans. Amer. Math. Soc. 116 (1965) 474–491; B.W. Gnedenko, Ann. Math., 44 (1943) 432–453; B. Kopocinski, Zarys Teorii Odnowy i Niezawodnosci, PWN, Warszawa, 1973]. In [K. Kolowrocki, Appl. Math., 36 (1993) 56–69] a new four-element class of limit reliability functions for a regular, homogeneous series–parallel system in which the number of series components of the system has the order of logarithm of the number of parallel components is fixed. This paper presents the solution of the problem of domains of attraction for these asymptotic reliability functions. Moreover, some examples are given.

A note on some new renewal ageing notions

Many notions have been proposed to describe the ageing property of a lifetime. Some of the earlier developments can be found in Barlow and Proschan (Statistical Theory of Reliability and Life Testing, Holt, Reinhart and Winston, New York, 1975) and Gnedenko et al. (Mathematical Methods of Reliability Theory, Academic press, New York, London, 1969). Recently, Abouammoh et al. (Statist. Probab. Lett. 48 (2000) 189) introduced new classes arising from stochastic comparisons with the stationary renewal distribution. We show here that their new renewal better than used in expectation class contains only the exponential random variables.

ON THE MINIMAX VARIANCE ESTIMATORS OF SCALE IN TIME TO FAILURE MODELS

A scale parameter is the principal parameter to be estimated, since it corresponds to one of the main reliability characteristics, namely the average time to failure. To provide robustness of scale estimators to gross errors in the data, we apply the Huber minimax approach in time to failure models of the statistical reliability theory. The minimax valiance estimator of scale is obtained in the important particular case of the exponential distribution.

A remark on of limit reliability function of large series–parallel systems with assisting components

In the reliability investigation of large scale systems the problem of the complexity of their reliability functions arises. This problem may be approximately solved by assuming that the number of the system components tends to infinity and finding the limit reliability function of the system. The results for series, parallel, series–parallel and parallel–series systems are given in R.E. Barlow, F. Proschan, Statistical Theory of Reliability and Life Testing, Probability Models, Holt Rinehart and Winston, New York, 1975; K. Kołowrocki, On a Class of Limit Reliability Functions for Series–Parallel and Parallel–Series Systems, Monograph, Gdynia Maritime Academy Press, 1993; K. Kołowrocki, Reliability Engineering and System Safety 39 (1993) 11–23. In the paper as a partial extension of these results, theorems which determinate necessary and sufficient conditions for a reliability function to be an asymptotic reliability function of the series–parallel system with assisting components are given. The problem is solved for the system which the number of its series components is of order of the number of its parallel components. Moreover, some examples are presented.

On the Rate of Strong Consistency of the Total Time on Test Statistic

We give a complete description of the rate of strong consistency of the scaled and unscaled total time on test curves, which are fundamental notions in the statistical theory of reliability and life testing. The proof is crucially based on the general Vervaat process.