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

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

1. Weibull W. “A statistical theory of the strength of materials”. Royal Swedish Institute of Engineering Research (Ingenioersvetenskaps Akad. Handl.), 153, 1-55, 1939.
2. Jiang R, Murthy DNP. “A study of Weibull shape parameter: properties and significance”. Reliability Engineering and System Safety, 96(12), 1619-1626, 2011.
3. Weibull W. “A statistical distribution function of wide applicability”. Journal of Applied Mechanics, 103, 293-297, 1951.
4. Lu C, Danzer R, Fischer FD. “Influence of threshold stress on the estimation of the Weibull statistics”. Journal of the American Ceramic Society, 85(6), 1640-1642, 2002.
5. Alqam M, Bennett RM, Zureick AH. “Three-parameter vs. two-parameter Weibull distribution for pultruded composite material properties”. Composite Structures, 58 (4), 497-503, 2002.
6. Ambrozic M, Gorjan L, Gomilsek M. “Bend strength variation of ceramics in serial fabrication”. Journal of the European Ceramic Society, 34(7), 1873-1879, 2014.
7. Nohut S. “Influence of sample size on strength distribution of advanced ceramics”. Ceramics International, 40(3), 4285-4295, 2014.
8. Bütikofera L, Stawarczykb B, Roos M. “Two regression methods for estimation of a two-parameter Weibull distribution for reliability of dental materials”. Dental Materials, 31(2), 33-50, 2015.

9. Edwards DJ, Guess FM, Young TM. “Improved estimation of the lower percentiles of material properties”. Wood Science and Technology, 45(3), 533-546, 2011.
10. Fernandez-Saez J, Chao J, Duran J, Amo JM. “Estimating lower-bound fracture parameters for brittle materials”. Journal of Materials Science Letters, 12, 1493-1496, 1993.
11. Phan LD, McCool JI. “Exact confidence intervals for Weibull parameters and percentiles”. Proceedings of the Institution of Mechanical Engineers Part O: Journal of Risk and Reliability, 223(4), 387-394, 2009.
12. Birgören B. Effect of sample size and distribution parameters in estimation of confidence lower bounds for Weibull percentiles. Editör: Caruta BM. Ceramics and Composite Materials: New Research, 215-236, New York, USA, Nova Science Publishers, 2006.
13. Casella G, Berger RL. Statistical Inference. 2nd ed. Belmont, USA, Duxbury Press, 2001.
14. Rinne H. The Weibull Distribution: A Handbook. Boca Raton, USA, CRC Press, 2009.
15. Barbero E, Fernandez-Saez J, Navarro C. “On the estimation of percentiles of the Weibull distribution”. Journal of Materials Science Letters, 18(17), 1441-1443, 1999.
16. Barbero E, Fernandez-Saez J, Navarro C. “Statistical analysis of the mechanical properties of composite materials”. Composites: Part B, 31(5), 375-381, 2000.
17. Birgören B, Dirikolu MH. “A computer simulation for estimating lower-bound fracture strength of composites using Weibull distribution”. Composites Part B: Engineering, 35(3), 263-266, 2004.
18. McCool JI. “Estimation and inference in sets of Weibull samples”. Proceedings of the Institution of Mechanical Engineers Part O: Journal of Risk and Reliability, 228(2), 115-126, 2014.
19. Song L, Wu D, Li Y. “Optimal probability estimators for determining Weibull parameters”. Journal of Materıals Science Letters, 22(23), 1651-1653, 2003.
20. Wu D, Zhou J, Li Y. “Unbiased estimation of Weibull parameters with the linear regression method”. Journal of the European Ceramic Society, 26 (7), 1099-1105, 2006.
21. Heo JH, Salas JD, Kim KD. “Estimation of confidence intervals of quantiles for the Weibull distribution”. Stochastic Environmental Research and Risk Assessment, 15(4), 284-309, 2001.
22. Padgett WJ. Tomlinson M. “Lower confidence bounds for percentiles of Weibull and Birnbaum-saunders distributions”. Journal of Statistical Computation and Simulation, 73(6), 429-443, 2003.
23. Ho LL, Silva AF. “Unbiased estimators for mean time to failure and percentiles in a Weibull regression model”. The International Journal of Quality & Reliability Management, 23(3), 323-339, 2006.
24. Yang Z, Xie M, Wong ACM. “A unified confidence interval for reliability-related quantities of two-parameter Weibull distribution”. Journal of Statistical Computation and Simulation, 77(5), 365-378, 2007.
25. Hudak D, Tiryakioğlu M. “On estimating percentiles of the Weibull distribution by the linear regression method”. Journal of Materials Science, 44(8), 1959-1964, 2009.
26. Lv S, Niu Z, Wang G, Qua L, Hea Z. “Lower percentile estimation of accelerated life tests with nonconstant scale parameter”. Quality and Reliability Engineering International, 33(7), 1437-1446, 2017.
27. Young TM, León RV, Chen CH, Chen W, Guess FM, Edwards DJ. “Robustly estimating lower percentiles when observations are costly”. Quality Engineering, 27(3), 361-373, 2015.
28. Yalçınkaya M, Birgören B, “Confidence interval estimation of Weibull lower percentiles in small samples via bayesian inference”. Journal of the European Ceramic Society, 37, 2983-2990, 2017.
29. Trustrum K, Jayatilaka S. “On estimating the Weibull modulus for on estimating the Weibull modulus for a brittle material”. Journal of Materials Science, 14, 1080-1084, 1979.
30. Khalili A, Kromp K. “Statistical properties of Weibull estimators”. Journal of Materials Science, 26(24), 6741-6752, 1991.
31. Abernethy R. The new Weibull handbook. 5th ed. Florida, USA, Dr. Robert B. Abernethy, 2009.
32. McCabe JF, Carrick TE. “A statistical approach to the mechanical testing of dental materials”. Dental Materials, 2(4), 139-142, 1986.
33. Menon MV. “Estimation of the shape and scale parameters of the Weibull distribution”. Technometrics, 5(2), 175-182, 1963.
34. Shao J. Mathematical Statistics. New York, USA, Springer, 2003.