Aditi Chaturvedi , Surinder Kumar
ARTICLE

(English) PDF

ABSTRACT

In this paper, we consider Kumaraswamy-G distributions and derive a Uniformly Minimum Variance Unbiased Estimator (UMVUE) and a Maximum Likelihood Estimator (MLE) of the two measures of reliability, namely R(t) = P(X > t) and P = P(X > Y) under Type II censoring scheme and sampling scheme of Bartholomew (1963). We also develop interval estimates of the reliability measures. A comparative study of the different methods of point estimation has been conducted on the basis of simulation studies. An analysis of a real data set has been presented for illustration purposes.

KEYWORDS

interval estimation, Kumaraswamy-G distributions, Monte-Carlo simulation, point estimation.

REFERENCES

Bartholomew, D. J., (1963). The sampling distribution of an estimate arising in life testing, Technometrics, 5, pp. 361–374.

Canavos, G., Tsokos, C. P., (1971). A Study of an Ordinary and Empirical Bayes Approach of Estimation of Reliability in the Gamma Life Testing Model. Proceedings of IEEE Symposium on Reliability, pp. 1–17.

Chaturvedi, A., Bhatnagar, A., (2020). Development of Preliminary Test Estimators and Preliminary Test Confidence Intervals for Measures of Reliability of Kumaraswamy-G Distributions Based on Progressive Type-II Censoring. Journal of Statistical Theory and Practice, 14(3), pp. 1–28.

Chaturvedi, A., Tomer, S. K., (2003). UMVU estimation of the reliability function of the generalized life distributions. Statistical Papers, 44(3), pp. 301–313.

Cordeiro, G. M., De Castro, M., (2011). A new family of generalized distributions. Journal of Statistical Computation and Simulation, 81(7), pp. 883–898.

Dey, S., Mazucheli, J., Anis, M. S., (2017). Estimation of reliability of multicomponent stress–strength for a Kumaraswamy distribution. Communications in Statistics - Theory and Methods, 46(4), pp. 1560–1572, DOI: 10.1080/03610926.2015.1022457.

Dey, S., Mazucheli, J., Nadarajah, S., (2018). Kumaraswamy distribution: different methods of estimation. Computational and Applied Mathematics, 37(2), pp. 2094–2111. DOI: https://doi.org/10.1007/s40314-017-0441-1

Eldin, M. M., Khalil, N., Amein, M., (2014). Estimation of Parameters of the Kumaraswamy Distribution Based on General Progressive Type II Censoring. American Journal of Theoretical and Applied Statistics, 3(6), pp. 217–222, DOI: 10.11648/j.ajtas. 20140306.17

Fletcher, S.G., Ponnambalam, K., (1996). Estimation of reservoir yield and storage distribution using moments analysis. Journal of Hydrology, 182, pp. 259–275.

Ganji, A., Ponnambalam, K., Khalili, D.,Karamouz, M., (2006). Grain yield reliability analysis with crop water demand uncertainty. Stochastic Environmental Research and Risk Assessment , 20, pp. 259–277, DOI:https://doi.org/10.1007/s00477-005-0020-7.

Garg, M., (2009). On generalized order statistics from Kumaraswamy distribution. Tamsui Oxford Journal of Mathematical Sciences, 25(2), pp. 153–166.

Hassan, A. S., Sabry, M. A., Elsehetry, A. M., (2020). A New Family of Upper-Truncated Distributions: Properties and Estimation. Thailand Statistician, 18(2), pp. 196–214.

Johnson, N. L., & Kotz, S., (1970). Distributions in Statistics, vol. III: Continuous Univariate Distributions. New York: Houghton-Mifflin.

Jones, M. C., (2009). Kumaraswamy’s distribution: A beta-type distribution with some tractability advantages. Statistical Methodology, 6(1), pp. 70–81, DOI: https://doi. org/10.1016/j.stamet.2008.04.001.

Kizilaslan, F., Nadar, M., (2016). Estimation and prediction of the Kumaraswamy distribution based on record values and inter-record times. Journal of Statistical Computation and Simulation, 86(12), pp. 2471–2493, DOI: 10.1080/00949655.2015.111 9832.

Kumaraswamy, P., (1976). Sinepower probability density function. Journal of Hydrology, 31, pp. 181–184.

Kumaraswamy, P., (1978). Extended sinepower probability density function. Journal of Hydrology, 37, pp. 81–89.

Kumaraswamy, P., (1980). A generalized probability density function for double-bounded random process. Journal of Hydrology, 46, pp. 79–88.

Kundu, A., Chowdhury, S., (2018). Ordering properties of sample minimum from Kumaraswamy- G random variables. Statistics, 52(1), pp. 133–146, DOI: 10.1080/02 331888.2017.1353516.

Kumari T., Chaturvedi, A., Pathak, A., (2019). Estimation and Testing Procedures for the Reliability Functions of Kumaraswamy-G Distributions and a Characterization Based on Records. Journal of Statistical Theory and Practice, 13(1), DOI:10.1007/s42519- 018-0014-7.

Mameli, V., Musio, M., (2013). A Generalization of the Skew-Normal Distribution: The Beta Skew-Normal. Communications in Statistics - Theory and Methods, 42(12), pp. 2229–2244, DOI: 10.1080/03610926.2011.607530.

Mameli, V., (2015). The Kumaraswamy skew-normal distribution. Statistics & Probability Letters, 104, pp. 75–81.

Nadar, M., Kizilaslan, F., Papadopoulos, A., (2014). Classical and Bayesian estimation of P(Y < X) for Kumaraswamy’s distribution. Journal of Statistical Computation and Simulation, 84:7, pp. 1505–1529, DOI: 10.1080/00949655.2012.750658.

Nadarajah, S., (2008). On the distribution of Kumaraswamy. Journal of Hydrology, 348(3- 4), pp. 568–569, DOI: 10.1016/j.jhydrol.2007.09.008.

Nadarajah, S., Cordeiro, G. M., Ortega, E. M. M., (2012). General results for the Kumaraswamy- G distribution. Journal of Statistical Computation and Simulation, 82(7), pp. 951–979, DOI: 10.1080/00949655.2011.562504.

Patel, J. K., Kapadia, C. H., Owen D. B., (1976). Handbook of Statistical Distributions. Marcel Dekker, New York.

Ponnambalam, K., Seifi, A., Vlach, J., (2001). Probabilistic design of systems with general distributions of parameters. International Journal of Circuit Theory and Applications, 29, pp. 527–536, DOI: https://doi.org/10.1002/cta.173.

Proschan, F., (1963). Theoretical explanation of observed decreasing failure rate. Technometrics, 15, pp. 375–383.

Rohatgi, V. K., Saleh, A. K. Md. E., (2012). An introduction to probability and statistics. Wiley, New York.

Seifi, A., Ponnambalam, K., Vlach, J., (2000). Maximization of Manufacturing Yield of Systems with Arbitrary Distributions of Component Values. Annals of Operations Research ,99, pp. 373–383, DOI: 10.1023/A:1019288220413.

Shrivastava, A., Chaturvedi, A., Bhatti, M. I., (2019). Robust Bayesian analysis of a multivariate dynamic model. Physica A: Statistical Mechanics and its Applications, 528, 121451.

Sindhu, T.N., Feroze, N., Aslam, M., (2013).Bayesian Analysis of the Kumaraswamy Distribution under Failure Censoring Sampling Scheme. International Journal of Advanced Science and Technology, 51, pp. 39–58.

Sundar, V., Subbiah, K., (1989). Application of double bounded probability density function for analysis of ocean waves. Ocean Engineering, 16(2), pp. 193–200.

Tamandi, M., Nadarajah, S., (2016). On the Estimation of Parameters of Kumaraswamy- G Distributions. Communications in Statistics - Simulation and Computation, 45(10), pp. 3811–3821, DOI: 10.1080/03610918.2014.957840.

Back to top
© 2019–2022 Copyright by Statistics Poland, some rights reserved. Creative Commons Attribution-ShareAlike 4.0 International Public License (CC BY-SA 4.0)