Asymptotic Expansions to Improve Large Sample Confidence Intervals for System Reliability

Winterbottom, Alan

doi:10.2307/2335477

Cited by 4 publications

(9 citation statements)

References 7 publications

(8 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We will now briefly review the method in this section. For more details, the readers may use the work of Winterbottom as a reference.…”

Section: Wcf Asymptotic Expansion and Its Implementationmentioning

confidence: 99%

“…Using the same notations used in the work of Winterbottom, we denote the evaluated values of G 1 , H , and G 3 at

\overset{θ}{true}

g_{1}

, h , and

g_{3}

. Equations in are then equivalent to the following set of functions:

({, \begin{matrix} h = {((), \lim_{n \to \infty} n normalE {(\overset{true^}{ϕ} - ϕ)}^{2})}^{- \frac{1}{2}} \\ {sans-serifg}_{1} = - \frac{3}{2} \lim_{n \to \infty} n E H (\overset{true^}{ϕ} - ϕ) + \frac{1}{6} \lim_{n \to \infty} n^{2} E H^{3} {((), \overset{ϕ}{true} - ϕ)}^{3} \\ {sans-serifg}_{3} = h^{2} (\frac{1}{2} \lim_{n \to \infty} n^{2} E H^{3} {((), \overset{ϕ}{true} - ϕ)}^{3} - \frac{1}{6} \lim_{n \to \infty} n^{2} E H^{3} {((), \overset{ϕ}{true} - ϕ)}^{3}) . \end{matrix})

…”

Section: Wcf Asymptotic Expansion and Its Implementationmentioning

confidence: 99%

“…An example is the AO method presented in the work of Mann . Among all such approximation methods, the Winterbottom‐Cornish‐Fisher (WCF) expansion method, proposed in the work of Winterbottom, has demonstrated the advantage of being highly accurate. In the cases where sample size is not large enough for a normal large sample‐based approximation method, Winterbottom introduced an asymptotic expansion method for system reliability assessment.…”

Section: Introductionmentioning

confidence: 99%

“…Among all such approximation methods, the Winterbottom‐Cornish‐Fisher (WCF) expansion method, proposed in the work of Winterbottom, has demonstrated the advantage of being highly accurate. In the cases where sample size is not large enough for a normal large sample‐based approximation method, Winterbottom introduced an asymptotic expansion method for system reliability assessment. When system structure is known and the system reliability function can be represented as a function of model parameters of the components, a correction can be made to improve the asymptotic confidence limits and reduce the bias.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Reliability assessment for load‐sharing systems with exponential components using statistical expansion as a correction

Xie

2019

Appl Stoch Models Bus & Ind

View full text Add to dashboard Cite

Among recent system models, one specific type of system is generally used to model the dependence among components. Components are connected parallel in such systems as they fail one by one and are supposed to share the system work load. The model is thus referred to as the load‐sharing system model. Despite the availability of extensive reliability assessment methods for different systems, load‐sharing systems have not received enough attention from the scholars who have studied reliability assessment so far. Load‐sharing systems are generally designed for high levels of reliability. Therefore, tests for such systems can be expensive and time consuming. Limitation on resources always leads to small test sample sizes. This increases the difficulties associated with obtaining an accurate and robust system reliability assessment result. This paper proposes a novel assessment method for a certain type of load‐sharing system with components following exponential lifetime distributions. Based on the parameter estimation of the system reliability model, we introduce the Winterbottom‐Cornish‐Fisher asymptotic expansion method for implementing a correction of normal approximation. We demonstrate the accuracy of our method through a series of examples and simulation studies.

show abstract

“…We will now briefly review the method in this section. For more details, the readers may use the work of Winterbottom as a reference.…”

Section: Wcf Asymptotic Expansion and Its Implementationmentioning

confidence: 99%

“…Using the same notations used in the work of Winterbottom, we denote the evaluated values of G 1 , H , and G 3 at

\overset{θ}{true}

g_{1}

, h , and

g_{3}

. Equations in are then equivalent to the following set of functions:

({, \begin{matrix} h = {((), \lim_{n \to \infty} n normalE {(\overset{true^}{ϕ} - ϕ)}^{2})}^{- \frac{1}{2}} \\ {sans-serifg}_{1} = - \frac{3}{2} \lim_{n \to \infty} n E H (\overset{true^}{ϕ} - ϕ) + \frac{1}{6} \lim_{n \to \infty} n^{2} E H^{3} {((), \overset{ϕ}{true} - ϕ)}^{3} \\ {sans-serifg}_{3} = h^{2} (\frac{1}{2} \lim_{n \to \infty} n^{2} E H^{3} {((), \overset{ϕ}{true} - ϕ)}^{3} - \frac{1}{6} \lim_{n \to \infty} n^{2} E H^{3} {((), \overset{ϕ}{true} - ϕ)}^{3}) . \end{matrix})

…”

Section: Wcf Asymptotic Expansion and Its Implementationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Reliability assessment for load‐sharing systems with exponential components using statistical expansion as a correction

Xie

2019

Appl Stoch Models Bus & Ind

View full text Add to dashboard Cite

show abstract

“…Specifically, sometimes their results may produce negative values over certain intervals (see Changand Thompson [4]). Lampkin and Winterbottom [8] and Winterbottom [27] apply a Cornish-Fisher expansion to obtain percentiles of the system reliabilityy for a series system, but use an Edgeworth expansion for the posterior reliability pdf. Both expansions are in terms of orthogonal polynomials or functions.…”

Section: Related Literaturementioning

confidence: 99%

Exact bayesian estimation of system reliability from component test data

Tang

Moskowitz

1997

Naval Research Logistics

View full text Add to dashboard Cite

Asymptotic Expansions for Confidence Limits in the Presence of Nuisance Parameters, with Applications

Peers

Iqbal

1985

Journal of the Royal Statistical Society Series B: Statistical Methodology

View full text Add to dashboard Cite

SUMMARY Series expansions for confidence limits based on consistent estimators are obtained in the presence of nuisance parameters. The maximum likelihood estimator is treated as a special case and the necessary characteristics of its sampling distribution are obtained in terms of basic quantities derivable from the likelihood function. The theory is applied to the problem of setting separate confidence limits for the scale and shape parameters of the Weibull distribution. Comparisons are made with earlier simulation studies.

show abstract

Asymptotic Expansions to Improve Large Sample Confidence Intervals for System Reliability

Cited by 4 publications

References 7 publications

Reliability assessment for load‐sharing systems with exponential components using statistical expansion as a correction

Reliability assessment for load‐sharing systems with exponential components using statistical expansion as a correction

Exact bayesian estimation of system reliability from component test data

Asymptotic Expansions for Confidence Limits in the Presence of Nuisance Parameters, with Applications

Contact Info

Product

Resources

About