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AIMS Mathematics

2023, Volume 8, Issue 9: 21968-21992. doi: 10.3934/math.20231120
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A new least squares method for estimation and prediction based on the cumulative Hazard function

  • 1.
    Department of Mathematics, Faculty of Science, Helwan University, Ain Helwan, Cairo, Egypt
  • 2.
    Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt
  • 3.
    Department of Accounting, College of Business Administration in Hawtat Bani Tamim, Prince Sattam Bin Abdulaziz University, Saudi Arabia
  • Received: 17 March 2023 Revised: 24 June 2023 Accepted: 04 July 2023 Published: 11 July 2023

  • MSC : 60G70, 62E20, 62F10, 62G30, 62G32, 62N05

  • In this paper, the cumulative hazard function is used to solve estimation and prediction problems for generalized ordered statistics (defined in a general setup) based on any continuous distribution. The suggested method makes use of Rényi representation. The method can be used with type Ⅱ right-censored data as well as complete data. Extensive simulation experiments are implemented to assess the efficiency of the proposed procedures. Some comparisons with the maximum likelihood (ML) and ordinary weighted least squares (WLS) methods are performed. The comparisons are based on both the root mean squared error (RMSE) and Pitman's measure of closeness (PMC). Finally, two real data sets are considered to investigate the applicability of the presented methods.

    Citation: Amany E. Aly, Magdy E. El-Adll, Haroon M. Barakat, Ramy Abdelhamid Aldallal. A new least squares method for estimation and prediction based on the cumulative Hazard function[J]. AIMS Mathematics, 2023, 8(9): 21968-21992. doi: 10.3934/math.20231120

    Related Papers:

  • In this paper, the cumulative hazard function is used to solve estimation and prediction problems for generalized ordered statistics (defined in a general setup) based on any continuous distribution. The suggested method makes use of Rényi representation. The method can be used with type Ⅱ right-censored data as well as complete data. Extensive simulation experiments are implemented to assess the efficiency of the proposed procedures. Some comparisons with the maximum likelihood (ML) and ordinary weighted least squares (WLS) methods are performed. The comparisons are based on both the root mean squared error (RMSE) and Pitman's measure of closeness (PMC). Finally, two real data sets are considered to investigate the applicability of the presented methods.


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    [1] N. Balakrishnan, E. Cramer, The art of progressive censoring: Applications to reliability and quality, Boston, MA: Birkhäuser, 2014. https://doi.org/10.1007/978-0-8176-4807-7
    [2] G. Casella, R. L. Berger, Statistical inference, Duxbury Press, Pacific Grove, CA., 2002.
    [3] E. L. Lehmann, G. Casella, Theory of point estimation, 2 Eds., New York: Springer-Verlag, 1998.
    [4] M. Ahsanullah, Linear prediction of record values for the two parameter exponential distribution, Ann. Inst. Stat. Math., 32 (1980), 363–368. https://doi.org/10.1007/BF02480340 doi: 10.1007/BF02480340
    [5] E. K. AL-Hussaini, Predicting observables from a general class of distributions, J. Stat. Plan. Infer., 79 (1999), 79–91. https://doi.org/10.1016/S0378-3758(98)00228-6 doi: 10.1016/S0378-3758(98)00228-6
    [6] E. K. AL-Hussaini, A. B. Ahmad, On Bayesian predictive distributions of generalized order statistics, Metrika, 57 (2003), 165–176. https://doi.org/10.1007/s001840200207 doi: 10.1007/s001840200207
    [7] A. E. Aly, Prediction of the exponential fractional upper record values, Math. Slovaca, 72 (2022), 491–506. http://dx.doi.org/10.1515/ms-2022-0032 doi: 10.1515/ms-2022-0032
    [8] A. E. Aly, Predictive inference of dual generalized order statistics from the inverse Weibull distribution, Stat. Pap., 64 (2023), 139–160. http://dx.doi.org/10.1007/s00362-022-01312-0 doi: 10.1007/s00362-022-01312-0
    [9] M. Amleh, M. Raqab, Prediction of censored Weibull lifetimes in a simple step-stress plan with Khamis-Higgins model, Stat. Optim. Inf. Comput., 10 (2022), 658–677. http://dx.doi.org/ 10.19139/soic-2310-5070-1069 doi: 10.19139/soic-2310-5070-1069
    [10] H. M. Barakat, O. M. Khaled, H. A. Ghanem, New method for prediction of future order statistics, Qual. Technol. Quant M., 18 (2021), 101–116.
    [11] H. M. Barakat, M. E. El-Adll, A. E. Aly, Two-sample nonparametric prediction intervals based on random number of generalized order statistics, Commun. Stat. Theor. M., 50 (2021), 4571–4586. http://dx.doi.org/10.1080/03610926.2020.1719421. doi: 10.1080/03610926.2020.1719421
    [12] M. E. El-Adll, H. M. Barakat, A. E. Aly, Asymptotic prediction for future observations of a random sample of unknown continuous distribution, Complexity, 2022 (2022), 4571–4586. http://dx.doi.org/10.1155/2022/4073799 doi: 10.1155/2022/4073799
    [13] K. S. Kaminsky, L. S. Rhodin, Maximum likelihood prediction, Ann. Inst. Stat. Math., 37 (1985), 507–517. https://doi.org/10.1007/BF02481119 doi: 10.1007/BF02481119
    [14] J. F. Lawless, A prediction problem concerning samples from the exponential distribution with applications in life testing, Technometrics, 13 (1971), 725–730. http://dx.doi.org/10.2307/1266949 doi: 10.2307/1266949
    [15] G. S. Lingappaiah, Prediction in exponential life testing, Can. J. Stat., 1 (1973), 113–117. http://dx.doi.org/10.2307/3314650 doi: 10.2307/3314650
    [16] Z. M. Raqab, Optimal prediction-intervals for the exponential distribution based on generalized order statistics, IEEE Trans. Reliab., 50 (2001), 112–115. https://doi.org/10.1109/24.935025 doi: 10.1109/24.935025
    [17] J. Aitcheson, I. Dunsmore, Statistical prediction analysis, Cambridge University Press, Cambridge, 1975. https://doi.org/10.1017/CBO9780511569647
    [18] H. A. David, H. N. Nagaraja, Order statistics, 3 Eds., Wiley, NJ, 2003. https://doi.org/10.1002/0471722162
    [19] S. Geisser, Predictive inference: An introduction, Chapman and Hall: London, 1993.
    [20] K. S. Kaminsky, P. I. Nelson, Prediction intervals, In: Handbook of Statistics, Balakrishnan, N. and C. R. Rao (Eds.), Amesterdam, North Holland, 50 (1998), 431–450. https://doi.org/10.7788/bue-1997-jg33
    [21] H. M. Barakat, M. E. El-Adll, M. E. Sobh, Bootstrapping $m$-generalized order statistics with variable rank, AIMS Math., 7 (2022), 13704–13732. http://dx.doi.org/10.3934/math.2022755 doi: 10.3934/math.2022755
    [22] U. Kamps, A concept of generalized order statistics, Teubner, Stuttgart, 1995. https://doi.org/10.1007/978-3-663-09196-7
    [23] U. Kamps, E. Cramer, On distributions of generalized order statistics, In: Handbook of Statistics, Balakrishnan, N. and C. R. Rao (Eds.), Amesterdam, North Holland, 35 (2001), 269–280. http://dx.doi.org/10.1080/02331880108802736
    [24] J. Swain, S. Venkatraman, J. Wilson, Least squares estimation of distribution function in Johnson's translation system, J. Stat. Comput. Simul., 29 (1988), 271–297. https://doi.org/10.1080/00949658808811068 doi: 10.1080/00949658808811068
    [25] R. D. Gupta, D. Kundu, Generalized exponential distribution: Different method of etimations, J. Stat. Comput. Simul., 69 (2001), 315–337. https://doi.org/10.1080/00949650108812098 doi: 10.1080/00949650108812098
    [26] D. Kundu, M. Z. Raqab, Generalized Rayleigh distribution: Different methods of estimation, Comput. Stat. Data Anal., 49 (2005), 187–200. https://doi.org/10.1016/j.csda.2004.05.008 doi: 10.1016/j.csda.2004.05.008
    [27] M. E. El-Adll, A. E. Aly, Prediction intervals for future observations of pareto distribution based on generalized order statistics, J. Appl. Statist. Sci., 22 (2016), 111–125.
    [28] H. M. Barakat, E. M. Nigm, M. E. El-Adll, M. Yusuf, Prediction of future exponential lifetime based on random number of generalized order statistics under a general set-up, Stat. Pap., 59 (2018), 605–631. http://dx.doi.org/10.1007/s00362-016-0779-2 doi: 10.1007/s00362-016-0779-2
    [29] A. Rényi, On the theory of order statistics, Acta Math. Hungarica, 4 (1953), 191–231. https://doi.org/10.1007/BF02127580 doi: 10.1007/BF02127580
    [30] E. J. G. Pitman, J. Wishart, The closest estimates of statistical parameters, Math. Proc. Cambridge Philos. Soc., 59 (1937), 212–222. https://doi.org/10.1017/S0305004100019563 doi: 10.1017/S0305004100019563
    [31] J. P. Keating, R. L. Mason, P. K. Sen, Pitman's measure of closeness: A comparison of statistical estimators, SIAM, Philadelphia: Society for Industrial and Applied Mathematics, 1993.
    [32] N. Balakrishnan, K. F. Davies, J. P. Keating, R. L. Mason, Pitman closeness of best linear unbiased and invariant predictors for exponential distribution in one- and two-sample situations, Commun. Stat.-Theor. M., 41 (2012), 1–15. https://doi.org/10.1080/03610920903537301 doi: 10.1080/03610920903537301
    [33] H. N. Nagaraja, Comparison of estimators and predictors from two-parameter exponential distribution, Sankhya Ser B., 48 (1986), 10–18.
    [34] Z. M. Raqab, A. L. Alkhalfan, N. Balakrishnan, Pitman comparisons of predictors of censored observations from progressively censored samples for exponential distribution, J. Stat. Comput. Simul., 86 (2016), 1539–1558. https://doi.org/10.1080/00949655.2015.1071820 doi: 10.1080/00949655.2015.1071820
    [35] M. E. El-Adll, Inference for the two-parameter exponential distribution with generalized order statistics, Math. Popul. Stud., 28 (2021), 1–23. http://dx.doi.org/10.1080/08898480.2019.1681187 doi: 10.1080/08898480.2019.1681187
    [36] A. A. Al-Babtain, M. K. Shakhatreh, M. Nassar, A. Z. Afify, A new modified Kies family: Properties, estimation under complete and type-Ⅱ censored samples, and engineering applications, Mathematics, 8 (2020), 1345. http://dx.doi.org/10.3390/math8081345 doi: 10.3390/math8081345
    [37] A. M. Abd El-Raheem, E. M. Almetwally, M. S. Mohamed, E. H. Hafez, Accelerated life tests for modified Kies exponential lifetime distribution: Binomial removal, transformers turn insulation application and numerical results, AIMS Math., 6 (2021), 5222–5255. https://doi.org/10.3934/math.2021310 doi: 10.3934/math.2021310
    [38] H. M. Barakat, M. E. El-Adll, A. E. Aly, Prediction of future generalized order statistics based on two-parameter exponential distribution for large samples, Qual. Technol. Quant. Manag., 19 (2022), 259–275. http://dx.doi.org/10.1080/16843703.2022.2034261 doi: 10.1080/16843703.2022.2034261
    [39] D. N. P. Murthy, M. Xie, R. Jiang, Weibull models, Wiley: Hoboken, NJ, USA, 2004.
    [40] D. G. Hoel, A representation of mortality data by competing risks, Biometrics, 28 (1972), 475–488. https://doi.org/10.2307/2556161 doi: 10.2307/2556161
    [41] W. S. Abu El Azm, R. A. Aldallal, H. M. Aljohani, S. G. Nassr, Estimations of competing lifetime data from inverse Weibull distribution under adaptive progressively hybrid censored, Math. Biosci. Eng., 19 (2022), 6252–6275. https://doi.org/10.3934/mbe.2022292 doi: 10.3934/mbe.2022292
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