scholarly journals Weibull Failure Probability Estimation Based on Zero-Failure Data

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Ping Jiang ◽  
Yunyan Xing ◽  
Xiang Jia ◽  
Bo Guo
Keyword(s):  
Weibull Distribution ◽  
Failure Probability ◽  
Development Time ◽  
Shape Parameter ◽  
Distribution Curve ◽  
Small Sample ◽  
Reliability Testing ◽  
Failure Data ◽  
Curve Method ◽  
Small Sample Sizes

Reliability testing is often carried out with small sample sizes and short duration because of increasing costs and the restriction of development time. Therefore, for highly reliable products, zero-failure data are often collected in such tests, which could not be used to evaluate reliability by traditional methods. To cope with this problem, the match distribution curve method was proposed by some researchers. The key step needed to exercise this method is to estimate the failure probability, which has yet to be solved in the case of the Weibull distribution. This paper presents a method to estimate the intervals of failure probability for the Weibull distribution by using the concavity or convexity and property of the distribution function. Furthermore, to use the method in practice, this paper proposes using the approximate value of the shape parameter determined by either engineering experience or by hypothesis testing through apvalue. The estimation of the failure probability is thus calculated using a Bayesian approach. A numerical example is presented to validate the effectiveness and robustness of the method.

scholarly journals A Simple Normal Approximation for Weibull Distribution with Application to Estimation of Upper Prediction Limit

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
H. V. Kulkarni ◽  
S. K. Powar
Keyword(s):  
Monte Carlo ◽  
Weibull Distribution ◽  
Shape Parameter ◽  
Normal Approximation ◽  
Random Variable ◽  
Industrial Applications ◽  
Small Sample ◽  
Prediction Limit ◽  
Coverage Probabilities ◽  
Small Sample Sizes

We propose a simple close-to-normal approximation to a Weibull random variable (r.v.) and consider the problem of estimation of upper prediction limit (UPL) that includes at leastlout ofmfuture observations from a Weibull distribution at each ofrlocations, based on the proposed approximation and the well-known Box-Cox normal approximation. A comparative study based on Monte Carlo simulations revealed that the normal approximation-based UPLs for Weibull distribution outperform those based on the existing generalized variable (GV) approach. The normal approximation-based UPLs have markedly larger coverage probabilities than GV approach, particularly for small unknown shape parameter where the distribution is highly skewed, and for small sample sizes which are commonly encountered in industrial applications. Results are illustrated with a real dataset for practitioners.

Modelling macroparasite aggregation using a nematode-sheep system: the Weibull distribution as an alternative to the Negative Binomial distribution?

Parasitology ◽  
2005 ◽  
Vol 131 (3) ◽  
pp. 393-401 ◽  
Author(s):  
S. GABA ◽  
V. GINOT ◽  
J. CABARET
Keyword(s):  
Weibull Distribution ◽  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Negative Binomial ◽  
Information Criterion ◽  
Gastrointestinal Nematode ◽  
Small Sample ◽  
Parameter Estimations ◽  
Wide Range ◽  
Small Sample Sizes

Macroparasites are almost always aggregated across their host populations, hence the Negative Binomial Distribution (NBD) with its exponent parameter k is widely used for modelling, quantifying or analysing parasite distributions. However, many studies have pointed out some drawbacks in the use of the NBD, with respect to the sensitivity of k to the mean number of parasites per host or the under-representation of the heavily infected hosts in the estimate of k. In this study, we compare the fit of the NBD with 4 other widely used distributions on observed parasitic gastrointestinal nematode distributions in their sheep host populations (11 datasets). Distributions were fitted to observed data using maximum likelihood estimator and the best fits were selected using the Akaike's Information Criterion (AIC). A simulation study was also conducted in order to assess the possible bias in parameter estimations especially in the case of small sample sizes. We found that the NBD is seldom the best fit for gastrointestinal nematode distributions. The Weibull distribution was clearly more appropriate over a very wide range of degrees of aggregation, mainly because it was more flexible in fitting the heavily infected hosts. Moreover, the Weibull distribution estimates are less sensitive to sample size. Thus, when possible, we suggest to carefully check on observed data if the NBD is appropriate before conducting any further analysis on parasite distributions.

Small sample discrete reliability growth modeling using a grey systems model

2018 ◽  
Vol 8 (3) ◽  
pp. 246-271 ◽  
Author(s):  
Thomas Paul Talafuse ◽  
Edward A. Pohl
Keyword(s):  
Growth Model ◽  
Small Sample ◽  
System Level ◽  
Parameter Estimates ◽  
Reliability Growth ◽  
Sample Sizes ◽  
Content Type ◽  
Failure Data ◽  
Amsaa Model ◽  
Small Sample Sizes

PurposeWhen performing system-level developmental testing, time and expenses generally warrant a small sample size for failure data. Upon failure discovery, redesigns and/or corrective actions can be implemented to improve system reliability. Current methods for estimating discrete (one-shot) reliability growth, namely the Crow (AMSAA) growth model, stipulate that parameter estimates have a great level of uncertainty when dealing with small sample sizes. The purpose of this paper is to present an application of a modified GM(1,1) model for handling system-level testing constrained by small sample sizes.Design/methodology/approachThe paper presents a methodology for incorporating failure data into a modified GM(1,1) model for systems with failures following a poly-Weibull distribution. Notional failure data are generated for complex systems and characterization of reliability growth parameters is performed via both the traditional AMSAA model and the GM(1,1) model for purposes of comparing and assessing performance.FindingsThe modified GM(1,1) model requires less complex computational effort and provides a more accurate prediction of reliability growth model parameters for small sample sizes and multiple failure modes when compared to the AMSAA model. It is especially superior to the AMSAA model in later stages of testing.Originality/valueThis research identifies cost-effective methods for developing more accurate reliability growth parameter estimates than those currently used.

scholarly journals Estimation of the Shape Parameter of a Wear-Out Failure Period for a Three-Parameter Weibull Distribution in a Small Sample

2020 ◽  
Vol 9 (6) ◽  
pp. 39
Author(s):  
Toru Ogura ◽  
Takatoshi Sugiyama ◽  
Nariaki Sugiura
Keyword(s):  
Monte Carlo ◽  
Weibull Distribution ◽  
Sample Size ◽  
Unbiased Estimator ◽  
Shape Parameter ◽  
Mean Squared Error ◽  
Minimum Variance ◽  
Small Sample ◽  
Scale Parameters ◽  
Squared Error

We propose a method to estimate a shape parameter for a three-parameter Weibull distribution. The proposed method first derives an unbiased estimator for the shape parameter independent of the location and scale parameters and then estimates the shape parameter using a minimum-variance linear unbiased estimator. Since the proposed method is expressed using a hyperparameter, its optimal hyperparameter is searched using Monte Carlo simulations. The recommended hyperparameter used for estimating the shape parameter depends on the sample size, and this causes no problems since the sample size is known when data is obtained. The proposed method is evaluated using a bias and a root mean squared error, and the results are very promising when the population shape parameter is 2 or more in the Weibull distribution representing the wear-out failure period. A numerical dataset is analyzed to demonstrate the practical use of the proposed method.

scholarly journals Reliability Assessment for Very Few Failure Data and Weibull Distribution

2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Lulu Zhang ◽  
Guang Jin ◽  
Yang You
Keyword(s):  
Distribution Function ◽  
Confidence Interval ◽  
Weibull Distribution ◽  
Failure Probability ◽  
Evaluation Method ◽  
High Reliability ◽  
Point Estimation ◽  
Engineering Practice ◽  
Reliability Parameters ◽  
Failure Data

Only very few failure data can be obtained for the time censored test of high-reliability and long-life products. For very few failure data, the current methods fail to obtain both the point estimation and confidence interval of reliability parameters. If the point estimation and confidence interval of reliability parameters are obtained based on different methods, the results tend to be unreliable. In this study, based on the existing research, a Bayesian reliability evaluation method for very few failure data under the Weibull distribution was proposed. First, the range of failure probability was limited based on the convexity and self-features of the Weibull distribution function. Second, based on the background of the sample with very few failure data, the pretest distribution function and parameters were set and solved. The point estimation and confidence interval model of failure probability based on the Bayesian formula was established. The improved match distribution curve method was used to compute both the point estimation and confidence interval of reliability parameters. Furthermore, by comparing the results of numerical examples, the calculation results obtained by the proposed method were verified as being very reasonable. Finally, taking wet friction plates as an example, the results showed the effectiveness of this method in engineering practice.

scholarly journals Aproksimasi Numerik Belah Dua dan Newton-Raphson pada Estimasi Parameter Distribusi Weibul

CAUCHY ◽  
2013 ◽  
Vol 2 (4) ◽  
pp. 193
Author(s):  
Indira P Kinasih
Keyword(s):  
Weibull Distribution ◽  
Numerical Approximation ◽  
Shape Parameter ◽  
Likelihood Function ◽  
Life Time ◽  
Life Distribution ◽  
Approximation Result ◽  
Failure Data ◽  
Newton Raphson ◽  
The Right

Life time distribution of warranted product, for example like automobile and photocopier is an interesting studies. Based on the fact that mostly, life distribution of warranted product is somewhat skewed to the right, this suggests that usual failure distribution such as Weibull distribution may provide a reasonable fit to the data. Related to these studies, parameter estimation is directly applicable since life distribution function is a function of parameters. There are no closed solution for Weibull likelihood function maximation. Therefore, numerical approximation could be an alternative solution. In this paper, Newton-Raphson and bisection method were employed to estimate the scale and shape parameter of Weibull distribution for photocopier failure data for since 4,5 years. Performance of those method were compared based on their approximation result and how fast they get into the solution. Data were taken from Bulmer and Eccleston (2003) research about photocopier realibility model.

Integrating Visual and Bayesian Statistical Analyses in Single Case Experimental Research to Evaluate the Effectiveness and Magnitude of a Comprehensive Behavioral Intervention

2018 ◽  
Author(s):  
Prathiba Natesan ◽  
Smita Mehta
Keyword(s):  
Behavioral Intervention ◽  
Effect Size ◽  
Rate Ratio ◽  
Visual Analysis ◽  
Single Case ◽  
Small Sample ◽  
Statistical Analyses ◽  
Data Types ◽  
Ratio Effect ◽  
Small Sample Sizes

Single case experimental designs (SCEDs) have become an indispensable methodology where randomized control trials may be impossible or even inappropriate. However, the nature of SCED data presents challenges for both visual and statistical analyses. Small sample sizes, autocorrelations, data types, and design types render many parametric statistical analyses and maximum likelihood approaches ineffective. The presence of autocorrelation decreases interrater reliability in visual analysis. The purpose of the present study is to demonstrate a newly developed model called the Bayesian unknown change-point (BUCP) model which overcomes all the above-mentioned data analytic challenges. This is the first study to formulate and demonstrate rate ratio effect size for autocorrelated data, which has remained an open question in SCED research until now. This expository study also compares and contrasts the results from BUCP model with visual analysis, and rate ratio effect size with nonoverlap of all pairs (NAP) effect size. Data from a comprehensive behavioral intervention are used for the demonstration.

No Evidence that Experiencing Physical Warmth Promotes Interpersonal Warmth: Two Failures to Replicate Williams and Bargh (2008)

2018 ◽  
Author(s):  
Christopher Chabris ◽  
Patrick Ryan Heck ◽  
Jaclyn Mandart ◽  
Daniel Jacob Benjamin ◽  
Daniel J. Simons
Keyword(s):  
Null Hypothesis ◽  
Small Sample ◽  
Sample Sizes ◽  
Double Blind ◽  
Bayesian Analyses ◽  
Physical Warmth ◽  
Small Sample Sizes ◽  
Interpersonal Warmth

Williams and Bargh (2008) reported that holding a hot cup of coffee caused participants to judge a person’s personality as warmer, and that holding a therapeutic heat pad caused participants to choose rewards for other people rather than for themselves. These experiments featured large effects (r = .28 and .31), small sample sizes (41 and 53 participants), and barely statistically significant results. We attempted to replicate both experiments in field settings with more than triple the sample sizes (128 and 177) and double-blind procedures, but found near-zero effects (r = –.03 and .02). In both cases, Bayesian analyses suggest there is substantially more evidence for the null hypothesis of no effect than for the original physical warmth priming hypothesis.

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