Unit-Weibull Distribution: Different Method of Estimations

Yıl 2023, Cilt: 13 Sayı: 2, 547 - 560, 15.06.2023

Hasan Hüseyin Gül

https://doi.org/10.31466/kfbd.1239446

Öz

Recently, the unit-Weibull (UW) distribution is used quite effectively in analyzing lifetime data. The main goal of this article is to investigate the performance of seven estimation methods, namely maximum likelihood (ML), least square (LS), weighted least square (WLS), Anderson-Darling (AD), right-tail Anderson-Darling (RAD), Cramer-von-Mises (CVM) and percentile (PCE) for parameter estimation. An extensive Monte Carlo simulation study is considered to compare the performances of these methods through biases and mean square errors (MSEs). The numerical results show that the PCE estimator has significantly smaller MSE value for different sample sizes and parameter values in most cases. In addition, the ML and LS estimators have lower bias values than the other estimators in general. Finally, a real data set is presented for illustrative purposes.

Anahtar Kelimeler

Unit-Weibull distribution, Estimation methods, Maximum likelihood, Anderson-Darling, Least square

Unit-Weibull Dağılımı: Tahmin Metotları

Öz

Son zamanlarda Unit-Weibull (UW) dağılımı yaşam zamanı verilerin analizinde oldukça etkin bir şekilde kullanılmaktadır. Bu makalenin temel amacı, en çok olabilirlik (ML), en küçük kareler (LS), ağırlıklı en küçük kareler (WLS), Anderson-Darling (AD), sağ kuyruklu Anderson-Darling (RAD), Cramer-von-Mises (CVM) and percentile (PCE) olmak üzere yedi tahmin yönteminin performansını karşılaştırmaktır. Bu yöntemlerin performanslarını yan ve hata kare ortalaması (MSE'ler) aracılığıyla karşılaştırmak için kapsamlı bir Monte Carlo simülasyon çalışması düşünülmüştür. Sayısal sonuçlar, PCE tahmin edicisinin çoğu durumda farklı örneklem büyüklükleri ve parametre değerleri için önemli ölçüde daha küçük MSE değerine sahip olduğunu göstermektedir. Ayrıca ML ve LS tahmin edicileri genel olarak diğer tahmin edicilere göre daha düşük yan değerlerine sahiptir. Son olarak, açıklama amacıyla gerçek bir veri seti sunulmuştur.

Anahtar Kelimeler

Unit-Weibull dağılımı, En çok olabilirlik, Anderson-Darling, En küçük kareler

Kaynakça

• Cordeiro, G. M., Ortega, E. M., and Nadarajah, S. (2010). The Kumaraswamy Weibull distribution with application to failure data. Journal of The Franklin Institute, 347(8), 1399-1429.
• Dumonceaux, R., and Antle, C. E. (1973). Discrimination between the log-normal and the Weibull distributions. Technometrics, 15(4), 923-926.
• D’Agostino, R. B., and Stephens, M. A. (1986). Goodness-of-Fit Techniques. Dekker, New York, NY.
• Kao, J. H. (1958). Computer methods for estimating Weibull parameters in reliability studies. IRE Transactions on Reliability and Quality Control, 15-22.
• Kao, J. H. (1959). A graphical estimation of mixed Weibull parameters in life-testing of electron tubes. Technometrics, 1(4), 389-407.
• Khan, M. S., and King, R. (2013). Transmuted modified Weibull distribution: A generalization of the modified Weibull probability distribution. European Journal of Pure and Applied Mathematics, 6(1), 66-88.
• Khan, M. S., King, R., and Hudson, I. L. (2017). Transmuted Weibull distribution: Properties and estimation. Communications in Statistics-Theory and Methods, 46(11), 5394-5418.
• Lee, C., Famoye, F., and Olumolade, O. (2007). Beta-Weibull distribution: some properties and applications to censored data. Journal of Modern Applied Statistical Methods, 6(1), 17.
• Luceño, A. (2006). Fitting the generalized Pareto distribution to data using maximum goodness-of-fit estimators. Computational Statistics & Data Analysis, 51(2), 904-917.
• Marshall, A. W., and Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika, 84(3), 641-652.
• Mazucheli, J., Menezes, A. F. B., and Dey, S. (2018). Improved maximum-likelihood estimators for the parameters of the unit-gamma distribution. Communications in Statistics-Theory and Methods, 47(15), 3767-3778.
• Mazucheli, J., Menezes, A. F. B., and Dey, S. (2018). The unit-Birnbaum-Saunders distribution with applications. Chilean Journal of Statistics, 9(1), 47-57.
• Mazucheli, J., Menezes, A. F. B., and Ghitany, M. E. (2018). The unit-Weibull distribution and associated inference. Journal of Applied Probability and Statistics, 13(2), 1-22.
• Mazucheli, J., Menezes, A. F. B., and Dey, S. (2019). Unit-Gompertz distribution with applications. Statistica, 79(1), 25-43.
• Mazucheli, J., Menezes, A. F. B., and Chakraborty, S. (2019). On the one parameter unit-Lindley distribution and its associated regression model for proportion data. Journal of Applied Statistics, 46(4), 700-714.
• Mudholkar, G. S., and Srivastava, D. K. (1993). Exponentiated Weibull family for analyzing bathtub failure-rate data. IEEE Transactions on Reliability, 42(2), 299-302.
• Rinne, H. (2008). The Weibull distribution: a handbook. Chapman and Hall/CRC.
• Silva, G. O., Ortega, E. M., and Cordeiro, G. M. (2010). The beta modified Weibull distribution. Lifetime Data Analysis, 16(3), 409-430.
• Singla, N., Jain, K., and Sharma, S. K. (2012). The beta generalized Weibull distribution: properties and applications. Reliability Engineering & System Safety, 102, 5-15.
• Swain, J. J., Venkatraman, S., and Wilson, J. R. (1998). Least-squares estimation of distribution functions in Johnson's translation system. Journal of Statistical Computation and Simulation, 29(4), 271-297.
• Weibull, W. (1951). A statistical distribution function of wide applicability. Journal of Applied Mechanics, 18(3), 293-297.
• Xie, M., Tang, Y., and Goh, T. N. (2002). A modified Weibull extension with bathtub-shaped failure rate function. Reliability Engineering & System Safety, 76(3), 279-285.
• Zhang, T., and Xie, M. (2011). On the upper truncated Weibull distribution and its reliability implications. Reliability Engineering & System Safety, 96(1), 194-200.