Abstract
In this paper, some properties of the order dependent time-homogeneous load-sharing model are obtained, including an algorithmic procedure to simulate samples from this model. Then, the problem of how to get predictions of the future failure times is analysed in a sample from censored data (early failures). Punctual predictions based on the median, the mean and the convolutions of exponential distributions are proposed and prediction bands are obtained. Some illustrative examples show how to apply the theoretical results. An application in the study of lifetimes of coherent systems is proposed as well.
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References
Akkouchi M (2008) On the convolution of exponential distributions. J Chungcheong Math Soc 21:501–510
Cox DR (1972) Regression models and life tables (with discussion). J R Stat Soc B 34:187–220
De Santis E, Spizzichino F (2022) Construction of aggregation paradoxes through load-sharing models. Adv Appl Probab. https://doi.org/10.1017/apr.2022.17
De Santis E, Malinovsky Y, Spizzichino FL (2021) Stochastic precedence and minima among dependent variables. Methodol Comput Appl Probab 23:187–205
Deshpande JV, Dewan I, Naik-Nimbalkar UV (2010) A family of distributions to model load sharing systems. J Stat Plan Inference 140:1441–1451
Foschi R, Nappo G, Spizzichino FL (2021) Diagonal sections of copulas, multivariate conditional hazard rates and distributions of order statistics for minimally stable lifetimes. Depend Model 9:394–423
Levy E (2022) On the density for sums of independent exponential, Erlangand gamma variates. Stat Pap 63:693–721
Navarro J, Buono F (2022) Predicting future failure times by using quantile regression. Metrika. https://doi.org/10.1007/s00184-022-00884-z
Navarro J, Sarabia J (2022) Copula representations for the sum of dependent risks: models and comparisons. Probab Eng Inf Sci 36:320–340
Norros I (1985) Systems weakened by failures. Stoch Process Appl 20:181–196
Norros I (1986) A compensator representation of multivariate life length distributions, with applications. Scand J Stat 13:99–112
Ross SM (1984) A model in which component failure rates depend on the working set. Naval Res Logist Q 31:297–300
Ross S (2007) Introduction to probability models, 9th edn. Academic Press, New York
Schechner Z (1984) A load-sharing model: the linear breakdown rule. Naval Res Logist Q 31:137–144
Shaked M, Shanthikumar JG (1988) Multivariate conditional hazard rates and the MIFRA and MIFR properties. J Appl Probab 25:150–168
Shaked M, Shanthikumar JG (2015) Multivariate conditional hazard rate functions—an overview. Appl Stoch Model Bus Ind 31:285–296
Spizzichino FL (2018) Reliability, signature, and relative quality functions of systems under time-homogeneous load-sharing models. Appl Stoch Model Bus Ind 35:158–176
Yun WY, Cha JH (2010) A stochastic model for a general load-sharing system under overload condition. Appl Stoch Model Bus Ind 26:624–38
Zhang Z, Yang Y, Ji X (2020) On stochastic behaviors of load-sharing parallel systems. Commun Stat 49:5392–5405
Acknowledgements
We would like to thank the anonymous reviewers for several helpful suggestions that have served to add clarity and breadth to the earlier version of this paper. FB is member of the research group GNAMPA of INdAM (Istituto Nazionale di Alta Matematica) and is partially supported by MIUR-PRIN 2017, Project “Stochastic Models for Complex Systems”, No. 2017 JFFHSH. JN is partially supported by “Ministerio de Ciencia e Innovación” of Spain under Grant PID2019-103971GB-I00/AEI/10.13039/501100011033.
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Buono, F., Navarro, J. Simulations and predictions of future values in the time-homogeneous load-sharing model. Stat Papers 65, 661–685 (2024). https://doi.org/10.1007/s00362-023-01404-5
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DOI: https://doi.org/10.1007/s00362-023-01404-5