The problem of estimating lifetime distribution parameters under progressively Type-II censoring originated in the context of reliability. But traditionally it is assumed that the available data from this censoring scheme are performed in exact numbers. However, some collected lifetime data might be imprecise and are represented in the form of fuzzy numbers. Thus, it is necessary to generalize classical statistical estimation methods for real numbers to fuzzy numbers. This paper deals with the estimation of lifetime distribution parameters under progressively Type-II censoring scheme when the lifetime observations are reported by means of fuzzy numbers. A new method is proposed to determine the maximum likelihood estimates of the parameters of interest. The methodology is illustrated with two popular models in lifetime analysis, the Rayleigh and Lognormal lifetime distributions.
Balakrishnan, N., and Aggarwala, R. (2000), Progressive Censoring: Theory, Methods and Applications. Birkhauser, Boston.
Balakrishnan, N., and Asgharzadeh, A. (2005). Inference for the scaled half-logistic distribution based on progressively Type II censored samples. Communications in Statistics-Theory and Methods, 34, 73-87.
Balakrishnan, N., and Kannan, N. (2000). Point and interval estimation for the parameters of the logistic distribution based on progressively Type-II censored samples. In N. Balakrishnan, and C. R. Rao (Eds.), Handbook of statistics: Vol. 20 (pp. 431-456).
Balakrishnan, N., Kannan, N., Lin, C. T., and Ng, H. K. T. (2003). Point and interval estimation for the normal distribution based on progressively Type-II censored samples. IEEE Transactions on Reliability, 52, 90-95.
Balakrishnan, N. and Sandhu, R. A. (1995). A simple algorithm for generating progressively Type-II censored samples. The American Statistician, 49(2), 229-230.
Cohen, A. C. (1963). Progressively censored samples in life testing. Tecnometrics, Volume 5, 327-329.
Coppi, R., Gil, M.A. and Kiers, H.A.L., (2006). The fuzzy approach to statistical analysis. Computational Statistics and Data Analysis, 51(1), 114.
Denoeux, T. (2011). Maximum likelihood estimation from fuzzy data using the EM algorithm, Fuzzy Sets and Systems. 183(1), 72-91.
Dempster, A.P., Laird, N.M., and Rubin, D.B., (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B, 39, 1-38.
Dubois, D. and Prade, H.(1980). Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York.
Dyer, D. D. and Whisenand, C. W. (1973). Best linear estimator of the parameter of the Rayleigh distribution-Part I: Small sample theory for censored order statistics. IEEE Transactions on Reliability, 22, 27-34.
Gebhardt, J., Gil M.A. and Kruse R., (1998). Fuzzy set-theoretic methods in statistics, in: R. Slowinski(Ed.), Fuzzy Sets in Decision Analysis, Operations Research and Statistics, Kluwer Academic Publishers, Boston, pp.311-347.
Huang, H., Zuo, M. and Sun, Z., (2006). Bayesian reliability analysis for fuzzy lifetime data. Fuzzy Sets and Systems, 157, 16741686.
Kim C. and Han K. (2009). Estimation of the scale parameter of the Rayleigh distribution under general progressive censoring, Journal of the Korean Statistical Society, 38, 239-246.
Mann, N. R. (1971). Best linear invariant estimator for Weibull parameters under progressive censoring. Technometrics, 13, 521-533.
Pak, A., Parham, G.H. and Saraj, M., (2013). On estimation of Rayleigh scale parameter under doubly Type-II censoring from imprecise data. Journal of Data Science, 11, 303-320.
Pak, A., Parham, G.H. and Saraj, M., (2014). Inferences on the Competing Risk Reliability Problem for Exponential Distribution Based on Fuzzy Data. IEEE Transactions on reliability, 63(1), 2-13.
Polovko, A. M. (1968), Fundamentals of Reliability Theory. New York: Academic Press.
Pradhan B., and Kundu D. (2009). On progressively censored generalized exponential distribution, Test, 18, 497-515.
Raqab, M. Z. and Madi, M. T. (2002). Bayesian prediction of the total time on test using doubly censored Rayleigh data. Journal of Statistical Computation and Simulation, 72, 781-789.
Singpurwalla, N.D. and Booker, J.M. (2004). Membership functions and probability measures of fuzzy sets. Journal of the American Statistical Association, 99(467), 867877.
Thomas D. R., Wilson W. M. (1972) Linear order statistic estimation for the two-parameter Weibull and extreme value distributions from Type-II progressively censored samples., Technometrics, 14, 679-691.
Viveros, R., and Balakrishnan, N. (1994). Interval estimation of life characteristics from progressively censored data. Technometrics, 36, 84-91.
Zadeh, L. A. (1968). Probability measures of fuzzy events, Journal of Mathematical Analysis and Applications10, 421-427.
Zimmermann, H. J. (1991). Fuzzy set teory and its application, Kluwer, Dordrecht.