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Statistical evaluation of multiple process data in geometric processes with exponential failures

Yıl 2025, , 738 - 761, 28.04.2025

Ömer Altındağ

https://doi.org/10.15672/hujms.1498557

Öz

The geometric process is a monotonic stochastic process commonly used to model some sort of processes having monotonic trend in time. The statistical inference problem for a geometric process has been well studied in the literature. However, existing studies only cover single process data obtained throughout a single realization of a geometric process. This study presents how multiple process data for a geometric process can arise and considers its statistical evaluation by assuming that all processes are homogeneous and the inter-arrival times follow an exponential distribution. Two data structures for multiple process data are introduced: one consists of complete samples, while the other includes both complete and censored samples. The maximum likelihood and modified maximum likelihood estimators for the parameters of the geometric process are derived on the basis of these data structures. The Expectation-Maximization algorithm is used to compute the maximum likelihood estimators in the case of censored data. The asymptotic properties of the estimators are also derived. Test statistics are proposed based on the asymptotic results of the estimators to distinguish a geometric process from a renewal process and to test the homogeneity of the processes. A simulation study is conducted to demonstrate the performance of the inferential procedures. Finally, both artificial and real data analyzes are presented for illustration.

Anahtar Kelimeler

geometric process maximum likelihood modified maximum likelihood EM algorithm

Teşekkür

The author is very grateful to the editor and three anonymous referees for their valuable suggestions. The author also thanks Prof. Dr. Birdal Şenoğlu for his suggestion about the MML methodology used in the article.

Kaynakça

  • [1] R. Arnold, S. Chukova, Y. Hayakawa and S. Marshall, Warranty cost analysis with an alternating geometric process, Proc. of the Inst. of Mech. Eng., Part O: J. of Risk and Reliab. 233 (4), 698-715, 2019.
  • [2] H. Ascher and H. Feingold, Repairable systems reliability: modeling, inference, misconceptions and their causes, Wiley, 1984.
  • [3] H. Aydoğdu, B. Şenoğlu and M. Kara, Parameter estimation in geometric process with Weibull distribution, Appl. Math. and Comp. 217 (6), 2657-2665, 2010.
  • [4] G.K. Bhattacharyya, The asymptotics of maximum likelihood and related estimators based on type II censored data, J. of the American Stat. Assoc. 80 (390), 398-404, 1985.
  • [5] C. Biçer, H.S. Bakouch and H.D. Biçer, Inference on parameters of a geometric process with scaled Muth distribution, Fluct. and Noise Let. 20 (01), 2150006, 2021.
  • [6] H.D. Biçer, C. Biçer and H.S. Bakouch, A geometric process with Hjorth marginal: Estimation, discrimination, and reliability data modeling, Qual. and Reliab. Eng. Int. 38 (5), 2795-2819, 2022.
  • [7] J.S.K. Chan, Y. Lam and D.Y. Leung, Statistical inference for geometric processes with gamma distributions, Comp. Stat. & Data Analy. 47 (3), 565-581, 2004.
  • [8] J.S.K. Chan, C.P. Lam, P.L.H. Yu, S.T.B. Choy, and C.W.S. Chen, A Bayesian conditional autoregressive geometric process model for range data, Comp. Stat. & Data Analy. 56 (11), 3006-3019, 2012.
  • [9] J.S.K. Chan, W.Y. Wan and P.L.H. Yu, Poisson geometric process approach for predicting drop-out and committed first-time blood donors, J. of Appl. Stat. 41 (7), 1486-1503, 2014.
  • [10] J.S.K. Chan, P.L. Yu, Y. Lam and A.P. Ho, Modelling SARS data using threshold geometric process, Stat. in Med. 25 (11), 1826-1839, 2006.
  • [11] D.R. Cox and P.A. Lewis, The statistical analysis of series of events, Springer, 1966.
  • [12] A.H.S Garmabaki, A. Ahmadi, Y.A. Mahmood and A. Barabadi, Reliability modelling of multiple repairable units, Qual. and Reliab. Eng. Int. 32 (7), 2329-2343, 2016
  • [13] A.H.S. Garmabaki, A. Ahmadi, J. Block, H. Pham and U. Kumar, A reliability decision framework for multiple repairable units, Reliab. Eng. & Syst. Saf. 150, 78-88, 2016.
  • [14] M. Kara, Parameter estimation in geometric processes, M.Sc. Thesis, Ankara University, 2009.
  • [15] M. Kara, G. Güven, B. Şenoğlu and H. Aydoğdu, Estimation of the parameters of the gamma geometric process, J. of Stat. Comp. and Sim. 92 (12), 2525-2535, 2022.
  • [16] J.T. Kvaløy and B.H. Lindqvist, TTT-based tests for trend in repairable systems data, Reliab. Eng. & Syst. Saf. 60 (1), 13-28, 1998.
  • [17] Y. Lam, Geometric processes and replacement problem, Acta Math. Appl. Sinica 4 (4), 366-377, 1988.
  • [18] Y. Lam, A note on the optimal replacement problem, Adv. in Appl. Prob. 20 (2), 479-482, 1988.
  • [19] Y. Lam, An optimal repairable replacement model for deteriorating systems, J. of Appl. Prob. 28 (4), 843-851, 1991.
  • [20] Y. Lam, Nonparametric inference for geometric processes, Comm. in Stat. Theo. and Meth. 21 (7), 2083-2105, 1992.
  • [21] Y. Lam, The geometric process and its applications, World Scientific, 2007.
  • [22] Y. Lam, A geometric process maintenance model with preventive repair, Euro. J. of Op. Res. 182 (2), 806-819, 2007.
  • [23] Y. Lam and J.S.K. Chan, Statistical inference for geometric processes with lognormal distribution, Comp. Stat. & Data Analy. 27 (1), 99-112, 1998.
  • [24] Y. Lam and Y.L. Zhang, Analysis of a two‐component series system with a geometric process model, Nav. Res. Log. 43 (4), 491-502, 1996.
  • [25] Y. Lam and Y.L. Zhang, A geometric-process maintenance model for a deteriorating system under a random environment, IEEE Trans. on Reliab. 52 (1), 83-89, 2003.
  • [26] S.A. Lone, I. Alam and A. Rahman, Statistical analysis under geometric process in accelerated life testing plans for generalized exponential distribution, Ann. of Data Sci. 10 (6), 1653-1665, 2023.
  • [27] I.Y. Na and W. Chang, Multi‐system reliability trend analysis model using incomplete data with application to tank maintenance, Qual. and Reliab. Eng. Int. 33 (8),2385- 2395, 2017.
  • [28] S. Park, N. Balakrishnan and G. Zheng, Fisher information in hybrid censored data, Stat. & Prob. Let. 78 (16), 2781-2786, 2008.
  • [29] M.H. Pekalp and H. Aydoğdu, An integral equation for the second moment function of a geometric process and its numerical solution, Nav. Res. Log. 65 (2), 176-184, 2018.
  • [30] M.H. Pekalp, H. Aydoğdu and K.F. Türkman, Discriminating between some lifetime distributions in geometric counting processes, Comm. in Stat. Sim. and Comp. 51 (3), 715-737, 2020.
  • [31] M.H. Pekalp and H. Aydoğdu, Power series expansions for the probability distribution, mean value and variance functions of a geometric process with gamma interarrival times, J. of Comp. and Appl. Math. 388, 113287, 2021.
  • [32] M.H. Pekalp and H. Aydoğdu, Parametric estimations of the mean value and variance functions in geometric process. J. of Comp. and Appl. Math. 449, 115969, 2024.
  • [33] H. Rasay, F. Azizi and F. Naderkhani, A mathematical maintenance model for a production system subject to deterioration according to a stochastic geometric process. . Ann. of Op. Res. 1-28, 2024
  • [34] Y.Y. Tang and Y. Lam, A $\delta$-shock maintenance model for a deteriorating system, Euro. J. of Op. Res. 168 (2), 541-556, 2006.
  • [35] M.L. Tiku, Estimating the mean and standard deviation from a censored normal sample, Biometrika 54 (1/2), 155-165, 1967.
  • [36] M.L. Tiku, Estimating the parameters of log-normal distribution from censored samples, J. of the Ame. Stat. Assoc. 63 (321), 134-140, 1968.
  • [37] M.L. Tiku, W.Y. Tan and N. Balakrishnan, Robust Inference, Marcel Dekker, 1986.
  • [38] M.L. Tiku and A.D. Akkaya, Robust estimation and hypothesis testing, New Age International, 2004.
  • [39] I. Usta, Bayesian estimation for geometric process with the Weibull distribution, Comm. in Stat. Sim. and Comp. 53 (5), 2527-2553, 2024.
  • [40] D.C. Vaughan and M.L. Tiku, Estimation and hypothesis testing for a nonnormal bivariate distribution with applications, Math. and Comp. Mod. 32 (1-2), 53-67, 2000.
  • [41] W.Y. Wan and J.S.K. Chan, A new approach for handling longitudinal count data with zero‐inflation and overdispersion: Poisson geometric process model, Biometrical Journal: J. of Math. Meth. in Biosci. 51 (4), 556-570, 2009.
  • [42] Y. Wang, Z. Lu, and S. Xiao, Parametric bootstrap confidence interval method for the power law process with applications to multiple repairable systems, IEEE Access 6, 49157-49169, 2018.
  • [43] J.S. White, The moments of log-Weibull order statistics, Technometrics 11 (2), 373- 386, 1969.
  • [44] Y.L. Zhang and G.J. Wang, A geometric process warranty model using a combination policy, Comm. in Stat. Theo. and Meth. 48 (6), 1493-1505, 2019.
  • [45] A. Yılmaz, Bayesian Parameter Estimation for Geometric Process with Rayleigh Distribution, Bitlis Eren Üni. Fen Bil. Der. 13 (2), 482-491, 2024.
  • [46] G. Zhen and J.L. Gastwirth, On the Fisher information in randomly censored data, Stat. & Prob. Let. 52 (4), 421-426, 2001.

Yıl 2025, , 738 - 761, 28.04.2025

Ömer Altındağ

https://doi.org/10.15672/hujms.1498557

Öz

Kaynakça

  • [1] R. Arnold, S. Chukova, Y. Hayakawa and S. Marshall, Warranty cost analysis with an alternating geometric process, Proc. of the Inst. of Mech. Eng., Part O: J. of Risk and Reliab. 233 (4), 698-715, 2019.
  • [2] H. Ascher and H. Feingold, Repairable systems reliability: modeling, inference, misconceptions and their causes, Wiley, 1984.
  • [3] H. Aydoğdu, B. Şenoğlu and M. Kara, Parameter estimation in geometric process with Weibull distribution, Appl. Math. and Comp. 217 (6), 2657-2665, 2010.
  • [4] G.K. Bhattacharyya, The asymptotics of maximum likelihood and related estimators based on type II censored data, J. of the American Stat. Assoc. 80 (390), 398-404, 1985.
  • [5] C. Biçer, H.S. Bakouch and H.D. Biçer, Inference on parameters of a geometric process with scaled Muth distribution, Fluct. and Noise Let. 20 (01), 2150006, 2021.
  • [6] H.D. Biçer, C. Biçer and H.S. Bakouch, A geometric process with Hjorth marginal: Estimation, discrimination, and reliability data modeling, Qual. and Reliab. Eng. Int. 38 (5), 2795-2819, 2022.
  • [7] J.S.K. Chan, Y. Lam and D.Y. Leung, Statistical inference for geometric processes with gamma distributions, Comp. Stat. & Data Analy. 47 (3), 565-581, 2004.
  • [8] J.S.K. Chan, C.P. Lam, P.L.H. Yu, S.T.B. Choy, and C.W.S. Chen, A Bayesian conditional autoregressive geometric process model for range data, Comp. Stat. & Data Analy. 56 (11), 3006-3019, 2012.
  • [9] J.S.K. Chan, W.Y. Wan and P.L.H. Yu, Poisson geometric process approach for predicting drop-out and committed first-time blood donors, J. of Appl. Stat. 41 (7), 1486-1503, 2014.
  • [10] J.S.K. Chan, P.L. Yu, Y. Lam and A.P. Ho, Modelling SARS data using threshold geometric process, Stat. in Med. 25 (11), 1826-1839, 2006.
  • [11] D.R. Cox and P.A. Lewis, The statistical analysis of series of events, Springer, 1966.
  • [12] A.H.S Garmabaki, A. Ahmadi, Y.A. Mahmood and A. Barabadi, Reliability modelling of multiple repairable units, Qual. and Reliab. Eng. Int. 32 (7), 2329-2343, 2016
  • [13] A.H.S. Garmabaki, A. Ahmadi, J. Block, H. Pham and U. Kumar, A reliability decision framework for multiple repairable units, Reliab. Eng. & Syst. Saf. 150, 78-88, 2016.
  • [14] M. Kara, Parameter estimation in geometric processes, M.Sc. Thesis, Ankara University, 2009.
  • [15] M. Kara, G. Güven, B. Şenoğlu and H. Aydoğdu, Estimation of the parameters of the gamma geometric process, J. of Stat. Comp. and Sim. 92 (12), 2525-2535, 2022.
  • [16] J.T. Kvaløy and B.H. Lindqvist, TTT-based tests for trend in repairable systems data, Reliab. Eng. & Syst. Saf. 60 (1), 13-28, 1998.
  • [17] Y. Lam, Geometric processes and replacement problem, Acta Math. Appl. Sinica 4 (4), 366-377, 1988.
  • [18] Y. Lam, A note on the optimal replacement problem, Adv. in Appl. Prob. 20 (2), 479-482, 1988.
  • [19] Y. Lam, An optimal repairable replacement model for deteriorating systems, J. of Appl. Prob. 28 (4), 843-851, 1991.
  • [20] Y. Lam, Nonparametric inference for geometric processes, Comm. in Stat. Theo. and Meth. 21 (7), 2083-2105, 1992.
  • [21] Y. Lam, The geometric process and its applications, World Scientific, 2007.
  • [22] Y. Lam, A geometric process maintenance model with preventive repair, Euro. J. of Op. Res. 182 (2), 806-819, 2007.
  • [23] Y. Lam and J.S.K. Chan, Statistical inference for geometric processes with lognormal distribution, Comp. Stat. & Data Analy. 27 (1), 99-112, 1998.
  • [24] Y. Lam and Y.L. Zhang, Analysis of a two‐component series system with a geometric process model, Nav. Res. Log. 43 (4), 491-502, 1996.
  • [25] Y. Lam and Y.L. Zhang, A geometric-process maintenance model for a deteriorating system under a random environment, IEEE Trans. on Reliab. 52 (1), 83-89, 2003.
  • [26] S.A. Lone, I. Alam and A. Rahman, Statistical analysis under geometric process in accelerated life testing plans for generalized exponential distribution, Ann. of Data Sci. 10 (6), 1653-1665, 2023.
  • [27] I.Y. Na and W. Chang, Multi‐system reliability trend analysis model using incomplete data with application to tank maintenance, Qual. and Reliab. Eng. Int. 33 (8),2385- 2395, 2017.
  • [28] S. Park, N. Balakrishnan and G. Zheng, Fisher information in hybrid censored data, Stat. & Prob. Let. 78 (16), 2781-2786, 2008.
  • [29] M.H. Pekalp and H. Aydoğdu, An integral equation for the second moment function of a geometric process and its numerical solution, Nav. Res. Log. 65 (2), 176-184, 2018.
  • [30] M.H. Pekalp, H. Aydoğdu and K.F. Türkman, Discriminating between some lifetime distributions in geometric counting processes, Comm. in Stat. Sim. and Comp. 51 (3), 715-737, 2020.
  • [31] M.H. Pekalp and H. Aydoğdu, Power series expansions for the probability distribution, mean value and variance functions of a geometric process with gamma interarrival times, J. of Comp. and Appl. Math. 388, 113287, 2021.
  • [32] M.H. Pekalp and H. Aydoğdu, Parametric estimations of the mean value and variance functions in geometric process. J. of Comp. and Appl. Math. 449, 115969, 2024.
  • [33] H. Rasay, F. Azizi and F. Naderkhani, A mathematical maintenance model for a production system subject to deterioration according to a stochastic geometric process. . Ann. of Op. Res. 1-28, 2024
  • [34] Y.Y. Tang and Y. Lam, A $\delta$-shock maintenance model for a deteriorating system, Euro. J. of Op. Res. 168 (2), 541-556, 2006.
  • [35] M.L. Tiku, Estimating the mean and standard deviation from a censored normal sample, Biometrika 54 (1/2), 155-165, 1967.
  • [36] M.L. Tiku, Estimating the parameters of log-normal distribution from censored samples, J. of the Ame. Stat. Assoc. 63 (321), 134-140, 1968.
  • [37] M.L. Tiku, W.Y. Tan and N. Balakrishnan, Robust Inference, Marcel Dekker, 1986.
  • [38] M.L. Tiku and A.D. Akkaya, Robust estimation and hypothesis testing, New Age International, 2004.
  • [39] I. Usta, Bayesian estimation for geometric process with the Weibull distribution, Comm. in Stat. Sim. and Comp. 53 (5), 2527-2553, 2024.
  • [40] D.C. Vaughan and M.L. Tiku, Estimation and hypothesis testing for a nonnormal bivariate distribution with applications, Math. and Comp. Mod. 32 (1-2), 53-67, 2000.
  • [41] W.Y. Wan and J.S.K. Chan, A new approach for handling longitudinal count data with zero‐inflation and overdispersion: Poisson geometric process model, Biometrical Journal: J. of Math. Meth. in Biosci. 51 (4), 556-570, 2009.
  • [42] Y. Wang, Z. Lu, and S. Xiao, Parametric bootstrap confidence interval method for the power law process with applications to multiple repairable systems, IEEE Access 6, 49157-49169, 2018.
  • [43] J.S. White, The moments of log-Weibull order statistics, Technometrics 11 (2), 373- 386, 1969.
  • [44] Y.L. Zhang and G.J. Wang, A geometric process warranty model using a combination policy, Comm. in Stat. Theo. and Meth. 48 (6), 1493-1505, 2019.
  • [45] A. Yılmaz, Bayesian Parameter Estimation for Geometric Process with Rayleigh Distribution, Bitlis Eren Üni. Fen Bil. Der. 13 (2), 482-491, 2024.
  • [46] G. Zhen and J.L. Gastwirth, On the Fisher information in randomly censored data, Stat. & Prob. Let. 52 (4), 421-426, 2001.
Toplam 46 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular İstatistiksel Analiz
Bölüm İstatistik
Yazarlar

Ömer Altındağ 0000-0002-7035-9612

Erken Görünüm Tarihi 14 Mart 2025
Yayımlanma Tarihi 28 Nisan 2025
Gönderilme Tarihi 10 Haziran 2024
Kabul Tarihi 3 Mart 2025
Yayımlandığı Sayı Yıl 2025

Kaynak Göster

APA Altındağ, Ö. (2025). Statistical evaluation of multiple process data in geometric processes with exponential failures. Hacettepe Journal of Mathematics and Statistics, 54(2), 738-761. https://doi.org/10.15672/hujms.1498557
AMA Altındağ Ö. Statistical evaluation of multiple process data in geometric processes with exponential failures. Hacettepe Journal of Mathematics and Statistics. Nisan 2025;54(2):738-761. doi:10.15672/hujms.1498557
Chicago Altındağ, Ömer. “Statistical Evaluation of Multiple Process Data in Geometric Processes With Exponential Failures”. Hacettepe Journal of Mathematics and Statistics 54, sy. 2 (Nisan 2025): 738-61. https://doi.org/10.15672/hujms.1498557.
EndNote Altındağ Ö (01 Nisan 2025) Statistical evaluation of multiple process data in geometric processes with exponential failures. Hacettepe Journal of Mathematics and Statistics 54 2 738–761.
IEEE Ö. Altındağ, “Statistical evaluation of multiple process data in geometric processes with exponential failures”, Hacettepe Journal of Mathematics and Statistics, c. 54, sy. 2, ss. 738–761, 2025, doi: 10.15672/hujms.1498557.
ISNAD Altındağ, Ömer. “Statistical Evaluation of Multiple Process Data in Geometric Processes With Exponential Failures”. Hacettepe Journal of Mathematics and Statistics 54/2 (Nisan 2025), 738-761. https://doi.org/10.15672/hujms.1498557.
JAMA Altındağ Ö. Statistical evaluation of multiple process data in geometric processes with exponential failures. Hacettepe Journal of Mathematics and Statistics. 2025;54:738–761.
MLA Altındağ, Ömer. “Statistical Evaluation of Multiple Process Data in Geometric Processes With Exponential Failures”. Hacettepe Journal of Mathematics and Statistics, c. 54, sy. 2, 2025, ss. 738-61, doi:10.15672/hujms.1498557.
Vancouver Altındağ Ö. Statistical evaluation of multiple process data in geometric processes with exponential failures. Hacettepe Journal of Mathematics and Statistics. 2025;54(2):738-61.

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