Profile Likelihood Confidence Intervals for the Parameters of a Nonhomogeneous Poisson Process with Linear Rate

Orawo Luke Akongo

doi:doi:10.11648/j.ajtas.20221103.14

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Profile Likelihood Confidence Intervals for the Parameters of a Nonhomogeneous Poisson Process with Linear Rate

Orawo Luke Akongo

Published in American Journal of Theoretical and Applied Statistics (Volume 11, Issue 3)

Received: 9 April 2022 Accepted: 9 May 2022 Published: 27 June 2022

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Abstract

Nonhomogeneous Poisson Processes (NHPP) are commonly used to model count data where the rate of occurrence of events in a given time period is dependent on time. Examples exist in the literature where NHPP has been used to model real life count data for the purpose of parameter estimation and prediction. The most common methods used to obtain the point estimates of the parameters of the NHPP are the method maximum likelihood and the ordinary least squares method. The commonly used Wald-type confidence intervals are based on the assumption of asymptotic normality and are inaccurate when this assumption is violated This study considers an alternative method based the profile likelihood function to construct approximate confidence intervals for the parameters of a nonhomogeneous Poisson process with linear rate λ(t)=α+βt, based on the number of counts in measurement subintervals. Such a linear rate function is applicable in situations where piecewise-linear approximation to a general rate function is adequate. The profile likelihood confidence intervals for the two parameters are constructed from the graphs of their respective relative profile likelihood functions, which are obtained numerically from the joint relative likelihood function. Simulations were used to compare the profile likelihood and Wald confidence intervals on the basis of coverage probability and mean length. The effects of sample size (number of subintervals) on the interval estimates of the parameters were also investigated. The results of the simulation study show that the profile likelihood method is superior to the Wald method since it yields shorter confide intervals containing plausible values of each of the two parameters.

Published in	American Journal of Theoretical and Applied Statistics (Volume 11, Issue 3)
DOI	10.11648/j.ajtas.20221103.14
Page(s)	102-108
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), 2024. Published by Science Publishing Group

Keywords

Relative Likelihood Function, Profile Likelihood Function, Profile Likelihood Confidence Intervals, Wald Confidence Intervals

References

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[2]	Sumiati, I., Rahmani, U., Supian, S., Subiyanto (2019). Application of the Nonhomogeneous Poisson Process for Counting Earthquake. World Scientific News, 127 (3), 163-176.
[3]	Albert Orwa Akuno, Timothy Mutunga Ndonye, Janiffer Mwende Nthiwa, Luke Akong’o Orawo. Regression Approach to Parameter Estimation of an Exponential Software Reliability Model. American Journal of Theoretical and Applied Statistics. Vol. 5, No. 3, 2016, pp. 80-86. doi: 10.11648/j.ajtas.20160503.11.
[4]	Grabski, Franciszek. "Nonhomogeneous Stochastic Processes Connected to Poisson Process" Scientific Journal of Polish Naval Academy, vol. 213, no. 2, 2018, pp. 5-15. https://doi.org/10.2478/sjpna-2018-0009
[5]	Jain, M., Gupta, K. P. (2012). Reliability Analysis of a Software With Non Homogeneous Poisson Process (NHPP) Failure Intensity. Journal of Bioinformatics and Intelligent Control. 1 (1): 1-19 DOI: 10.1166/jbic.2013.1024.
[6]	Frenkel, I., Gertsbakh, I. B., & Khvatskin, L. (2003). Parameter Estimation and Hypotheses Testing for Nonhomogeneous Poisson Process. Transport and Telecommunication Journal. 4 (2), 9-17.
[7]	Meyfroyt, P. H. A. (2012) Parameter Estimation for Software Reliability Models. Ph.D. Thesis, Universidad Carlos III de Madrid, Madrid.
[8]	Stringfellow, C., & Andrews, A. A. (2004). An Empirical Method for Selecting Software Reliability Growth Models. Empirical Software Engineering, 7 (4), 319-343. DOI: 10.1023/A:1020515105175.
[9]	Chalmers, R. P., Pek, J., Yang, L. (2017) Profile-likelihood Confidence Intervals in Item Response Theory Models, Multivariate Behavioral Research, 52: 5, 533-550, DOI: 10.1080/00273171.2017.1329082.
[10]	Gimenez, O., Choquet, R., Lamor, L., Scofield, P., Fletcher, D., Lebreton, J., & Pradel, R. (2005). Efficient profile-likelihood confidence intervals for capture-recapture models. Journal of Agricultural, Biological, and Environmental Statistics, 10 (2), 184-196. DOI: 10.1198/108571105X46462.
[11]	Reich, G., & Judd, K. L. (2019). Efficient Likelihood Ratio Confidence Intervals using Constrained Optimization. Econometrics: Mathematical Methods & Programming eJournal. http://dx.doi.org/10.2139/ssrn.3455484
[12]	Evans, M. A., Kim, H.-M., & O’Brien, T. E. (1996). An Application of Profile-Likelihood Based Confidence Interval to Capture: Recapture Estimators. Journal of Agricultural, Biological, and Environmental Statistics, 1 (1), 131. doi: 10.2307/1400565.
[13]	Kalbfleisch, J. G. (1985) Probability and Statistical Inference, Volume 2: Statistical Inference. 2nd Edition, Springer Verlag, New York. https://doi.org/10.1007/978-1-4612-5136-1.
[14]	Tiangiang Yan (2019) Nonhomogeneous Poisson Process Models with a Generalized Bathtub Intensity Function for Repairable System. MSc Thesis: Russ College of Engineering and Technology, Ohio University-USA.
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Orawo Luke Akongo. (2022). Profile Likelihood Confidence Intervals for the Parameters of a Nonhomogeneous Poisson Process with Linear Rate. American Journal of Theoretical and Applied Statistics, 11(3), 102-108. https://doi.org/10.11648/j.ajtas.20221103.14

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Orawo Luke Akongo. Profile Likelihood Confidence Intervals for the Parameters of a Nonhomogeneous Poisson Process with Linear Rate. Am. J. Theor. Appl. Stat. 2022, 11(3), 102-108. doi: 10.11648/j.ajtas.20221103.14

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AMA Style

Orawo Luke Akongo. Profile Likelihood Confidence Intervals for the Parameters of a Nonhomogeneous Poisson Process with Linear Rate. Am J Theor Appl Stat. 2022;11(3):102-108. doi: 10.11648/j.ajtas.20221103.14

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@article{10.11648/j.ajtas.20221103.14,
  author = {Orawo Luke Akongo},
  title = {Profile Likelihood Confidence Intervals for the Parameters of a Nonhomogeneous Poisson Process with Linear Rate},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {11},
  number = {3},
  pages = {102-108},
  doi = {10.11648/j.ajtas.20221103.14},
  url = {https://doi.org/10.11648/j.ajtas.20221103.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20221103.14},
  abstract = {Nonhomogeneous Poisson Processes (NHPP) are commonly used to model count data where the rate of occurrence of events in a given time period is dependent on time. Examples exist in the literature where NHPP has been used to model real life count data for the purpose of parameter estimation and prediction. The most common methods used to obtain the point estimates of the parameters of the NHPP are the method maximum likelihood and the ordinary least squares method. The commonly used Wald-type confidence intervals are based on the assumption of asymptotic normality and are inaccurate when this assumption is violated This study considers an alternative method based the profile likelihood function to construct approximate confidence intervals for the parameters of a nonhomogeneous Poisson process with linear rate λ(t)=α+βt, based on the number of counts in measurement subintervals. Such a linear rate function is applicable in situations where piecewise-linear approximation to a general rate function is adequate. The profile likelihood confidence intervals for the two parameters are constructed from the graphs of their respective relative profile likelihood functions, which are obtained numerically from the joint relative likelihood function. Simulations were used to compare the profile likelihood and Wald confidence intervals on the basis of coverage probability and mean length. The effects of sample size (number of subintervals) on the interval estimates of the parameters were also investigated. The results of the simulation study show that the profile likelihood method is superior to the Wald method since it yields shorter confide intervals containing plausible values of each of the two parameters.},
 year = {2022}
}

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TY  - JOUR
T1  - Profile Likelihood Confidence Intervals for the Parameters of a Nonhomogeneous Poisson Process with Linear Rate
AU  - Orawo Luke Akongo
Y1  - 2022/06/27
PY  - 2022
N1  - https://doi.org/10.11648/j.ajtas.20221103.14
DO  - 10.11648/j.ajtas.20221103.14
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  - 102
EP  - 108
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20221103.14
AB  - Nonhomogeneous Poisson Processes (NHPP) are commonly used to model count data where the rate of occurrence of events in a given time period is dependent on time. Examples exist in the literature where NHPP has been used to model real life count data for the purpose of parameter estimation and prediction. The most common methods used to obtain the point estimates of the parameters of the NHPP are the method maximum likelihood and the ordinary least squares method. The commonly used Wald-type confidence intervals are based on the assumption of asymptotic normality and are inaccurate when this assumption is violated This study considers an alternative method based the profile likelihood function to construct approximate confidence intervals for the parameters of a nonhomogeneous Poisson process with linear rate λ(t)=α+βt, based on the number of counts in measurement subintervals. Such a linear rate function is applicable in situations where piecewise-linear approximation to a general rate function is adequate. The profile likelihood confidence intervals for the two parameters are constructed from the graphs of their respective relative profile likelihood functions, which are obtained numerically from the joint relative likelihood function. Simulations were used to compare the profile likelihood and Wald confidence intervals on the basis of coverage probability and mean length. The effects of sample size (number of subintervals) on the interval estimates of the parameters were also investigated. The results of the simulation study show that the profile likelihood method is superior to the Wald method since it yields shorter confide intervals containing plausible values of each of the two parameters.
VL  - 11
IS  - 3
ER  -

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Author Information

Orawo Luke Akongo

Mathematics Department, Egerton University, Njoro, Kenya

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Orawo Luke Akongo. (2022). Profile Likelihood Confidence Intervals for the Parameters of a Nonhomogeneous Poisson Process with Linear Rate. American Journal of Theoretical and Applied Statistics, 11(3), 102-108. https://doi.org/10.11648/j.ajtas.20221103.14

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ACS Style

Orawo Luke Akongo. Profile Likelihood Confidence Intervals for the Parameters of a Nonhomogeneous Poisson Process with Linear Rate. Am. J. Theor. Appl. Stat. 2022, 11(3), 102-108. doi: 10.11648/j.ajtas.20221103.14

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AMA Style

Orawo Luke Akongo. Profile Likelihood Confidence Intervals for the Parameters of a Nonhomogeneous Poisson Process with Linear Rate. Am J Theor Appl Stat. 2022;11(3):102-108. doi: 10.11648/j.ajtas.20221103.14

Copy | Download

@article{10.11648/j.ajtas.20221103.14,
  author = {Orawo Luke Akongo},
  title = {Profile Likelihood Confidence Intervals for the Parameters of a Nonhomogeneous Poisson Process with Linear Rate},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {11},
  number = {3},
  pages = {102-108},
  doi = {10.11648/j.ajtas.20221103.14},
  url = {https://doi.org/10.11648/j.ajtas.20221103.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20221103.14},
  abstract = {Nonhomogeneous Poisson Processes (NHPP) are commonly used to model count data where the rate of occurrence of events in a given time period is dependent on time. Examples exist in the literature where NHPP has been used to model real life count data for the purpose of parameter estimation and prediction. The most common methods used to obtain the point estimates of the parameters of the NHPP are the method maximum likelihood and the ordinary least squares method. The commonly used Wald-type confidence intervals are based on the assumption of asymptotic normality and are inaccurate when this assumption is violated This study considers an alternative method based the profile likelihood function to construct approximate confidence intervals for the parameters of a nonhomogeneous Poisson process with linear rate λ(t)=α+βt, based on the number of counts in measurement subintervals. Such a linear rate function is applicable in situations where piecewise-linear approximation to a general rate function is adequate. The profile likelihood confidence intervals for the two parameters are constructed from the graphs of their respective relative profile likelihood functions, which are obtained numerically from the joint relative likelihood function. Simulations were used to compare the profile likelihood and Wald confidence intervals on the basis of coverage probability and mean length. The effects of sample size (number of subintervals) on the interval estimates of the parameters were also investigated. The results of the simulation study show that the profile likelihood method is superior to the Wald method since it yields shorter confide intervals containing plausible values of each of the two parameters.},
 year = {2022}
}

Copy | Download

TY  - JOUR
T1  - Profile Likelihood Confidence Intervals for the Parameters of a Nonhomogeneous Poisson Process with Linear Rate
AU  - Orawo Luke Akongo
Y1  - 2022/06/27
PY  - 2022
N1  - https://doi.org/10.11648/j.ajtas.20221103.14
DO  - 10.11648/j.ajtas.20221103.14
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  - 102
EP  - 108
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20221103.14
AB  - Nonhomogeneous Poisson Processes (NHPP) are commonly used to model count data where the rate of occurrence of events in a given time period is dependent on time. Examples exist in the literature where NHPP has been used to model real life count data for the purpose of parameter estimation and prediction. The most common methods used to obtain the point estimates of the parameters of the NHPP are the method maximum likelihood and the ordinary least squares method. The commonly used Wald-type confidence intervals are based on the assumption of asymptotic normality and are inaccurate when this assumption is violated This study considers an alternative method based the profile likelihood function to construct approximate confidence intervals for the parameters of a nonhomogeneous Poisson process with linear rate λ(t)=α+βt, based on the number of counts in measurement subintervals. Such a linear rate function is applicable in situations where piecewise-linear approximation to a general rate function is adequate. The profile likelihood confidence intervals for the two parameters are constructed from the graphs of their respective relative profile likelihood functions, which are obtained numerically from the joint relative likelihood function. Simulations were used to compare the profile likelihood and Wald confidence intervals on the basis of coverage probability and mean length. The effects of sample size (number of subintervals) on the interval estimates of the parameters were also investigated. The results of the simulation study show that the profile likelihood method is superior to the Wald method since it yields shorter confide intervals containing plausible values of each of the two parameters.
VL  - 11
IS  - 3
ER  -

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