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Adaptive Type-II Hybrid Progressive Schemes Based on Maximum Product of Spacings for Parameter Estimation of Kumaraswamy Distribution

Received: 7 June 2022     Accepted: 18 July 2022     Published: 17 August 2022
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Abstract

The present article aims to illustrate how the Adaptive Type-II Progressive Hybrid censoring scheme can be used to make statistical inferences regarding the shape parameters of the Kumaraswamy distribution. By adopting this scheme, one can reduce the total testing time and the cost associated with the failure of the units. Best of all, one can increase the effectiveness of the statistical analysis while reducing the total test time. The maximum product of spacings method (MPS) in classical estimation settings is highly effective. According to several authors, this method is a superior alternative to the maximum likelihood estimation method (MLE), which delivers more accurate estimates than the maximum likelihood estimation method. Our goal in this article is to estimate the shape parameters of the Kumaraswamy distribution by utilizing the MPS method. Asymptotic normality properties of the estimators are implemented to obtain approximate confidence intervals. In addition, bootstrap confidence intervals are calculated. Monte Carlo simulations have been carried out to compare the MPS and MLE methods. In order to assess the effectiveness of the proposed procedure, a numerical example based on real data is presented.

Published in Applied and Computational Mathematics (Volume 11, Issue 4)
DOI 10.11648/j.acm.20221104.13
Page(s) 102-115
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References
[1] Almetwally, E. M., & Almongy, H. M. (2019). Maximum product spacing and Bayesian method for parameter estimation for generalized power Weibull distribution under censoring scheme. Journal of Data Science, 17 (2), 407-444.
[2] Anatolyev, S., & Kosenok, G. (2005). An alternative to maximum likelihood based on spacings. Econometric Theory, 21 (2), 472-476.
[3] Balakrishnan, N., & Asgharzadeh, A. (2005). Inference for the scaled half-logistic distribution based on progressively Type-II censored samples. Communications in Statistics-Theory and Methods, 34 (1), 73-87.
[4] Balakrishnan, N., & Cramer, E. (2014). The art of progressive censoring. New York: Springer.
[5] Brito, RS (2009). Study of asymptotic expansions, numerical evaluation of moments of generalized beta distributions, applications in regression models and discriminant analysis. Master’s thesis, Federal Rural University of Pernambuco.
[6] Cheng, R. C. H., & Amin, N. A. K. (1979). Maximum product-of-spacings estimation with applications to the lognormal distribution. Math Report, 791.
[7] Coolen, F. P. A., & Newby, M. J. (1991). A note on the use of the product of spacings in Bayesian inference. Kwantitatieve Methoden, 37, 19-32.
[8] Cui, W., Yan, Z., & Peng, X. (2019). Statistical Analysis for Constant-Stress Accelerated Life Test With Weibull Distribution Under Adaptive Type-II Hybrid Censored Data. IEEE Access, 7, 165336-165344.
[9] David, H. A., & Nagaraja, H. N. (2003). Order statistics, third edition, Wiley: New York, NY.
[10] Efron, B. (1979). The bootstrap and modern statistics. Journal of the American Statistical Association, 95 (452), 1293-1296.
[11] El-Sherpieny, E. S. A., Almetwally, E. M., & Muhammed, H. Z. (2020). Progressive Type-II hybrid censored schemes based on maximum prod- uct spacing with application to Power Lomax distribution. Physica A: Statistical Mechanics and its Applications, 553, 124251.
[12] Fletcher, S. G., & Ponnambalam, K. (1996). Estimation of reservoir yield and storage distribution using moments analysis. Journal of hydrology, 182 (1-4), 259-275.
[13] Ghosh, K., & Jammalamadaka, S. R. (2001). A general estimation method using spacings. Journal of Statistical Planning and Inference, 93 (1-2), 71-82.
[14] Harter, H. L., and A. H. Moore. 1965. Maximum- Likelihood Estimation of the Parameters of Gamma and Weibull Populations from Complete and from Censored Samples. Technometrics 7 (4), 639-643.
[15] Helu, A., & Samawi, H. (2017). On Marginal Distributions under Progressive Type-II Censoring: Similarity/Dissimilarity Properties. Open Journal of Statistics, 7 (4), 633-644.
[16] Helu, A., Samawi, H., Rochani, H., Yin, J., & Vogel, R. (2020). Kernel density estimation based on progressive Type-II censoring. Journal of the Korean Statistical Society, 49 (2), 475-498.
[17] Huzurbazar, V. S. (1947). The likelihood equation, consistency and the maxima of the likelihood function. Annals of Eugenics, 14 (1), 185-200.
[18] Jones, M. C. (2009). Kumaraswamy distribution: A beta-type distribution with some tractability advantages. Statistical methodology, 6 (1), 70-81.
[19] Kim, C., Jung, J., & Chung, Y. (2011). Bayesian estimation for the exponentiated Weibull model under Type II progressive censoring. Statistical Papers, 52 (1), 53-70.
[20] Kohansal, A., & Shoaee, S. (2019). Bayesian and classical estimation of reliability in a multicomponent stress-strength model under adaptive hybrid progressive censored data. Statistical Papers, 1-51.
[21] Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes. Journal of hydrology, 46 (1-2), 79-88.
[22] Kundu, D., & Joarder, A. (2006). Analysis of Type- II progressively hybrid censored data. Computational Statistics & Data Analysis, 50 (10), 2509-2528.
[23] Nadarajah, S. (2008). On the distribution of Kumaraswamy. Journal of Hydrology, 348 (3), 568-569.
[24] Ng, Hon Keung Tony, Debasis Kundu, and Ping Shing Chan. Statistical analysis of exponential lifetimes under an adaptive Type-II progressive censoring scheme. Naval Research Logistics (NRL) 56, no. 8 (2009): 687-698.
[25] Nassar, M., Alotaibi, R., & Dey, S. (2022). Estimation Based on Adaptive Progressively Censored under Competing Risks Model with Engineering Applications. Mathematical Problems in Engineering, 2022.
[26] Panahi, H. (2017). Estimation methods for the generalized inverted exponential distribution under Type ii progressively hybrid censoring with application to spreading of micro-drops data. Communications in Mathematics and Statistics, 5 (2), 159-174.
[27] Panahi, H., & Asadi, S. (2021). On adaptive progressive hybrid censored Burr Type III distribution: application to the nano droplet dispersion data. Quality Technology & Quantitative Management, 18 (2), 179-201.
[28] Pitman, E. J. (1979). Some basic theory for statistical inference: Monographs on applied probability and statistics. Chapman and Hall/CRC.
[29] Ponnambalam, K., Seifi, A., & Vlach, J. (2001). Probabilistic design of systems with general distributions of parameters. International journal of circuit theory and applications, 29 (6), 527-536.
[30] Pyke, R. (1965). Spacings. Journal of the Royal Statistical Society: Series B (Methodological), 27 (3), 395-436.
[31] Ranneby, B. (1984). The maximum spacing method. An estimation method related to the maximum likelihood method. Scandinavian Journal of Statistics, 93-112.
[32] Seifi, A., Ponnambalam, K., & Vlach, J. (2000). Maximization of manufacturing yield of systems with arbitrary distributions of component values. Annals of Operations research, 99 (1), 373-383.
[33] Shao, Y., & Hahn, M. G. (1999). Maximum product of spacings method: A unified formulation with illustration of strong consistency. Illinois Journal of Mathematics, 43 (3), 489-499.
[34] Singh, U., Singh, S. K., & Singh, R. K. (2014). Product spacings as an alternative to likelihood for Bayesian inferences. J Stat Appl Probab, 3 (2), 179-188.
[35] Singh, R. K., Singh S. K., & Singh, U. (2016). Maximum product spacings method for the estimation of parameters of generalized inverted exponential distribution under Progressive Type-II Censoring. Journal of Statistics and Management Systems, 19 (2), 219-245.
[36] Sundar, V., & Subbiah, K. (1989). Application of double bounded probability density function for analysis of ocean waves. Ocean engineering, 16 (2), 193-200.
[37] Titterington, D. M. (1985). Comment on Estimating parameters in continuous univariate distributions. Journal of the Royal Statistical Society: Series B (Methodological), 47 (1), 115-116.
[38] Wang, L. (2018). Inference for Weibull competing risks data under generalized progressive hybrid censoring. IEEE Transactions on Reliability, 67 (3), 998-1007.
[39] Yan, Z., & Wang, N. (2020). Statistical analysis based on adaptive progressive hybrid censored sample from alpha power generalized exponential distribution. IEEE Access, 8, 54691-54697.
[40] Ye, Z. S., Chan, P. S., Xie, M., & Ng, H. K. T. (2014). Statistical inference for the extreme value distribution under adaptive Type-II Progressive censoring schemes. Journal of Statistical Computation and Simulation, 84 (5), 1099-1114.
[41] Zheng, G., & Shi, Y. M. (2013). Statistical analysis in constant-stress accelerated life tests for generalized exponential distribution based on adaptive Type-II Progressive hybrid censored data. Chinese Journal of Applied Probability and Statistics, 29 (4), 363-80.
Cite This Article
  • APA Style

    Amal Helu. (2022). Adaptive Type-II Hybrid Progressive Schemes Based on Maximum Product of Spacings for Parameter Estimation of Kumaraswamy Distribution. Applied and Computational Mathematics, 11(4), 102-115. https://doi.org/10.11648/j.acm.20221104.13

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    Amal Helu. Adaptive Type-II Hybrid Progressive Schemes Based on Maximum Product of Spacings for Parameter Estimation of Kumaraswamy Distribution. Appl. Comput. Math. 2022, 11(4), 102-115. doi: 10.11648/j.acm.20221104.13

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    AMA Style

    Amal Helu. Adaptive Type-II Hybrid Progressive Schemes Based on Maximum Product of Spacings for Parameter Estimation of Kumaraswamy Distribution. Appl Comput Math. 2022;11(4):102-115. doi: 10.11648/j.acm.20221104.13

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  • @article{10.11648/j.acm.20221104.13,
      author = {Amal Helu},
      title = {Adaptive Type-II Hybrid Progressive Schemes Based on Maximum Product of Spacings for Parameter Estimation of Kumaraswamy Distribution},
      journal = {Applied and Computational Mathematics},
      volume = {11},
      number = {4},
      pages = {102-115},
      doi = {10.11648/j.acm.20221104.13},
      url = {https://doi.org/10.11648/j.acm.20221104.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20221104.13},
      abstract = {The present article aims to illustrate how the Adaptive Type-II Progressive Hybrid censoring scheme can be used to make statistical inferences regarding the shape parameters of the Kumaraswamy distribution. By adopting this scheme, one can reduce the total testing time and the cost associated with the failure of the units. Best of all, one can increase the effectiveness of the statistical analysis while reducing the total test time. The maximum product of spacings method (MPS) in classical estimation settings is highly effective. According to several authors, this method is a superior alternative to the maximum likelihood estimation method (MLE), which delivers more accurate estimates than the maximum likelihood estimation method. Our goal in this article is to estimate the shape parameters of the Kumaraswamy distribution by utilizing the MPS method. Asymptotic normality properties of the estimators are implemented to obtain approximate confidence intervals. In addition, bootstrap confidence intervals are calculated. Monte Carlo simulations have been carried out to compare the MPS and MLE methods. In order to assess the effectiveness of the proposed procedure, a numerical example based on real data is presented.},
     year = {2022}
    }
    

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    SN  - 2328-5613
    UR  - https://doi.org/10.11648/j.acm.20221104.13
    AB  - The present article aims to illustrate how the Adaptive Type-II Progressive Hybrid censoring scheme can be used to make statistical inferences regarding the shape parameters of the Kumaraswamy distribution. By adopting this scheme, one can reduce the total testing time and the cost associated with the failure of the units. Best of all, one can increase the effectiveness of the statistical analysis while reducing the total test time. The maximum product of spacings method (MPS) in classical estimation settings is highly effective. According to several authors, this method is a superior alternative to the maximum likelihood estimation method (MLE), which delivers more accurate estimates than the maximum likelihood estimation method. Our goal in this article is to estimate the shape parameters of the Kumaraswamy distribution by utilizing the MPS method. Asymptotic normality properties of the estimators are implemented to obtain approximate confidence intervals. In addition, bootstrap confidence intervals are calculated. Monte Carlo simulations have been carried out to compare the MPS and MLE methods. In order to assess the effectiveness of the proposed procedure, a numerical example based on real data is presented.
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Author Information
  • Department of Mathematics, The University of Jordan, Amman, Jordan

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