Estimating confidence lower bounds of Weibull lower percentiles with small samples in material reliability analysis

Yıl 2020, Cilt: 26 Sayı: 1, 184 - 194, 20.02.2020

Meryem Yalçınkaya Burak Birgören

Öz

Weibull distribution is widely used in the modeling of mechanical properties such as tensile strength of ceramic and composite materials. The 95% one-sided confidence lower bounds on the 1st and 10th Weibull percentiles, namely A-basis and B-basis material properties, are important in reliability studies for understanding early failures and reducing risks. These lower bounds are generally estimated by small samples due to the high costs of the experiments, hence the precision of estimation remain low. Therefore, in the literature, many exact and approximate interval estimation methods for Weibull percentiles have been proposed for achieving better performance. In this study, a comprehensive comparison of the exact methods with Monte-Carlo simulations has been made. In addition, some methods developed for Weibull parameters are also included in this comparison since they can be used for exact lower bound estimation but have never used for this purpose in the literature. In the study, the lower bounds have been estimated by the maximum likelihood method, the Menon method and 25 different models of weighted/ unweighted least squares methods (such as improved estimators, interchanged axes), and average false coverage probabilities are used for the comparison criterion. According to the simulation results, the maximum likelihood and the weighted least squares method with Faucher & Tyson weight factors have very similar performances for sample sizes less than 8; and the maximum likelihood method has always shown the best performance for sample sizes greater than or equal to 20. However, it is emphasized that linear regression methods are more practical in terms of ease of calculation when performance differences are negligible

Anahtar Kelimeler

Weibull lower percentiles, A and B basis material properties, Reliability

Malzeme güvenilirlik analizinde Weibull alt yüzdeliklerinin alt güven sınırının küçük örneklemlerle tahmini

Öz

Weibull dağılımı seramik ve kompozit malzemelerin kopma mukavemeti gibi mekanik özelliklerinin modellenmesinde yaygın şekilde kullanılır. Mekanik özelliklerin %95 güven düzeyinde 1. yüzdelik ve 10. yüzdelik alt güven sınırları, diğer adıyla A-Temel ve B-Temel malzeme özellikleri, güvenilirlik çalışmalarında erken arızaları anlama ve risk azaltma için önemlidir. Bu alt sınırlar, deneylerin yüksek maliyetleri nedeniyle genellikle küçük örneklemlerle, dolayısıyla düşük hassasiyet düzeyleriyle tahmin edilir. Bu nedenle literatürde daha iyi tahminler gerçekleştirmeyi amaçlayan birçok kesin ve yaklaşık alt sınır tahmin yöntemi önerilmiştir. Bu çalışmada kesin yöntemlerin Monte-Carlo benzetimleriyle kapsamlı bir karşılaştırması yapılmıştır. Ayrıca, Weibull parametreleri için geliştirilen bazı yöntemler kesin alt sınır tahmininde kullanılabilir olmakla birlikte literatürde bu amaçla hiç kullanılmadığından bu karşılaştırma kapsamına alınmıştır. Çalışmada alt sınırlar, maksimum benzerlik, Menon yöntemi ve ağırlıklı/ağırlıksız en küçük kareler yöntemlerinin 25 farklı modeli (geliştirilmiş tahminleyiciler, değiştirilmiş eksenler gibi) ile tahmin edilmiş ve sonuçlar, ortalama yanlış kapsama olasılığı kriterine göre karşılaştırılmıştır. Benzetim sonuçlarına göre, hacmi 20’den küçük örneklemlerde maksimum benzerlik ile Faucher & Tyson ağırlık faktörlü en küçük kareler yöntemlerinin çok benzer performansa sahip olduğu, hacmi 20 ve daha büyük örneklemlerde ise maksimum benzerlik yönteminin her zaman daha iyi olduğu gözlemlenmiştir. Bununla birlikte, performans farklarının ihmal edilebilir olduğu durumlarda hesaplama kolaylığı açısından en küçük kareler yöntemlerinin daha pratik olduğu vurgulanmıştır.

Anahtar Kelimeler

Weibull alt yüzdelikler, A-Temel ve B-Temel malzeme özellikleri, Güvenilirlik

Kaynakça

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