Bayesian Interval Estimation in a Non-Homogeneous Poisson Process with Delayed S-Shaped Intensity Function

Otieno Collins; Orawo Luke Akong’o; Matiri George Munene; Justin Obwoge Okenye

doi:doi:10.11648/j.ajtas.20231203.12

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Bayesian Interval Estimation in a Non-Homogeneous Poisson Process with Delayed S-Shaped Intensity Function

Otieno Collins, Orawo Luke Akong’o, Matiri George Munene, Justin Obwoge Okenye

Published in American Journal of Theoretical and Applied Statistics (Volume 12, Issue 3)

Received: 11 April 2023 Accepted: 16 May 2023 Published: 5 June 2023

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Abstract

Software reliability assessment has been explored by many researchers over the past decades. With the increasing development of new complex software systems, accurate methods for estimating reliability model parameters are needed. Facilitated by the increasing use of computer systems in various sectors such as air traffic control, banking, industrial processes, and government operations, developing accurate reliability assessment methods is indispensable. The Delayed S-shaped software reliability model is one of the non-homogeneous Poisson process (NHPP) software reliability models proposed for capturing error detection and removal processes in software reliability testing. Many researchers have fitted the model to software failure data and performed estimation using the Maximum Likelihood method and Bayesian approach, however, construction of Bayesian credible sets for the parameters of this model and comparison of their efficiencies with the Wald confidence intervals using simulation have not been explored. The Bayesian interval estimation was conducted with three different joint prior distributions assigned to the parameters α and β of the model, namely the gamma distributed informative prior and, 1/α, and 1/αβ as non-informative priors. The Bayesian credible intervals and Wald confidence intervals for the two parameters were compared on the basis of interval lengths and coverage probabilities. The simulation was assumed to emulate the end-user environment and can generate inter-failure times data for the study. The Delayed S-shaped reliability model variables were simulated with fixed parameters set at (α, β)=(20, 0.5). The hyperparameters for the informative prior were chosen such that they have minimal effect on the results. In other words, the prior information does not swamp the information from the data. The Bayesian method yields superior results, as evidenced by shorter interval lengths and higher coverage probabilities in Table 1.

Published in	American Journal of Theoretical and Applied Statistics (Volume 12, Issue 3)
DOI	10.11648/j.ajtas.20231203.12
Page(s)	43-50
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2023. Published by Science Publishing Group

Keywords

Non-Homogeneous Poisson Process, Intensity Function, Software Reliability Model, Informative Priors, Bayesian Method, Wald Intervals, Maximum Likelihood

References

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[3]	Yamada, S., Ohba, M., & Osaki, S. (1984). S-shaped software reliability growth models and their applications. IEEE Transactions on Reliability, 33 (4), 289-292. doi: 10.1109/TR.1984.5221826.
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[5]	Hanagal, D. D., & Bhalerao, N. N. (2018). Analysis of Delayed s shaped software reliability growth model with time dependent fault content rate function. "Journal of Data Science, 16 (4), 857-878. doi: 10.6339/JDS.201810_16(4).00010.
[6]	Yin, Liang & Trivedi Kishor S. (1999). Confidence interval estimation of NHPP-based software reliability models. In Proceedings 10th International Symposium on Software Reliability Engineering (Cat. No. PR00443) (pp. 6-11). IEEE. doi: 10.1109/ISSRE.1999.809305.
[7]	Lai, R., & Garg, M. (2012). A detailed study of NHPP software reliability models. J. Softw., 7 (6), 1296-1306.
[8]	Lee, T. Q., Yeh, C. W., & Fang, C. C. (2014). Bayesian software reliability prediction based on Yamada Delayed S-Shaped model. In Applied Mechanics and Materials, 490, 1267-1278. Trans Tech Publications. https://doi.org/10.4028/www.scientific.net/AMM.490-491.1267
[9]	Cheruiyot, N., Orawo, L. A. O., & Islam, A. S. (2019). Predictive Analyses of Logarithmic Non–Homogeneous Poisson Process in Software Reliability Using Bayesian Approach with Informative Priors. American Journal of Mathematics and Statistics, 9 (2), 57-65. doi: 10.5923/j.ajms.20190902.0.
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[12]	Dodwell, T. J., Ketelsen, C., Scheichl, R., & Teckentrup, A. L. (2015). A hierarchical multilevel Markov chain Monte Carlo algorithm with applications to uncertainty quantification in subsurface flow. SIAM/ASA Journal on Uncertainty Quantification, 3 (1), 1075-1108. https://doi.org/10.1137/130915005
[13]	van de Schoot, R., Depaoli, S., King, R., Kramer, B., Märtens, K., Tadesse, M. G.,... & Yau, C. (2021). Bayesian statistics and modelling. Nature Reviews Methods Primers, 1 (1), 1-26. https://doi.org/10.1038/s43586-020-00003-0
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APA Style

Otieno Collins, Orawo Luke Akong’o, Matiri George Munene, Justin Obwoge Okenye. (2023). Bayesian Interval Estimation in a Non-Homogeneous Poisson Process with Delayed S-Shaped Intensity Function. American Journal of Theoretical and Applied Statistics, 12(3), 43-50. https://doi.org/10.11648/j.ajtas.20231203.12

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ACS Style

Otieno Collins; Orawo Luke Akong’o; Matiri George Munene; Justin Obwoge Okenye. Bayesian Interval Estimation in a Non-Homogeneous Poisson Process with Delayed S-Shaped Intensity Function. Am. J. Theor. Appl. Stat. 2023, 12(3), 43-50. doi: 10.11648/j.ajtas.20231203.12

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AMA Style

Otieno Collins, Orawo Luke Akong’o, Matiri George Munene, Justin Obwoge Okenye. Bayesian Interval Estimation in a Non-Homogeneous Poisson Process with Delayed S-Shaped Intensity Function. Am J Theor Appl Stat. 2023;12(3):43-50. doi: 10.11648/j.ajtas.20231203.12

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@article{10.11648/j.ajtas.20231203.12,
  author = {Otieno Collins and Orawo Luke Akong’o and Matiri George Munene and Justin Obwoge Okenye},
  title = {Bayesian Interval Estimation in a Non-Homogeneous Poisson Process with Delayed S-Shaped Intensity Function},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {12},
  number = {3},
  pages = {43-50},
  doi = {10.11648/j.ajtas.20231203.12},
  url = {https://doi.org/10.11648/j.ajtas.20231203.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20231203.12},
  abstract = {Software reliability assessment has been explored by many researchers over the past decades. With the increasing development of new complex software systems, accurate methods for estimating reliability model parameters are needed. Facilitated by the increasing use of computer systems in various sectors such as air traffic control, banking, industrial processes, and government operations, developing accurate reliability assessment methods is indispensable. The Delayed S-shaped software reliability model is one of the non-homogeneous Poisson process (NHPP) software reliability models proposed for capturing error detection and removal processes in software reliability testing. Many researchers have fitted the model to software failure data and performed estimation using the Maximum Likelihood method and Bayesian approach, however, construction of Bayesian credible sets for the parameters of this model and comparison of their efficiencies with the Wald confidence intervals using simulation have not been explored. The Bayesian interval estimation was conducted with three different joint prior distributions assigned to the parameters α and β of the model, namely the gamma distributed informative prior and, 1/α, and 1/αβ as non-informative priors. The Bayesian credible intervals and Wald confidence intervals for the two parameters were compared on the basis of interval lengths and coverage probabilities. The simulation was assumed to emulate the end-user environment and can generate inter-failure times data for the study. The Delayed S-shaped reliability model variables were simulated with fixed parameters set at (α, β)=(20, 0.5). The hyperparameters for the informative prior were chosen such that they have minimal effect on the results. In other words, the prior information does not swamp the information from the data. The Bayesian method yields superior results, as evidenced by shorter interval lengths and higher coverage probabilities in Table 1.},
 year = {2023}
}

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TY  - JOUR
T1  - Bayesian Interval Estimation in a Non-Homogeneous Poisson Process with Delayed S-Shaped Intensity Function
AU  - Otieno Collins
AU  - Orawo Luke Akong’o
AU  - Matiri George Munene
AU  - Justin Obwoge Okenye
Y1  - 2023/06/05
PY  - 2023
N1  - https://doi.org/10.11648/j.ajtas.20231203.12
DO  - 10.11648/j.ajtas.20231203.12
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 43
EP  - 50
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20231203.12
AB  - Software reliability assessment has been explored by many researchers over the past decades. With the increasing development of new complex software systems, accurate methods for estimating reliability model parameters are needed. Facilitated by the increasing use of computer systems in various sectors such as air traffic control, banking, industrial processes, and government operations, developing accurate reliability assessment methods is indispensable. The Delayed S-shaped software reliability model is one of the non-homogeneous Poisson process (NHPP) software reliability models proposed for capturing error detection and removal processes in software reliability testing. Many researchers have fitted the model to software failure data and performed estimation using the Maximum Likelihood method and Bayesian approach, however, construction of Bayesian credible sets for the parameters of this model and comparison of their efficiencies with the Wald confidence intervals using simulation have not been explored. The Bayesian interval estimation was conducted with three different joint prior distributions assigned to the parameters α and β of the model, namely the gamma distributed informative prior and, 1/α, and 1/αβ as non-informative priors. The Bayesian credible intervals and Wald confidence intervals for the two parameters were compared on the basis of interval lengths and coverage probabilities. The simulation was assumed to emulate the end-user environment and can generate inter-failure times data for the study. The Delayed S-shaped reliability model variables were simulated with fixed parameters set at (α, β)=(20, 0.5). The hyperparameters for the informative prior were chosen such that they have minimal effect on the results. In other words, the prior information does not swamp the information from the data. The Bayesian method yields superior results, as evidenced by shorter interval lengths and higher coverage probabilities in Table 1.
VL  - 12
IS  - 3
ER  -

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Author Information

Otieno Collins

Department of Mathematics, Egerton University, Egerton, Kenya
Orawo Luke Akong’o

Department of Mathematics, Egerton University, Egerton, Kenya
Matiri George Munene

Department of Mathematics, Egerton University, Egerton, Kenya
Justin Obwoge Okenye

Department of Mathematics, Egerton University, Egerton, Kenya

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APA Style

Otieno Collins, Orawo Luke Akong’o, Matiri George Munene, Justin Obwoge Okenye. (2023). Bayesian Interval Estimation in a Non-Homogeneous Poisson Process with Delayed S-Shaped Intensity Function. American Journal of Theoretical and Applied Statistics, 12(3), 43-50. https://doi.org/10.11648/j.ajtas.20231203.12

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ACS Style

Otieno Collins; Orawo Luke Akong’o; Matiri George Munene; Justin Obwoge Okenye. Bayesian Interval Estimation in a Non-Homogeneous Poisson Process with Delayed S-Shaped Intensity Function. Am. J. Theor. Appl. Stat. 2023, 12(3), 43-50. doi: 10.11648/j.ajtas.20231203.12

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AMA Style

Otieno Collins, Orawo Luke Akong’o, Matiri George Munene, Justin Obwoge Okenye. Bayesian Interval Estimation in a Non-Homogeneous Poisson Process with Delayed S-Shaped Intensity Function. Am J Theor Appl Stat. 2023;12(3):43-50. doi: 10.11648/j.ajtas.20231203.12

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@article{10.11648/j.ajtas.20231203.12,
  author = {Otieno Collins and Orawo Luke Akong’o and Matiri George Munene and Justin Obwoge Okenye},
  title = {Bayesian Interval Estimation in a Non-Homogeneous Poisson Process with Delayed S-Shaped Intensity Function},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {12},
  number = {3},
  pages = {43-50},
  doi = {10.11648/j.ajtas.20231203.12},
  url = {https://doi.org/10.11648/j.ajtas.20231203.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20231203.12},
  abstract = {Software reliability assessment has been explored by many researchers over the past decades. With the increasing development of new complex software systems, accurate methods for estimating reliability model parameters are needed. Facilitated by the increasing use of computer systems in various sectors such as air traffic control, banking, industrial processes, and government operations, developing accurate reliability assessment methods is indispensable. The Delayed S-shaped software reliability model is one of the non-homogeneous Poisson process (NHPP) software reliability models proposed for capturing error detection and removal processes in software reliability testing. Many researchers have fitted the model to software failure data and performed estimation using the Maximum Likelihood method and Bayesian approach, however, construction of Bayesian credible sets for the parameters of this model and comparison of their efficiencies with the Wald confidence intervals using simulation have not been explored. The Bayesian interval estimation was conducted with three different joint prior distributions assigned to the parameters α and β of the model, namely the gamma distributed informative prior and, 1/α, and 1/αβ as non-informative priors. The Bayesian credible intervals and Wald confidence intervals for the two parameters were compared on the basis of interval lengths and coverage probabilities. The simulation was assumed to emulate the end-user environment and can generate inter-failure times data for the study. The Delayed S-shaped reliability model variables were simulated with fixed parameters set at (α, β)=(20, 0.5). The hyperparameters for the informative prior were chosen such that they have minimal effect on the results. In other words, the prior information does not swamp the information from the data. The Bayesian method yields superior results, as evidenced by shorter interval lengths and higher coverage probabilities in Table 1.},
 year = {2023}
}

Copy | Download

TY  - JOUR
T1  - Bayesian Interval Estimation in a Non-Homogeneous Poisson Process with Delayed S-Shaped Intensity Function
AU  - Otieno Collins
AU  - Orawo Luke Akong’o
AU  - Matiri George Munene
AU  - Justin Obwoge Okenye
Y1  - 2023/06/05
PY  - 2023
N1  - https://doi.org/10.11648/j.ajtas.20231203.12
DO  - 10.11648/j.ajtas.20231203.12
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 43
EP  - 50
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20231203.12
AB  - Software reliability assessment has been explored by many researchers over the past decades. With the increasing development of new complex software systems, accurate methods for estimating reliability model parameters are needed. Facilitated by the increasing use of computer systems in various sectors such as air traffic control, banking, industrial processes, and government operations, developing accurate reliability assessment methods is indispensable. The Delayed S-shaped software reliability model is one of the non-homogeneous Poisson process (NHPP) software reliability models proposed for capturing error detection and removal processes in software reliability testing. Many researchers have fitted the model to software failure data and performed estimation using the Maximum Likelihood method and Bayesian approach, however, construction of Bayesian credible sets for the parameters of this model and comparison of their efficiencies with the Wald confidence intervals using simulation have not been explored. The Bayesian interval estimation was conducted with three different joint prior distributions assigned to the parameters α and β of the model, namely the gamma distributed informative prior and, 1/α, and 1/αβ as non-informative priors. The Bayesian credible intervals and Wald confidence intervals for the two parameters were compared on the basis of interval lengths and coverage probabilities. The simulation was assumed to emulate the end-user environment and can generate inter-failure times data for the study. The Delayed S-shaped reliability model variables were simulated with fixed parameters set at (α, β)=(20, 0.5). The hyperparameters for the informative prior were chosen such that they have minimal effect on the results. In other words, the prior information does not swamp the information from the data. The Bayesian method yields superior results, as evidenced by shorter interval lengths and higher coverage probabilities in Table 1.
VL  - 12
IS  - 3
ER  -

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