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.

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 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.

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.

scholarly journals Functional moments estimators analysis by the Monte-Carlo method for model of mixture with varying concentrations

2018 ◽  
Vol 14 (1) ◽  
pp. 7446-7451
Author(s):  
Kubaychuk Oksana
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Small Sample ◽  
Linear Estimator ◽  
Fixed Weight ◽  
The Monte Carlo Method ◽  
Random Weight ◽  
Improved Estimators ◽  
Small Sample Sizes ◽  
Moments Estimators

The functional moments estimation by the sample from the mixture with varying concentrations is studied. The problem of efficiency the simple linear estimator with fixed weight against the adaptive or improved estimators with random weight is considered. By the Monte-Carlo method it is shown that simple linear estimator is better for small sample sizes, but for large samples the adaptive and improved estimators are more efficient.

scholarly journals Stress-Strength Weibull Analysis with Different Shape Parameter β and Probabilistic Safety Factor

DYNA ◽  
2020 ◽  
Vol 87 (215) ◽  
pp. 28-33 ◽  
Author(s):  
Manuel Baro ◽  
Manuel Roman Piña Monarrez ◽  
Baldomero Villa
Keyword(s):  
Weibull Distribution ◽  
Safety Factor ◽  
Shape Parameter ◽  
Random Variable ◽  
Strength Analysis ◽  
Weibull Analysis ◽  
Shape Parameters ◽  
The Relationship ◽  
Close Solution ◽  
Probabilistic Safety

Since products are subjected to a random variable stress-strength, their reliability must be determined using the stress-strength analysis. Unfortunately, when both, stress and strength, follow a Weibull distribution with different shape parameters, the reliability stress-strength has not a close solution. Therefore, in this paper, the formulation to perform the analysis stress-strength Weibull with different shape parameters is derived. Furthermore, the formulation to determine the safety factor that corresponds to the designed reliability is also given. And because the relationship between the derived safety factor and the designed reliability is unique, then because reliability is random, the derived safety factor is random.

Evaluation methods for estimation of Weibull parameters used in Monte Carlo simulations for safety analysis of pressure vessels

2021 ◽  
Vol 63 (4) ◽  
pp. 379-385
Author(s):  
Bin Wang ◽  
Faisal Islam ◽  
Georg W. Mair
Keyword(s):  
Monte Carlo ◽  
Weibull Distribution ◽  
Monte Carlo Simulations ◽  
Sample Size ◽  
Least Squares ◽  
Test Data ◽  
Pressure Vessels ◽  
Small Sample ◽  
Least Squares Regression ◽  
Weibull Parameters

Abstract The test data for static burst strength and load cycle fatigue strength of pressure vessels can often be well described by Gaussian normal or Weibull distribution functions. There are various approaches which can be used to determine the parameters of the Weibull distribution function; however, the performance of these methods is uncertain. In this study, six methods are evaluated by using the criterion of OSL (observed significance level) from Anderson-Darling (AD) goodness of Fit (GoF), These are: a) the norm-log based method, b) least squares regression, c) weighted least squares regression, d) a linear approach based on good linear unbiased estimators, e) maximum likelihood estimation and f) method of moments estimation. In addition, various approaches of ranking function are considered. The results show that there are no outperforming methods which can be identified clearly, primarily due to the limitation of the small sample size of the test data used for Weibull analysis. This randomness resulting from the sampling is further investigated by using Monte Carlo simulations, concluding that the sample size of the experimental data is more crucial than the exact method used to derive Weibull parameters. Finally, a recommendation is made to consider the uncertainties of the limitations due to the small size for pressure vessel testing and also for general material testing.

Application of Confidence Regions to Ice Ridge Keel Data Statistical Assessment

2017 ◽  
Author(s):  
Petr Zvyagin ◽  
Jaakko Heinonen
Keyword(s):  
Confidence Region ◽  
Random Variable ◽  
Confidence Regions ◽  
Small Sample ◽  
Unknown Parameters ◽  
Minimum Area ◽  
Statistical Assessment ◽  
Gulf Of Bothnia ◽  
The Mean ◽  
Small Sample Sizes

Sets of measurements of underwater ridge parts usually contain a limited amount of data. Outcomes need to be made while relying on small sample sizes. In this event, the chance of making inaccurate estimations increases. This paper proposes to use stochastic confidence regions in the estimation of the unknown parameters of keel depths. A model for a random variable with a lognormal distribution for keel depths is assumed. Regions for the mean and standard deviation of keel depths are obtained from Mood’s and minimum-area confidence regions for parameters of the normally distributed random variable. Conservative safety probability of non-exceeding the critical keel depth in one random interaction of the ridge with structure is estimated. An algorithm for statistically assessment of ice ridge keel data by means of confidence region building is here offered. The assessment of a set of ridge keel depths for the Gulf of Bothnia (Baltic Sea) is performed.

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