Moments and estimation of reduced Kies distribution based on progressive type-II right censored order statistics

Year 2019, Volume: 48 Issue: 1, 332 - 350, 01.02.2019

Sanku Dey Mazen Nassar Devendra Kumar

Abstract

Based on progressive type-II censored samples, we first derive the  recurrence relations for the single and product moments and then use these results to compute the means and  variances  of reduced Kies  distribution (RKD), a new  distribution, recently introduced by [21]. Next, we obtain the maximum likelihood estimators of the unknown parameter and the approximate confidence interval of the RKD. Finally, we consider Bayes estimation under the symmetric and asymmetric loss functions using gamma prior for the shape  parameter. We have also derived two-sided Bayes probability interval (TBPI) and the highest posterior density (HPD) credible intervals of this distribution. Monte Carlo simulations are performed to compare the performances of the proposed methods, and a data set has been analyzed for illustrative purposes.

Keywords

Progressive type-II right censored order statistics, Single moments, Product moments, Recurrence relations, Reduced Kies distribution

References

• Aggarwala, R. and Balakrishnan, N. Recurrence relations for single and product moments of progressively Type-II censored order statistics from a exponential and truncated exponential distribution, Ann. Inst. Statist. Math., 48, 757-771, (1996).
• Arnold, B. C., Balakrishnan, N. and Nagaraja, H. N. A First Course in Order Statistics, John Wiley & Sons, New York, (1992).
• Balakrishnan, N. and Malik, H. J. Order statistics from the linear-exponential distribution, part I: Increasing hazard rate case, Communications in Statistics, Theory and Methods, 15, 179-203, (1986).
• Balakrishnan, N. and Sandhu, R. A. A simple simulational algorithm for generating progressive Type-II censored samples, Amer. Statist., 49, 229-230, (1995).
• Balakrishnan, N. and Aggarwala, R. Recurrence relations for single and product moments of order statistics from a generalized logistic distribution with applications to inference and generalizations to double truncation, Handbook of Statistics, 17, 85-126., (1998).
• Balakrishnan, N. and Sultan, K. S. Recurrence relations for single and product moments of progressively Type-II censored order statistics from a exponential and truncated exponential distribution, Ann. Inst. Statist. Math., 48, 757-771, (1998).
• Balakrishnan, N. and Sultan, K. S. Recurrence relations and identities for moments of order statistics, In: N. Balakrishnan and C. R. Rao, eds. Handbook of Statistics, 16, 149-228, Order Statistics: Theory and Methods, North-Holland, Amsterdam, The Netherlands., (1998).
• Balakrishnan, N. and Aggarwala, R. Progressive Censoring: Theory, Method and Applications Birkhauser, Boston., (2000).
• Balakrishnan N, Lin C. T. On the distribution of a test for exponentiality based on progressively type-II right censored spacings, J. Stat. Comput. Simul. 73, 277-283, (2003).
• Balakrishnan, N., Ng, H. K.T. and Kannan, N. A test of exponentiality based on spacings for progressively type-II censored data. In: Huber-Carol C et al (eds) Goodness-of-fit tests and model validity. Birkhauser, Boston, (2002).
• Balakrishnan, N., Kannan, N., Lin, C. T., and Wu, S. J. S. \textit{Inference for the extreme value distribution under progressive type-II censoring, Journal of Statistical Computation & Simulation, 74, 25-45, (2004).
• Balakrishnan, N. Progressive Censoring Methodology: An Appraisal Test, 16, 211-296 (with discussion), (2007).
• Balakrishnan, N., AL-Hussaini, E. K. and Saleh, H. M. Recurrence relations for moments of progressively censored order statistics from logistic distribution with applications to inference, Journal of Statistical Planning and Inference, 141(1), 17-30, (2011).
• Balakrishnan, N. and Saleh, H. M. Recurrence relations for single and product moments of progressively Type-II censored order statistics from a generalized halflogistic distribution with application to inference, Journal of Statistical Computation and Simulation , 83, 1704-1721, (2013).
• Cohen, A.C. Progressively censored samples in life testing, Technometrics, 5, 327-329, (1963).
• David, H. A. and Nagaraja, H. N. Order Statistics, Third Edition, John Wiley & Sons, New York. 119, 171-189, (2003).
• Fernandez, A. J. On estimating exponential parameters with general type II progressive censoring, Journal of Statistical Planning and Inference, 121, 135-147, (2004).
• Joshi, P. C. Recurrence relations between moments of order statistics from exponential and truncated exponential distributions, Sankhya Set. B 39, 362-371, (1978).
• Kies, J.A. The strength of glass performance. Naval Research Lab Report No. 5093, Washington, D.C., (1958).
• Kumar, D., Dey, S., and Nadarajah, S. Extended exponential distribution based on order statistics, Communications in Statistics- Theory and Methods, 46, 9166-9184, (2017).
• Kumar, C. S and Dharmaja, S. H. S. On reduced Kies distribution. Collection of Recent Statistical Methods and Applications, (C.S. Kumar, M. Chacko and E.I.A. Sathar, Eds.), 111-123, Department of Statistics, University of Kerala Publishers, Trivandrum. (2013).
• Kumar, C. S and Dharmaja, S. H. S. On some properties of Kies distribution, Metron, 72, 97-122, (2014).
• Kumar, C. S and Dharmaja, S. H. S. The Exponentiated Reduced Kies distribution- Properties and Applications, Communications in Statistics-Theory and Methods, 46(17), 8778-8790, (2016).
• Mahmoud, R. M., Sultan, K. S. and Saleh, H. M. Progressively censored data from the linear exponential distribution, moments and estimation, Metorn, LXIV(2), 199-215, (2006).
• Malik, M.R and Kumar, D. Relations for moments of progressively Type-II right censored order statistics from Erlang-truncated exponential distribution, Statistics in Transition New Series, 18, 651-668, (2017).
• Mann, N. R. Best linear invariant estimation for Weibull parameters under progressive censoring, Technometrics, 13, 521-534, (1971).
• Martz, H., and Waller, R. Bayesian Reliability Analysis, New York: John Wiley, (1982).
• Sultan, K. S., Mahmoud, M. R. and Saleh, H. M. Moments of estimation from progressively censored data of the half logistic distribution, International Journal of Reliability and Applications, 7(2), 187-201, (2006).
• Thomas, D. R. and Wilson, W. M. Linear order statistic estimation for the two-parameter Weibull and extreme-value distributions from Type-II progressively censored samples, Technometrics, 14, 679-691, (1972).
• Viveros, R. and Balakrishnan, N. Interval estimation of life characteristics from progressively censored data, Technometrics, 36, 84-91, (1994).
• Xu, Y., Cheng, L., Zhang, L., Yan, D. and You, C. Optimization of sample number for Weibull functions of brittle materials strength, Ceramics International, 27, 239-241, (2001).