Some Inferences about Gamma Parameters with an Application to a Reliability Problem

Lentner, M. M.; Buehler, Robert J.

doi:10.2307/2282715

Cited by 10 publications

(4 citation statements)

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“…In this section we exploit the theory of exponential families to remove nuisance parameters when the probability models are negative binomial. Other reliability applications of the same theory have been made by Lentner and Buehler [10] for gamma models and by Harris [5] .…”

Section: Distribution Theory For Estimation Of Products and Quotientsmentioning

confidence: 87%

Confidence Intervals for Some Functions of Several Bernoulli Parameters with Reliability Applications

Hwang¹,

Buehler

1973

Journal of the American Statistical Association

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Section: Distribution Theory For Estimation Of Products and Quotientsmentioning

confidence: 87%

Confidence Intervals for Some Functions of Several Bernoulli Parameters with Reliability Applications

Hwang¹,

Buehler

1973

Journal of the American Statistical Association

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“…. , u k−1 ) has monotone LR property in the sense that for λ (1) > λ (2) , h λ (1) (t|u 1 , u 2 , . . .…”

Section: If and Only Ifmentioning

confidence: 99%

“…(i) Lentner and Buehler[2] considered the case for k = 2 only and suggested using U =T 1 − T 2 ; V = T 1 . They derived the conditional pdf and cdf of V |U = u given as h(v|u, λ) and H (v|u, λ), respectively, which involve incomplete gamma functions and are quite complicated.…”

mentioning

confidence: 99%

On the reliability of a multicomponent series system

Arazyan

Pal

2006

Journal of Statistical Computation and Simulation

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Consider a series system consisting of several independent components where the components are assumed to have exponential distributions with unknown and possibly unequal failure rates. The objective here is to infer about the reliability of the whole system at a given time t based on independent samples from each component. In this article, we first consider testing a hypothesis on the reliability of the whole system against a suitable alternative and try to follow the likelihood ratio (LR) method. However, closed expressions for the maximum likelihood estimators (MLEs) of the model parameters are not available under the null (alternative) hypothesis, which hinders the development of the traditional LR test method. To avoid this problem, we have found approximate MLEs of the model parameters under the null (and alternative) hypothesis, which are then used to derive an approximate likelihood ratio (ALR) statistic. Using the ALR statistic, we then find a conditional test procedure whose cut-off point can be found numerically subject to the size condition. Finally, the acceptance region of the test procedure is inverted to obtain a confidence interval for the system reliability.

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“…confidence bounds on the reliability, R ( t m ) , at time, t m , the probability that the system will survive at least until time, t m . (See Lentner and Buehler [27] and El Mawaziny [12]). No such optimum bounds have been found for a model which is equivalent to this exponential-failure-time series-system model.…”

mentioning

confidence: 99%

Computer-aided selection of prior distributions for generating monte carlo confidence bounds on system reliability

Mann

1970

Naval Research Logistics

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A description is given of results of preliminary investigations (by a group at North American Rockwell Corporation) related to the Monte Carlo generation of lower Confidence bounds on the reliability of a logically complex system. In calculating system confidence bounds by use of a Monte Carlo procedure, one must generate the distribution of each independent subsystem reliability, given the life-test failure data for that subsystem. Therefore, an assumption of a specified a priori distribution for each subsystem reliability is implicit in the procedure.In order that clues may be obtained as to optimum prior assumptions to be used in calculating Monte Carlo bounds for a complex system, the model has been restricted to a series system wherein each independent subsystem has exponentially distributed failure time and prototypes of each subsystem are tested until a fixed (but not necessarily the same for each subsystem) number of failures occurs. For this model, optimum (uniformly most accurate unbiased) exact classical confidence bounds on the reliability R ( t m ) at a specified mission time, tm, are available, although not easily calculated (El Mawaziny [12] and Lenter and Buehler [27]). Computer programs for calculating the optimum classical bounds and the Bayesian Monte Carlo bounds were written, and a means of numerically comparing various forms of prior distributions against an optimum standard was thus provided. One prior distribution widely used in obtaining Monte Carlo and general Bayesian exact lower confidence bounds on system reliability is thereby shown numerically to yield bounds which are conservative in the classical sense for this series-system model. Another suggested prior distribution is shown to give bounds which are usually conservative but under certain conditions are liberal, and hence not truly confidence bounds. Moreover, it is demonstrated by a combination of numerical and analytical results, that for a series system containing more than one independent subsystem there exists no prior distribution for subsystem reliability which is independent of the data and which yields the optimum lower bounds. Other numerical results related to the selection of optimum methods for generating the bounds and evaluation of certain approximate methods are described. BACKGROUND AND APPROACH Review of Pertinent LiteratureIf it is possible to determine confidence bounds on system reliability solely from the testing of the subsystems of which the system is comprised, saving of expensive system testing can be effected. It may, in fact sometimes be infeasible to test the system as a whole. Furthermore, this method of obtaining system confidence bounds can be used for exploratory system design.The subject of confidence bounds for system reliability from subsystem testing is one about which much has been written, but not a great deal is known. Consider a series system in which the failure *

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Some Inferences about Gamma Parameters with an Application to a Reliability Problem

Cited by 10 publications

References 0 publications

Confidence Intervals for Some Functions of Several Bernoulli Parameters with Reliability Applications

Confidence Intervals for Some Functions of Several Bernoulli Parameters with Reliability Applications

On the reliability of a multicomponent series system

Computer-aided selection of prior distributions for generating monte carlo confidence bounds on system reliability

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