Generalized variables approach to generalized inverted exponential distribution reliability analyses with progressively type II censored data
Abstract
Reliability analysis plays a crucial role in various fields, including engineering, manufacturing, and qualitycontrol. It provides valuable insights into the failure behavior of systems and products. One commonly used distribution in reliability modeling is the generalized inverted exponential distribution (GIED). The GIED distribution is known for its flexibility and adaptability to a wide range of failure data. This paper presents a new method for estimating confidence intervals and testing hypotheses for GIED distribution reliability functions based on a generalized value approach. By transforming the reliability function into the generalized value domain and calculating the generalized lower confidence limit, the proposed method offers enhanced accuracy and precision. Furthermore, the generalized p-value approach for hypothesis testing provides a robust and computationally efficient method for analyzing reliability data. The results from a real data set and Monte Carlo simulations confirm the superiority of the proposed approach over classical methods. The proposed method offers improved accuracy and computational efficiency, making it a valuable tool for reliability analysis using GIED distributions.References
R.D. Gupta, and D. Kundu, Generalized exponential distributions, Aust. N. Z. J. Stat, vol. 41, pp. 173 – 188, 1999.
C.T. Lin, B.S. Duran, and T.O. Lewis, Inverted gamma as a life distribution, Microelectronics Reliability, vol. 29, no. 4, pp. 619 –626, 1989.
S. Dey, Inverted exponential distribution as a life distribution model from a Bayesian viewpoint, Data Science Journal, vol. 6, pp. 107 – 113, 2007.
A. M. Abouammoh, and A. M. Alshingiti, Reliability estimation of generalized inverted exponential distribution, Journal of Statistical Computation and Simulation, vol. 79, no. 11, pp. 1301 – 1315, 2009.
A. C. Cohen, Progressively Censored Samples in Life Testing, Technometrics, vol. 5, no. 3, pp. 327 – 339, 1963.
A. C. Cohen and N. J. Norgaard, Progressively Censored Sampling in the Three-Parameter Gamma Distribution, Technometrics, vol. 19, no. 3, pp. 327 – 339, 1977.
N. Balakrishnan, and R. Aggarwala, Progressive Censoring: Theory, Methods and Applications, Boston: Birkhauser, 2000.
N. Balakrishnan, and E. Cramer, The Art of Progressive Censoring: Applications to Reliability and Quality, New York: Birkhauser, 2014.
N. Balakrishnan, E. Cramer, and U. Kamps, Bounds for means and variances of progressive type II censored order statistics, Statistics & Probability Letters, vol. 54, no. 3, pp. 301 – 315, 2001.
M. A. M. Ali Mousa, Inference and prediction for Pareto progressively censored data, Journal of Statistical Computation and Simulation, vol. 71, pp. 163 – 181, 2001.
M. A. M. Ali Mousa, and S. A.AL-Sagheer, Bayesian prediction for progressively Type-II censored data from the Rayleigh model, Communications in Statistics Theory and Methods, vol. 34, pp. 2353 – 2361, 2005.
A. A. Soliman, A. Y. Al-Hossain, M. M. Al-Harbi, Predicting observables from Weibull model based on general progressive censored data with asymmetric loss, Statistical Methodology, vol. 8, no. 5, pp. 451 – 461, 2011.
S. S. Huang, and S. J. Wu, Bayesian estimation and prediction for Weibull model with progressive censoring, Journal of Statistical Computation and Simulation, vol. 82, no. 11, pp. 1607 – 1620, 2012.
S. Ghafouri, A. H. Rad, and F. Yousefzadeh, Two-sample prediction for progressively Type-II censored Weibull lifetimes,
Communications in Statistics, vol. 46, no. 2, pp. 1381 – 1400, 2017.
K. W. Tsui and S. Weerahandi, Generalized p-Values in Significance Testing of Hypotheses in the Presence of Nuisance Parameters, Journal of the American Statistical Association, vol. 84, pp. 602–607, 1989.
S. Weerahandi, Generalized Confidence Intervals, Journal of the American Statistical Association, vol. 88, no. 423, pp. 899–905,1993.
J. Hannig, H. Iyer, and P. Patterson, Fiducial Generalized Confidence Intervals, Journal of the American Statistical Association, vol. 101, no. 473, pp. 254–269, 2006.
W. H. Wu and H. N. Hsieh, Generalized confidence interval estimation for the mean of delta-lognormal distribution: an application to New Zealand trawl survey data, Journal of Applied Statistics, vol. 41, no. 7, pp. 1471–1485, 2006.
S. Weerahandi, Exact Statistical Methods for Data Analysis, NewYork:Springer-Verlag, 1995.
S. Weerahandi, Generalized Inference in Repeated Measures, NewYork::Wiley, 2004.
J. Lieblein and M. Zelen, Statistical investigation of the fatigue life of deep groove ball bearing, J. Res. Natl. Bur. Stand, vol. 57,pp. 273–316, 1956.
N.R. Mann, Exact three-order-statistic confidence bounds on reliable life for a Weibull model with progressive censoring, J. Am. Stat. Assoc., vol. 64, pp. 306–315, 1969.
N. Balakrishnan, and R.A. Sandhu, A Simple Simulational Algorithm for Generating Progressive Type-II Censored Samples, Communications in Statistics - Theory and Methods, vol. 49, pp. 229–230, 1995.
H.K. Yuen and S.K. Tse, Parameters estimation for Weibull distributed lifetimes under progressive censoring with random removals, J. Stat. Comput. Simul., vol. 55, pp. 37–71, 1996.
X. L. Meng, Posterior Predictive p-Values, The Annals of Statistics, vol. 22, no. 3, pp. 1142 – 1160, 1994.
S. Weerahandi, Testing regression equality with unequal variances, Econometrica, 1987.
S. Weerahandi, Generalized Point Estimation with Application to Small Response Estimation, Communications in Statistics - Theory and Methods, vol. 41, 2012.
S. Nkurunziza and F. Chen, Generalized confidence interval and p-value in location and scale family, Sankhya: The Indian Journal of Statistics, Series B, vol. 73, no. 2, 2011.
S. Gunasekera and M. M.A. Ananda, Generalized variable method inference for the location parameter of the general half-normal distribution, Journal of Statistical Computation and Simulation, vol. 85 no. 10, 2015.
N. Balakrishnan and R. Aggarwala, Progressive Censoring Theory, Methods, and Applications, Ann. Math. Statist., 2000.
S. Gunasekera and D. K. Wijekularathna, Generalized confidence limits for the performance index of the exponentially distributed lifetime, Communications in Statistics - Theory and Methods, vol. 0, pp. 1–19, 2018.
J. Gamage, T. Mathew and S. Weerahandi, Generalized p-values and generalized confidence regions for the multivariate BehrensFisher problem and MANOVA, Journal of Multivariate Analysis, pp. 177–189, 2004.
S. Ghosh, Generalized Inference in Repeated Measures: Exact Methods in MANOVA and Mixed Models, Technometrics, vol. 47, no. 2, pp. 233–233, 2006.
S. Weerahandi, Generalized confidence intervals, J. Amer. Statist., vol. 0, pp. 899–905, 2018.
B. Pradhan, and D. Kundu, On progressively censored generalized exponentia, Communications in Statistics - Theory and Methods, vol. 18, 2009.
H. Krishna and K. Kumar, Reliability estimation in generalized inverted exponential distribution with progressively type II censored sample, Journal of Statistical Computation and Simulation, vol. 83, no. 6, pp. 1007–1019, 2013.
K. Krishnamoorthy, Yin Lin, Yanping Xia, Confidence limits and prediction limits for a Weibull distribution based on the generalized variable approach, Journal of Statistical Planning and Inference, vol. 139, no. 8, pp. 2675–2684, 2009
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
