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Reliability inferences in a $1$-out-of-$n$:G multicomponent stress-strength system with unit gamma Gompertz-$G_0$ components

Year 2025, Volume: 54 Issue: 2, 575 - 598, 28.04.2025

Zohreh Pakdaman , Marzieh Shekari , Hossein Zamanı

https://doi.org/10.15672/hujms.1477060

Abstract

This paper considers reliability inferences in a system of stress-strength $1$ outside of $n$: G when the strength systems belong to the gamma Gompertz unit distribution family (UGG). Stochastic comparisons are obtained between the survival distribution functions of this model. Additionally, some stochastic comparisons are carried out with majorized shape parameters of the unit gamma Gompertz distribution. The asymptotic and several bootstrap confidence intervals of the reliability of the stress strength are studied. In addition, the efficiency of the asymptotic and bootstrap confidence intervals is analyzed by simulation. A numerical example based on real-life data is displayed as an illustration.

Keywords

bootstrap confidence interval majorization maximum likelihood estimator maximum spacing estimation reliability stress-strength stochastic order 1-out-of n:G stress-strength system

References

  • [1] Z.W. Birnbaum, On a use of the Mann-Whitney statistic, Proc. Third Berkeley Symp. Math. Stat. Probab. 1, 13-17, 1956.
  • [2] Z.W. Birnbaum and R. C. McCarty, A Distribution-Free Upper Confidence Bound for $\Pr\{Y<X\}$, Based on Independent Samples of X and Y , Ann. Math. Stat. 29, 558-562, 1958.
  • [3] R.A. Johnson, Stress-strength models for reliability. Handb. Stat. 7, 27-54, 1988.
  • [4] S. Kotz, Y. Lumelskii, and M. Pensky, The stress-strength model and its generalizations: theory and applications, World Scientific, 2003.
  • [5] Z. Pakdaman, J. Ahmadi, and M. Doostparast, Signature-based approach for stressstrength systems, Stat. Pap. 60, 1631-1647, 2019.
  • [6] Z. Pakdaman and M. Doostparast, Influence of environmental factors on stressstrength systems with replaceable components, Commun. Stat. Theory Methods 52 (18), 6487-6503, 2023.
  • [7] Z. Pakdaman and J. Ahmadi, Switching time of the standby component to the k-outof- n: G system in the stress-strength setup, Metrika 82 (2), 225-248, 2019.
  • [8] Z. Pakdaman and J. Ahmadi, Some Results on the Stress-Strength Reliability under the Distortion Functions, Int. J. Reliab. Qual. Saf. Eng. 25 (06), 1850028, 2018.
  • [9] Z. Pakdaman and R.A. Noughabi, On the study of the stress-strength reliability in Weibull-F Models, Hacet. J. Math. Stat. 53 (1), 269-288, 2024.
  • [10] Z. Pakdaman and M. Shekari, Comparing the StressStrength Reliability of Multicomponent Parallel Systems with Heterogeneous Exponentiated Half Logistic-F Components, Int. J. Reliab. Qual. Saf. Eng. 29 (02), 2150051, 2022.
  • [11] N. Joshi, S.R. Bapat, and R.N. Sengupta, Optimal estimation of reliability parameter for inverse Pareto distribution with application to insurance data, Int. J. Qual. Reliab. Manag. 2024.
  • [12] K. Singh, A.K. Mahto, Y. Tripathi, and L. Wang, Inference for reliability in a multicomponent stress-strength model for a unit inverse Weibull distribution under type-II censoring, Qual. Technol. Quant. Manag. 21 (2), 147-176, 2024.
  • [13] J.G. Ma, L. Wang, Y.M. Tripathi, and M.K. Rastogi, Reliability inference for stressstrength model based on inverted exponential Rayleigh distribution under progressive Type-II censored data, Commun. Stat. Simul. Comput. 52 (6), 2388-2407, 2023.
  • [14] L. Wang, S. Dey, Y.M. Tripathi, and S. J. Wu, Reliability inference for a multicomponent stress-strength model based on Kumaraswamy distribution, J. Comput. Appl. Math. 376, 112823, 2020.
  • [15] D.K. Al-Mutairi, M.E. Ghitany, and D. Kundu, Inferences on stress-strength reliability from weighted Lindley distributions, Commun. Stat. Theory Methods 44 (19), 4096–4113, 2015.
  • [16] S. Shoaee and E. Khorram, Stress-strength reliability of a two-parameter bathtubshaped lifetime distribution based on progressively censored samples, Commun. Stat. Theory Methods 44 (24), 5306-5328, 2015.
  • [17] J.B. Smith, A. Wong, and X. Zhou, Higher order inference for stress-strength reliability with independent Burr-type X distributions, J. Stat. Comput. Simul. 85 (15), 3092-3107, 2015.
  • [18] S. Babayi, E. Khorram, and F. Tondro, Inference of $R= P [X< Y]$ for generalized logistic distribution, Statistics 48 (4), 862-871, 2014.
  • [19] M. Nadar, F. Kzlaslan, and A. Papadopoulos, Classical and Bayesian estimation of $P (Y< X)$ for Kumaraswamy’s distribution, J. Stat. Comput. Simul. 84 (7), 1505- 1529, 2014.
  • [20] M. Basirat, S. Baratpour, and J. Ahmadi, Statistical inferences for stress-strength in the proportional hazard models based on progressive Type-II censored samples, J. Stat. Comput. Simul. 85 (3), 431-449, 2015.
  • [21] G.S. Rao, R.R.L. Kantam, K. Rosaiah, and J.P. Reddy, Estimation of reliability in multicomponent stress-strength based on inverse Rayleigh distribution, J. Stat. Appl. Prob. 2 (3), 261, 2013.
  • [22] F. Kizilaslan and M. Nadar, Classical and Bayesian estimation of reliability in multicomponent stress-strength model based on Weibull distribution, Rev. Colomb. Estad. 38 (2), 467-484, 2015.
  • [23] M.C. Korkmaz, The unit generalized half normal distribution: A new bounded distribution with inference and application, Univ. Politehnica Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 82 (2), 133-140, 2020.
  • [24] R.A. Bantan, F. Jamal, C. Chesneau, and M. Elgarhy, Theory and applications of the unit gamma/Gompertz distribution, Mathematics 9 (16), 1850, 2021.
  • [25] M. E. Ghitany, J. Mazucheli, A. F. B. Menezes, and F. Alqallaf, The unit-inverse Gaussian distribution: A new alternative to two-parameter distributions on the unit interval, Commun. Stat. Theory Methods 48 (14), 3423-3438, 2019.
  • [26] M.S. Shama, S. Dey , E. Altun , and A. Z. Afify, The Gamma-Gompertz distribution: Theory and applications, Math. Comput. Simul. 193, 689-712, 2022.
  • [27] R. Dykstra, S.C. Kochar, and J. Rojo, Stochastic comparisons of parallel systems of heterogeneous exponential components, J. Stat. Plan. Inference 65, 203–211, 1997.
  • [28] N. Misra and A. K. Misra, On comparison of reversed hazard rates of two parallel systems comprising of independent gamma components, Stat. Probab. Lett. 83, 1567- 1570, 2013.
  • [29] P. Zhao and N. Balakrishnan, New results on comparison of parallel systems with heterogeneous gamma components, Stat. Probab. Lett. 81, 36-44, 2011.
  • [30] N. Torrado and S. C. Kochar, Stochastic order relations among parallel systems from Weibull distributions, J. Appl. Prob. 52, 102-116, 2015.
  • [31] N. Torrado, On magnitude orderings between smallest order statistics From heterogeneous beta distributions, J. Math. Anal. Appl. 426, 824-835, 2015.
  • [32] A. Kundu and S. Chowdhury, Ordering properties of order statistics from heterogeneous exponentiated Weibull models, Stat. Probab. Lett. 114, 119-127, 2016.
  • [33] A. Kundu and S. Chowdhury, Ordering properties of sample minimum from Kumaraswamy-G random variables, Statistics 52 (1), 133-146, 2018.
  • [34] A. Kundu, S. Chowdhury, A.K. Nanda and N.K. Hazra, Some results on majorization and their applications, J. Comput. Appl. Math. 301, 161-177, 2016.
  • [35] H.B. Mann and D.R. Whitney, On a test of whether one of two random variables is stochastically larger than the other, Ann. Math. Stat. 18 (1), 50-60, 1947.
  • [36] F. Belzunce, C.M. Riquelme and J. Mulero, An Introduction to Stochastic Orders. Academic Press, 2015.
  • [37] A.W. Marshall, I. Olkin and B.C. Arnold, Inequalities: theory of majorization and its applications, New York, Academic press, 1979.
  • [38] M. Shaked and J.G. Shanthikumar, Stochastic Orders, New York, Springer, 2007.
  • [39] R.C.H. Cheng and N.A.K. Amin, Estimating parameters in continuous univariate distributions with a shifted origin, J. R. Stat. Soc. Ser. B (Methodol.) 45 (3), 394- 403, 1983.
  • [40] B.Ranneby, The maximum spacing method. An estimation method related to the maximum likelihood method, Scand. J. Stat. 11 (2), 93-112, 1984.
  • [41] G. Arslan and S.Y. Oncel, Parameter estimation of some Kumaraswamy-G type distributions, Math. Sci. 11 (2), 131-138, 2017.
  • [42] B. Efron, Bootstrap methods: Another look at the jackknife, The Annals of Statistics 7 (1), 1-26, 1979.
  • [43] D.K. Al-Mutairi and S.K. Agarwal, Distributions of the lifetimes of system components operating under an unknown common environment, J. Appl. Stat. 24, 85-96, 1997.
  • [44] R.C.H. Cheng and M.A. Stephens, A goodness-of-fit test using Morans statistic with estimated parameters, Biometrika 76 (2), 385-392, 1989.
  • [45] M. Crowder, Tests for a family of survival models based on extremes, Recent Advances in Reliability Theory: Methodology, Practice, and Inference, 307-321, 2000.
  • [46] F. Kzlaslan, Classical and Bayesian estimation of reliability in a multicomponent stress-strength model based on the proportional reversed hazard rate mode, Math. Comput. Simul. 136, 36-62, 2017.
  • [47] J. F. Lawless, Statistical Models and Methods for Lifetime Data, 2nd edition, Hoboken, John Wiley and Sons, New Jersey, 2003.
  • [48] G.S. Rao, Estimation of reliability in multicomponent stress-strength based on generalized inverted exponential distribution, Int. J. Curr. Res. Rev. 4 (21), 48, 2012.
  • [49] R.G. Srinivasa and K. Rrl, Estimation of reliability in multicomponent stress-strength model: Log-logistic distribution, Electron. J. Appl. Stat. Anal. 3 (2), 75-84, 2010.
  • [50] M. Teimouri and S. Nadarajah, MPS: Estimating Through the Maximum Product Spacing Approach, R package version 2.3.1, URL https://CRAN.R-project.org/package= MPS, 2019.
  • [51] P. Bickel and K.A. Doksum, Mathematical Statistics: Basic Ideas and Selected Topics, Vol 1, 2nd ed. Prentice Hall, 2001.
  • [52] B. Efron and R.J. Tibshirani, An Introduction to the Bootstrap, Chapman and Hall, New York, 1993.

Year 2025, Volume: 54 Issue: 2, 575 - 598, 28.04.2025

Zohreh Pakdaman , Marzieh Shekari , Hossein Zamanı

https://doi.org/10.15672/hujms.1477060

Abstract

References

  • [1] Z.W. Birnbaum, On a use of the Mann-Whitney statistic, Proc. Third Berkeley Symp. Math. Stat. Probab. 1, 13-17, 1956.
  • [2] Z.W. Birnbaum and R. C. McCarty, A Distribution-Free Upper Confidence Bound for $\Pr\{Y<X\}$, Based on Independent Samples of X and Y , Ann. Math. Stat. 29, 558-562, 1958.
  • [3] R.A. Johnson, Stress-strength models for reliability. Handb. Stat. 7, 27-54, 1988.
  • [4] S. Kotz, Y. Lumelskii, and M. Pensky, The stress-strength model and its generalizations: theory and applications, World Scientific, 2003.
  • [5] Z. Pakdaman, J. Ahmadi, and M. Doostparast, Signature-based approach for stressstrength systems, Stat. Pap. 60, 1631-1647, 2019.
  • [6] Z. Pakdaman and M. Doostparast, Influence of environmental factors on stressstrength systems with replaceable components, Commun. Stat. Theory Methods 52 (18), 6487-6503, 2023.
  • [7] Z. Pakdaman and J. Ahmadi, Switching time of the standby component to the k-outof- n: G system in the stress-strength setup, Metrika 82 (2), 225-248, 2019.
  • [8] Z. Pakdaman and J. Ahmadi, Some Results on the Stress-Strength Reliability under the Distortion Functions, Int. J. Reliab. Qual. Saf. Eng. 25 (06), 1850028, 2018.
  • [9] Z. Pakdaman and R.A. Noughabi, On the study of the stress-strength reliability in Weibull-F Models, Hacet. J. Math. Stat. 53 (1), 269-288, 2024.
  • [10] Z. Pakdaman and M. Shekari, Comparing the StressStrength Reliability of Multicomponent Parallel Systems with Heterogeneous Exponentiated Half Logistic-F Components, Int. J. Reliab. Qual. Saf. Eng. 29 (02), 2150051, 2022.
  • [11] N. Joshi, S.R. Bapat, and R.N. Sengupta, Optimal estimation of reliability parameter for inverse Pareto distribution with application to insurance data, Int. J. Qual. Reliab. Manag. 2024.
  • [12] K. Singh, A.K. Mahto, Y. Tripathi, and L. Wang, Inference for reliability in a multicomponent stress-strength model for a unit inverse Weibull distribution under type-II censoring, Qual. Technol. Quant. Manag. 21 (2), 147-176, 2024.
  • [13] J.G. Ma, L. Wang, Y.M. Tripathi, and M.K. Rastogi, Reliability inference for stressstrength model based on inverted exponential Rayleigh distribution under progressive Type-II censored data, Commun. Stat. Simul. Comput. 52 (6), 2388-2407, 2023.
  • [14] L. Wang, S. Dey, Y.M. Tripathi, and S. J. Wu, Reliability inference for a multicomponent stress-strength model based on Kumaraswamy distribution, J. Comput. Appl. Math. 376, 112823, 2020.
  • [15] D.K. Al-Mutairi, M.E. Ghitany, and D. Kundu, Inferences on stress-strength reliability from weighted Lindley distributions, Commun. Stat. Theory Methods 44 (19), 4096–4113, 2015.
  • [16] S. Shoaee and E. Khorram, Stress-strength reliability of a two-parameter bathtubshaped lifetime distribution based on progressively censored samples, Commun. Stat. Theory Methods 44 (24), 5306-5328, 2015.
  • [17] J.B. Smith, A. Wong, and X. Zhou, Higher order inference for stress-strength reliability with independent Burr-type X distributions, J. Stat. Comput. Simul. 85 (15), 3092-3107, 2015.
  • [18] S. Babayi, E. Khorram, and F. Tondro, Inference of $R= P [X< Y]$ for generalized logistic distribution, Statistics 48 (4), 862-871, 2014.
  • [19] M. Nadar, F. Kzlaslan, and A. Papadopoulos, Classical and Bayesian estimation of $P (Y< X)$ for Kumaraswamy’s distribution, J. Stat. Comput. Simul. 84 (7), 1505- 1529, 2014.
  • [20] M. Basirat, S. Baratpour, and J. Ahmadi, Statistical inferences for stress-strength in the proportional hazard models based on progressive Type-II censored samples, J. Stat. Comput. Simul. 85 (3), 431-449, 2015.
  • [21] G.S. Rao, R.R.L. Kantam, K. Rosaiah, and J.P. Reddy, Estimation of reliability in multicomponent stress-strength based on inverse Rayleigh distribution, J. Stat. Appl. Prob. 2 (3), 261, 2013.
  • [22] F. Kizilaslan and M. Nadar, Classical and Bayesian estimation of reliability in multicomponent stress-strength model based on Weibull distribution, Rev. Colomb. Estad. 38 (2), 467-484, 2015.
  • [23] M.C. Korkmaz, The unit generalized half normal distribution: A new bounded distribution with inference and application, Univ. Politehnica Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 82 (2), 133-140, 2020.
  • [24] R.A. Bantan, F. Jamal, C. Chesneau, and M. Elgarhy, Theory and applications of the unit gamma/Gompertz distribution, Mathematics 9 (16), 1850, 2021.
  • [25] M. E. Ghitany, J. Mazucheli, A. F. B. Menezes, and F. Alqallaf, The unit-inverse Gaussian distribution: A new alternative to two-parameter distributions on the unit interval, Commun. Stat. Theory Methods 48 (14), 3423-3438, 2019.
  • [26] M.S. Shama, S. Dey , E. Altun , and A. Z. Afify, The Gamma-Gompertz distribution: Theory and applications, Math. Comput. Simul. 193, 689-712, 2022.
  • [27] R. Dykstra, S.C. Kochar, and J. Rojo, Stochastic comparisons of parallel systems of heterogeneous exponential components, J. Stat. Plan. Inference 65, 203–211, 1997.
  • [28] N. Misra and A. K. Misra, On comparison of reversed hazard rates of two parallel systems comprising of independent gamma components, Stat. Probab. Lett. 83, 1567- 1570, 2013.
  • [29] P. Zhao and N. Balakrishnan, New results on comparison of parallel systems with heterogeneous gamma components, Stat. Probab. Lett. 81, 36-44, 2011.
  • [30] N. Torrado and S. C. Kochar, Stochastic order relations among parallel systems from Weibull distributions, J. Appl. Prob. 52, 102-116, 2015.
  • [31] N. Torrado, On magnitude orderings between smallest order statistics From heterogeneous beta distributions, J. Math. Anal. Appl. 426, 824-835, 2015.
  • [32] A. Kundu and S. Chowdhury, Ordering properties of order statistics from heterogeneous exponentiated Weibull models, Stat. Probab. Lett. 114, 119-127, 2016.
  • [33] A. Kundu and S. Chowdhury, Ordering properties of sample minimum from Kumaraswamy-G random variables, Statistics 52 (1), 133-146, 2018.
  • [34] A. Kundu, S. Chowdhury, A.K. Nanda and N.K. Hazra, Some results on majorization and their applications, J. Comput. Appl. Math. 301, 161-177, 2016.
  • [35] H.B. Mann and D.R. Whitney, On a test of whether one of two random variables is stochastically larger than the other, Ann. Math. Stat. 18 (1), 50-60, 1947.
  • [36] F. Belzunce, C.M. Riquelme and J. Mulero, An Introduction to Stochastic Orders. Academic Press, 2015.
  • [37] A.W. Marshall, I. Olkin and B.C. Arnold, Inequalities: theory of majorization and its applications, New York, Academic press, 1979.
  • [38] M. Shaked and J.G. Shanthikumar, Stochastic Orders, New York, Springer, 2007.
  • [39] R.C.H. Cheng and N.A.K. Amin, Estimating parameters in continuous univariate distributions with a shifted origin, J. R. Stat. Soc. Ser. B (Methodol.) 45 (3), 394- 403, 1983.
  • [40] B.Ranneby, The maximum spacing method. An estimation method related to the maximum likelihood method, Scand. J. Stat. 11 (2), 93-112, 1984.
  • [41] G. Arslan and S.Y. Oncel, Parameter estimation of some Kumaraswamy-G type distributions, Math. Sci. 11 (2), 131-138, 2017.
  • [42] B. Efron, Bootstrap methods: Another look at the jackknife, The Annals of Statistics 7 (1), 1-26, 1979.
  • [43] D.K. Al-Mutairi and S.K. Agarwal, Distributions of the lifetimes of system components operating under an unknown common environment, J. Appl. Stat. 24, 85-96, 1997.
  • [44] R.C.H. Cheng and M.A. Stephens, A goodness-of-fit test using Morans statistic with estimated parameters, Biometrika 76 (2), 385-392, 1989.
  • [45] M. Crowder, Tests for a family of survival models based on extremes, Recent Advances in Reliability Theory: Methodology, Practice, and Inference, 307-321, 2000.
  • [46] F. Kzlaslan, Classical and Bayesian estimation of reliability in a multicomponent stress-strength model based on the proportional reversed hazard rate mode, Math. Comput. Simul. 136, 36-62, 2017.
  • [47] J. F. Lawless, Statistical Models and Methods for Lifetime Data, 2nd edition, Hoboken, John Wiley and Sons, New Jersey, 2003.
  • [48] G.S. Rao, Estimation of reliability in multicomponent stress-strength based on generalized inverted exponential distribution, Int. J. Curr. Res. Rev. 4 (21), 48, 2012.
  • [49] R.G. Srinivasa and K. Rrl, Estimation of reliability in multicomponent stress-strength model: Log-logistic distribution, Electron. J. Appl. Stat. Anal. 3 (2), 75-84, 2010.
  • [50] M. Teimouri and S. Nadarajah, MPS: Estimating Through the Maximum Product Spacing Approach, R package version 2.3.1, URL https://CRAN.R-project.org/package= MPS, 2019.
  • [51] P. Bickel and K.A. Doksum, Mathematical Statistics: Basic Ideas and Selected Topics, Vol 1, 2nd ed. Prentice Hall, 2001.
  • [52] B. Efron and R.J. Tibshirani, An Introduction to the Bootstrap, Chapman and Hall, New York, 1993.
There are 52 citations in total.

Details

Primary Language English
Subjects Probability Theory, Stochastic Analysis and Modelling
Journal Section Statistics
Authors

Zohreh Pakdaman 0000-0003-1031-4698

Marzieh Shekari 0000-0001-7243-2185

Hossein Zamanı 0000-0003-1126-6288

Early Pub Date January 29, 2025
Publication Date April 28, 2025
Submission Date May 2, 2024
Acceptance Date January 13, 2025
Published in Issue Year 2025 Volume: 54 Issue: 2

Cite

APA Pakdaman, Z., Shekari, M., & Zamanı, H. (2025). Reliability inferences in a $1$-out-of-$n$:G multicomponent stress-strength system with unit gamma Gompertz-$G_0$ components. Hacettepe Journal of Mathematics and Statistics, 54(2), 575-598. https://doi.org/10.15672/hujms.1477060
AMA Pakdaman Z, Shekari M, Zamanı H. Reliability inferences in a $1$-out-of-$n$:G multicomponent stress-strength system with unit gamma Gompertz-$G_0$ components. Hacettepe Journal of Mathematics and Statistics. April 2025;54(2):575-598. doi:10.15672/hujms.1477060
Chicago Pakdaman, Zohreh, Marzieh Shekari, and Hossein Zamanı. “Reliability Inferences in a $1$-Out-of-$n$:G Multicomponent Stress-Strength System With Unit Gamma Gompertz-$G_0$ Components”. Hacettepe Journal of Mathematics and Statistics 54, no. 2 (April 2025): 575-98. https://doi.org/10.15672/hujms.1477060.
EndNote Pakdaman Z, Shekari M, Zamanı H (April 1, 2025) Reliability inferences in a $1$-out-of-$n$:G multicomponent stress-strength system with unit gamma Gompertz-$G_0$ components. Hacettepe Journal of Mathematics and Statistics 54 2 575–598.
IEEE Z. Pakdaman, M. Shekari, and H. Zamanı, “Reliability inferences in a $1$-out-of-$n$:G multicomponent stress-strength system with unit gamma Gompertz-$G_0$ components”, Hacettepe Journal of Mathematics and Statistics, vol. 54, no. 2, pp. 575–598, 2025, doi: 10.15672/hujms.1477060.
ISNAD Pakdaman, Zohreh et al. “Reliability Inferences in a $1$-Out-of-$n$:G Multicomponent Stress-Strength System With Unit Gamma Gompertz-$G_0$ Components”. Hacettepe Journal of Mathematics and Statistics 54/2 (April 2025), 575-598. https://doi.org/10.15672/hujms.1477060.
JAMA Pakdaman Z, Shekari M, Zamanı H. Reliability inferences in a $1$-out-of-$n$:G multicomponent stress-strength system with unit gamma Gompertz-$G_0$ components. Hacettepe Journal of Mathematics and Statistics. 2025;54:575–598.
MLA Pakdaman, Zohreh et al. “Reliability Inferences in a $1$-Out-of-$n$:G Multicomponent Stress-Strength System With Unit Gamma Gompertz-$G_0$ Components”. Hacettepe Journal of Mathematics and Statistics, vol. 54, no. 2, 2025, pp. 575-98, doi:10.15672/hujms.1477060.
Vancouver Pakdaman Z, Shekari M, Zamanı H. Reliability inferences in a $1$-out-of-$n$:G multicomponent stress-strength system with unit gamma Gompertz-$G_0$ components. Hacettepe Journal of Mathematics and Statistics. 2025;54(2):575-98.

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