Reliability Estimation for the Inverse Weibull Distribution Under Adaptive Type-II Progressive Hybrid Censoring: Comparative Study
Abstract
The aim of this study is to investigate different methods of estimating the stress-strength reliability parameter, $\theta =P(Y<X)$, when the strength (X) and the stress (Y) are independent random variables taken from the inverse Weibull distribution (IWD), with the same shape parameter and different scale parameters. Based on Adaptive Type-II Hybrid progressive censored samples, we employ classical and Bayesian approaches. In the classical approach, we use the maximum likelihood estimator (MLE), the approximate maximum likelihood estimator (AMLE), and the least squares estimator (LSE). In contrast, the Bayesian approach utilizes symmetric and asymmetric loss functions. Due to the absence of explicit forms for Bayes estimators, we propose using Lindley's approximation method for computing the Bayes estimators. We compare these estimators using extensive simulations and two criteria: the bias and the mean square error (MSE). Finally, two real-life data examples are provided for illustrations.References
Alotaibi, R., Nassar, M., and Elshahhat, A. (2022). Computational Analysis of XLindley Parameters Using Adaptive Type-II Progressive Hybrid Censoring with Applications in Chemical Engineering. Mathematics, 10(18), 3355.
Asadi, S., Panahi, H., Swarup, C., & Lone, S. A. (2022). Inference on adaptive progressive hybrid censored accelerated life test for Gompertz distribution and its evaluation for virus-containing micro droplets data. Alexandria Engineering Journal, 61(12), 10071-10084.
Balakrishnan, N. (1989a). Approximate maximum likelihood estimation of the mean and standard deviation of the normal distribution based on Type-II censored samples. Journal of Statistical Computation and Simulation 32 (3), 137C148.
Balakrishnan N. and Aggarwala R. (2000). Progressive censoring-Theory, Methods and Applications. Progressive censoring: theory, methods, and applications. Springer Science & Business Media.
Balakrishnan, N., & Cramer, E. (2014). The art of progressive censoring. Statistics for industry and technology.
Birnbaum, Z. W. (1956, January). On a use of the Mann-Whitney statistic. In Proceedings of the third Berkeley symposium on mathematical statistics and probability (Vol. 1, pp. 13-17). Berkeley, CA, USA: University of California Press.
Cacciari, M., & Montanari, G. C. (1987). A method to estimate the Weibull parameters for progressively censored tests. IEEE transactions on reliability, 36(1), 87-93.
Efron, B. (1988). Logistic regression, survival analysis, and the Kaplan-Meier curve. Journal of the American statistical Association, 83(402), 414-425.
Feynman, R.P. (1987). Mr. Feynman goes to Washigton. Engineering and science. California Institute of Technology, Pasadena, CA, 6-22.
Helu, A., & Samawi, H. (2021). Statistical analysis based on adaptive progressive hybrid censored data from Lomax distribution. Statistics, Optimization & Information Computing, 9(4), 789.
Helu, A., & Samawi, H. (2022). Inference on P(X < Y ) in Bivariate Lomax model based on progressive Type-II censoring. PloS one, 17(5), e0267981.
Hossain, A. and Zimmer,W. (2003). Comparison of estimation methods for theWeibull parameters: Complete and censored samples. Journal of Statistical Computation and Simulation, 73(2):145-153.
Johnson, N. L., Kotz, S., & Balakrishnan, N. (1994). Beta distributions. Continuous univariate distributions. 2nd ed. New York, NY: John Wiley and Sons, 221-235.
Kim, C., & Han, K. (2010). Estimation of the scale parameter of the half-logistic distribution under progressively Type-II censored sample. Statistical Papers, 51(2), 375-387.
Kotz, S., & Pensky, M. (2003). The stress-strength model and its generalizations: theory and applications. World Scientific.
Kumar, S. (2021). Estimation and Testing Procedures for the Reliability Characteristics of Chen Distribution Based on Type II Censoring and the Sampling Scheme of Bartholomew. Statistics, Optimization & Information Computing, 9(1), 99-122.
Kundu, D., & Joarder, A. (2006). Analysis of Type-II progressively hybrid censored data. Computational Statistics & Data Analysis, 50(10), 2509-2528.
Lindley, D. V. (1980). Approximate bayesian methods. Trabajos de estadstica y de investigacin operativa, 31(1), 223-245.
Makkar, P., Srivastava, P. K., Singh, R. S., & Upadhyay, S. K. (2014). Bayesian survival analysis of head and neck cancer data using lognormal model. Communications in Statistics-Theory and Methods, 43(2), 392-407.
Marusic, M., Markovic, D., & Jukic, D. (2010). Least squares fitting the three-parameter inverse Weibull density. Mathematical Communications, 15(2), 539-553.
Montanari, G. C., & Cacciari, M. (1988). Progressively-censored aging tests on XLPE-insulated cable models. IEEE Transactions on Electrical Insulation, 23(3), 365-372.
Musleh, R. M., & Helu, A. (2014). Estimation of the inverse Weibull distribution based on progressively censored data: Comparative
study. Reliability Engineering & System Safety, 131, 216-227.
Musleh, R., Helu, A., & Samawi, H. (2022). Kernel-based estimation of P(X < Y ) when X and Y are dependent random variables based on progressive Type-II censoring. Communications in Statistics-Theory and Methods, 51(8), 2368-2384.
Nelson, W. B. (2003). Applied life data analysis (Vol. 521). John Wiley & Sons.
Ng, H. K. T., Kundu, D., & Chan, P. S. (2009). Statistical analysis of exponential lifetimes under an adaptive Type-II progressive censoring scheme. Naval Research Logistics (NRL), 56(8), 687-698.
Panahi, H., & Asadi, P. (2022). Estimating the parameters of a generalized inverted exponential distribution based on adaptive Type-II hybrid progressive censoring with application. Journal of Statistics and Management Systems, 25(2), 433-455.
Pandey, B. N., N. Dwividi, and B. Pulastya (2011). Comparison between Bayesian and maximum likelihood estimation of the scale parameter in Weibull distribution with known shape under Linex loss function. Journal of Scientific Research 55, 163C172.
Sharma, V. K., Singh, S. K., Singh, U., & Agiwal, V. (2015). The inverse Lindley distribution: a stress-strength reliability model with application to head and neck cancer data. Journal of Industrial and Production Engineering, 32(3), 162-173.
Shawky, A. I., & Khan, K. (2022). Reliability Estimation in Multicomponent Stress-Strength Based on Inverse Weibull Distribution. Processes, 10(2), 226.
Varian, H. R. (1975). A Bayesian approach to real estate assessment. Studies in Bayesian econometric and statistics in Honor of Leonard J. Savage, 195-208.
Yadav, A. S., Singh, S. K., & Singh, U. (2018). Estimation of stressCstrength reliability for inverse Weibull distribution under progressive type-II censoring scheme. Journal of Industrial and Production Engineering, 35(1), 48-55.
Zellner, A. (1986). On assessing prior distributions and Bayesian regression analysis with g-prior distributions. Bayesian inference and decision techniques.
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