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Öğe Bayesian Confidence Interval Estimation of Weibull Modulus Under Increasing Failure Rate(Gazi Univ, 2021) Yalcinkaya, Meryem; Birgoren, BurakEstimating the confidence interval of the Weibull modulus is an important problem in the fracture strength modeling of ceramic and composite materials. It is particularly important in cases where the sample size is small due to high experimental costs. For this purpose, several classical methods, including the popular maximum likelihood method, and Bayesian methods have been developed in the literature. However, studies on Bayesian inference have remained very limited in the materials science literature. Recently a Bayesian Weibull model has been proposed for estimating confidence lower bounds for Weibull percentiles using the prior knowledge that the failure rates are increasing. This prior argument requires the Weibull modulus to be more than 1 due to wear-out failure. In this study, under the same prior information, two Bayesian Weibull models, one using the same prior argument and the other a relaxed version of it, have been developed for confidence interval estimation of the Weibull modulus. Their estimation performances have been compared against the maximum likelihood method with Monte Carlo simulations. The results show that the Bayesian Weibull models significantly outperform the maximum likelihood method for almost all Weibull modulus and sample size values.Öğe Confidence interval estimation of Weibull lower percentiles in small samples via Bayesian inference(Elsevier Sci Ltd, 2017) Yalcinkaya, Meryem; Birgoren, BurakWeibull distribution has been vastly used for modeling fracture strength of ceramic and composite materials. Confidence interval estimation of Weibull parameters and percentiles in small samples has been an important concern due to high experimental costs. It was shown previously that in classical inference the Maximum Likelihood Estimation Method is the best method among several alternatives for estimating 95% one-sided confidence lower bounds on the 1st and 10th Weibull percentiles, namely A-basis and B-basis material properties. This study proposes the Bayesian Weibull Method as an alternative using the information that ceramic and composite materials have increasing failure rates, which requires the Weibull shape parameter to be at least 1. Through Monte Carlo simulations, it is shown that the performance of the Bayesian Weibull Method is superior in that it achieves the precision levels of the Maximum Likelihood Estimation Method with significantly smaller sample sizes. (C) 2017 Elsevier Ltd. All rights reserved.Öğe Estimation of Confidence Intervals and Lower Bounds for Various Weibull Percentiles(World Scientific Publ Co Pte Ltd, 2023) Birgoren, Burak; Yalcinkaya, MeryemThe design allowables are derived statistically from measured material properties, and the Weibull distribution is one of the most commonly used distributions for statistical modeling. A- and B-basis design allowables are frequently used; they correspond to the confidence lower bounds for the 1st and 10th percentiles, respectively, with a confidence level of 95%. The maximum likelihood method is generally recommended and commonly used for parameter and confidence lower bound estimation. On the other hand, designers are also interested in confidence lower bounds for other percentiles, and in general, confidence intervals to specify uncertainty in percentile estimates. Monte-Carlo simulation methods have been proposed for this purpose; however, they are not easy to code and take a long time to run to obtain reliable results. As an easy-to-use alternative, this study proposes approximate polynomial functions of sample size for various percentiles and confidence levels. The coefficients of the functions are presented in tabular form for each combination of percentiles and confidence levels. They eliminate the need for simulations and provide precise confidence intervals and lower bounds for a large set of Weibull percentiles.Öğe Shortest Confidence Intervals of Weibull Modulus for Small Samples in Materials Reliability Analysis(Gazi Univ, 2023) Yalcinkaya, Meryem; Birgoren, BurakThe Weibull distribution has been widely used to model strength properties of brittle materials. Estimation of confidence intervals for Weibull shape parameter has been an important concern, since small sample sizes in materials science experiments bring about large intervals. Many methods have been proposed in the literature for constructing shorter intervals; the methods of maximum likelihood, least square, and Menon are among the most extensively studied methods. However, they all use an equal-tails approach. The pivotal quantities used for constructing confidence intervals have right-skewed and unimodal distributions, thus, they clearly do not produce the shortest intervals for a given confidence level in equal tail form. This study constructs the shortest confidence intervals for the three aforementioned methods and compares their performances by their equal-tails counterparts. To this end, a comprehensive simulation study has been conducted for the shape parameter values between 1 to 80 and the sample sizes between 3 to 20. The comparison criterion is chosen as the expected interval length. The results show that the shortest confidence intervals in each of three methods have yielded considerably narrower intervals. Further, the unknown parameter values are more centered in these intervals.
