Maximum Likelihood Estimation of Parameters for Poisson-exponential Distribution Under Progressive Type I Interval Censoring

Peter Tumwa Situma; Leo Odongo

doi:doi:10.11648/j.ajtas.20200902.11

| Peer-Reviewed

Maximum Likelihood Estimation of Parameters for Poisson-exponential Distribution Under Progressive Type I Interval Censoring

Peter Tumwa Situma, Leo Odongo

Published in American Journal of Theoretical and Applied Statistics (Volume 9, Issue 2)

Received: 13 March 2020 Accepted: 10 April 2020 Published: 23 April 2020

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

This paper considers the problem of estimating the parameters of Poisson-Exponential (PE) distribution under progressive type-I interval censoring scheme. PE is a two-parameter lifetime distribution having an increasing hazard function. It has been applied in complementary risks problems in latent risks, that is in scenarios where maximum lifetime values are observed but information concerning factors accounting for component failure is unavailable. Under progressive type-I interval censoring, observations are known within two consecutively pre-arranged times and items would be withdrawn at pre-scheduled time points. This scheme is most suitable in those cases where continuous examination is impossible. Maximum likelihood estimates of Poisson-Exponential parameters are obtained via Expectation-Maximization (EM) algorithm. The EM algorithm is preferred as it has been confirmed to be a more superior tool when dealing with incomplete data sets having missing values, or models having truncated distributions. Asymptotic properties of the estimates are studied through simulation and compared based on bias and the mean squared error under different censoring schemes and parameter values. It is concluded that for an increasing sample size, the estimated values of the parameters tend to the true value. Among the four censoring schemes considered, the third scheme provides the most precise and accurate results followed by fourth scheme, first scheme and finally the second scheme.

Published in	American Journal of Theoretical and Applied Statistics (Volume 9, Issue 2)
DOI	10.11648/j.ajtas.20200902.11
Page(s)	14-20
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), 2020. Published by Science Publishing Group

Keywords

Poisson-exponential, Progressive Type I Interval Censoring, Maximum Likelihood Estimation, EM Algorithm

References

[1]	Tomazella, V. L. D., Cancho, V. G., and Louzada, F. (2013). Bayesian reference analysis for the Poisson-Exponential lifetime distribution. Chilean Journal of Statistics; 4 (1): 99-113.
[2]	Gupta, R. D., and Kundu, D. (1999). Generalized exponential distributions. Australian and New Zealand Journal of Statistics; 41: 73-188.
[3]	Kus, C. (2007). A new lifetime distribution. Computational Statistics and Data Analysis; 51: 4497-4509.
[4]	Barreto, S. W., and Cribari, N. F. (2009). A generalization of Exponential-Poisson distribution. Statistics and Probability Letters; 79: 2493-2500.
[5]	Cancho, V. G., Louzada-Neto, F. and Barriga, G. D. C. (2011). The Poisson-Exponential lifetime distribution. Computational Statistics and Data Analysis; 55: 677-686.
[6]	Louzada-Neto, F., Cancho, V. G., and Barriga, G. D. C. (2011). The Poisson-Exponential distribution: a Bayesian approach. Journal of Applied Statistics; 38: 1239-1248.
[7]	Singh S. K., Singh, U. and Manoj, K. M. (2014). Estimation for the Parameter of Poisson-Exponential Distribution under Bayesian Paradigm, Journal of Data Science; 12: 157-173.
[8]	Rodrigues, G. C., Louzada, F., and Ramos, P. L. (2016). Poisson-exponential distribution: different methods of estimation. Journal of Applied Statistics; 45 (1): 128–144.
[9]	Gitahi, J. N., Kung’u, J., and Odongo, L. (2017). Estimation of the Parameters of Poisson-Exponential Distribution Based on Progressively Type II censoring using the Expectation-Maximization Algorithm. American Journal of Theoretical and Applied Statistics; 6: 141-149.
[10]	Belaghi, R. A., Asl, M. N., Alma, O. G., Singh, S., and Vasfi, M. (2018). Estimation and Prediction for the Poisson-Exponential Distribution Based on Type-II Censored Data. American Journal of Mathematical and Management Sciences. DOI: 10.1080/01966324.2018.1484827.
[11]	Cohen, A. C. (1963). Progressively censored samples in life testing. Technometrics; 5: 327-339.
[12]	Balakrishnan, N. (2007). Progressive censoring methodology: an appraisal. Research Papers in Economics; 16: 211-259.
[13]	Balakrishnan, N., and Cramer, E. (2014). The art of Progressive censoring: application to Reliability and Quality. Basel: Birkhauser.
[14]	Aggarwala, R. (2001). Progressive interval censoring: some mathematical results with applications to inference. Communication in Statistics Theory and Methods; 30: 1921–1935.
[15]	Ng, H. K. T., and Wang, Z. (2009). Statistical estimation for the parameters of Weibull distribution based on progressively type-I interval-censored sample. Journal of Statistical Computation and Simulation; 79: 145–159.
[16]	Cheng, C., Chen, J., and Li, Z. (2010). A new algorithm for maximum likelihood estimation with progressive Type-I interval-censored data. Communications in Statistics-Simulation and Computation; 39: 750–766.
[17]	Chen, D. G., and Lio, Y. L. (2010). Parameter estimations for generalized exponential distribution under progressive Type-I interval censoring. Computational Statistics and Data Analysis; 54: 1581–1591.
[18]	Lio, Y. L., Chen, D. G., and Tsai, T. R. (2011). Parameter estimations for generalized Rayleigh distribution under progressively Type-I interval-censored data. Open Journal of Statistics; 1: 46–57.
[19]	Lin, Y. J., and Lio, Y. L. (2012). Bayesian inference under progressive type-I interval censoring. Journal of Applied Statistics; 39: 1811–1824.
[20]	Teimouri, M., and Gupta, A. K. (2012). Estimation methods for the Gompertz-Makeham distribution under progressively Type-I interval censoring scheme. National Academy Science Letters; 35: 227–235.
[21]	Singh, S., and Tripathi, Y. M. (2018). Estimating the Parameters of an inverse Weibull distribution under progressive type-I interval censoring. Statistical Papers; 59 (1): 21-56.
[22]	Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm (with discussion). Journal of the Royal Statistical Society, B; 39: 1-39.
[23]	Ihaka, R., and Gentleman, R. (1996). A language for data analysis and graphics. Journal of Computational and Graphical Statistics; 5: 299-314.
[24]	Kemp, C. D., and Kemp, W. (1987). Rapid generation of frequency tables. Applied Statistics; 36: 277-282.

Cite This Article

Plain Text BibTeX RIS

APA Style

Peter Tumwa Situma, Leo Odongo. (2020). Maximum Likelihood Estimation of Parameters for Poisson-exponential Distribution Under Progressive Type I Interval Censoring. American Journal of Theoretical and Applied Statistics, 9(2), 14-20. https://doi.org/10.11648/j.ajtas.20200902.11

Copy | Download

ACS Style

Peter Tumwa Situma; Leo Odongo. Maximum Likelihood Estimation of Parameters for Poisson-exponential Distribution Under Progressive Type I Interval Censoring. Am. J. Theor. Appl. Stat. 2020, 9(2), 14-20. doi: 10.11648/j.ajtas.20200902.11

Copy | Download

AMA Style

Peter Tumwa Situma, Leo Odongo. Maximum Likelihood Estimation of Parameters for Poisson-exponential Distribution Under Progressive Type I Interval Censoring. Am J Theor Appl Stat. 2020;9(2):14-20. doi: 10.11648/j.ajtas.20200902.11

Copy | Download

@article{10.11648/j.ajtas.20200902.11,
  author = {Peter Tumwa Situma and Leo Odongo},
  title = {Maximum Likelihood Estimation of Parameters for Poisson-exponential Distribution Under Progressive Type I Interval Censoring},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {9},
  number = {2},
  pages = {14-20},
  doi = {10.11648/j.ajtas.20200902.11},
  url = {https://doi.org/10.11648/j.ajtas.20200902.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20200902.11},
  abstract = {This paper considers the problem of estimating the parameters of Poisson-Exponential (PE) distribution under progressive type-I interval censoring scheme. PE is a two-parameter lifetime distribution having an increasing hazard function. It has been applied in complementary risks problems in latent risks, that is in scenarios where maximum lifetime values are observed but information concerning factors accounting for component failure is unavailable. Under progressive type-I interval censoring, observations are known within two consecutively pre-arranged times and items would be withdrawn at pre-scheduled time points. This scheme is most suitable in those cases where continuous examination is impossible. Maximum likelihood estimates of Poisson-Exponential parameters are obtained via Expectation-Maximization (EM) algorithm. The EM algorithm is preferred as it has been confirmed to be a more superior tool when dealing with incomplete data sets having missing values, or models having truncated distributions. Asymptotic properties of the estimates are studied through simulation and compared based on bias and the mean squared error under different censoring schemes and parameter values. It is concluded that for an increasing sample size, the estimated values of the parameters tend to the true value. Among the four censoring schemes considered, the third scheme provides the most precise and accurate results followed by fourth scheme, first scheme and finally the second scheme.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Maximum Likelihood Estimation of Parameters for Poisson-exponential Distribution Under Progressive Type I Interval Censoring
AU  - Peter Tumwa Situma
AU  - Leo Odongo
Y1  - 2020/04/23
PY  - 2020
N1  - https://doi.org/10.11648/j.ajtas.20200902.11
DO  - 10.11648/j.ajtas.20200902.11
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  - 14
EP  - 20
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20200902.11
AB  - This paper considers the problem of estimating the parameters of Poisson-Exponential (PE) distribution under progressive type-I interval censoring scheme. PE is a two-parameter lifetime distribution having an increasing hazard function. It has been applied in complementary risks problems in latent risks, that is in scenarios where maximum lifetime values are observed but information concerning factors accounting for component failure is unavailable. Under progressive type-I interval censoring, observations are known within two consecutively pre-arranged times and items would be withdrawn at pre-scheduled time points. This scheme is most suitable in those cases where continuous examination is impossible. Maximum likelihood estimates of Poisson-Exponential parameters are obtained via Expectation-Maximization (EM) algorithm. The EM algorithm is preferred as it has been confirmed to be a more superior tool when dealing with incomplete data sets having missing values, or models having truncated distributions. Asymptotic properties of the estimates are studied through simulation and compared based on bias and the mean squared error under different censoring schemes and parameter values. It is concluded that for an increasing sample size, the estimated values of the parameters tend to the true value. Among the four censoring schemes considered, the third scheme provides the most precise and accurate results followed by fourth scheme, first scheme and finally the second scheme.
VL  - 9
IS  - 2
ER  -

Copy | Download

Author Information

Peter Tumwa Situma

Department of Mathematics and Actuarial Science, Kenyatta University (KU), Nairobi, Kenya
Leo Odongo

Department of Mathematics and Actuarial Science, Kenyatta University (KU), Nairobi, Kenya

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Peter Tumwa Situma, Leo Odongo. (2020). Maximum Likelihood Estimation of Parameters for Poisson-exponential Distribution Under Progressive Type I Interval Censoring. American Journal of Theoretical and Applied Statistics, 9(2), 14-20. https://doi.org/10.11648/j.ajtas.20200902.11

Copy | Download

ACS Style

Peter Tumwa Situma; Leo Odongo. Maximum Likelihood Estimation of Parameters for Poisson-exponential Distribution Under Progressive Type I Interval Censoring. Am. J. Theor. Appl. Stat. 2020, 9(2), 14-20. doi: 10.11648/j.ajtas.20200902.11

Copy | Download

AMA Style

Peter Tumwa Situma, Leo Odongo. Maximum Likelihood Estimation of Parameters for Poisson-exponential Distribution Under Progressive Type I Interval Censoring. Am J Theor Appl Stat. 2020;9(2):14-20. doi: 10.11648/j.ajtas.20200902.11

Copy | Download

@article{10.11648/j.ajtas.20200902.11,
  author = {Peter Tumwa Situma and Leo Odongo},
  title = {Maximum Likelihood Estimation of Parameters for Poisson-exponential Distribution Under Progressive Type I Interval Censoring},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {9},
  number = {2},
  pages = {14-20},
  doi = {10.11648/j.ajtas.20200902.11},
  url = {https://doi.org/10.11648/j.ajtas.20200902.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20200902.11},
  abstract = {This paper considers the problem of estimating the parameters of Poisson-Exponential (PE) distribution under progressive type-I interval censoring scheme. PE is a two-parameter lifetime distribution having an increasing hazard function. It has been applied in complementary risks problems in latent risks, that is in scenarios where maximum lifetime values are observed but information concerning factors accounting for component failure is unavailable. Under progressive type-I interval censoring, observations are known within two consecutively pre-arranged times and items would be withdrawn at pre-scheduled time points. This scheme is most suitable in those cases where continuous examination is impossible. Maximum likelihood estimates of Poisson-Exponential parameters are obtained via Expectation-Maximization (EM) algorithm. The EM algorithm is preferred as it has been confirmed to be a more superior tool when dealing with incomplete data sets having missing values, or models having truncated distributions. Asymptotic properties of the estimates are studied through simulation and compared based on bias and the mean squared error under different censoring schemes and parameter values. It is concluded that for an increasing sample size, the estimated values of the parameters tend to the true value. Among the four censoring schemes considered, the third scheme provides the most precise and accurate results followed by fourth scheme, first scheme and finally the second scheme.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Maximum Likelihood Estimation of Parameters for Poisson-exponential Distribution Under Progressive Type I Interval Censoring
AU  - Peter Tumwa Situma
AU  - Leo Odongo
Y1  - 2020/04/23
PY  - 2020
N1  - https://doi.org/10.11648/j.ajtas.20200902.11
DO  - 10.11648/j.ajtas.20200902.11
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  - 14
EP  - 20
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20200902.11
AB  - This paper considers the problem of estimating the parameters of Poisson-Exponential (PE) distribution under progressive type-I interval censoring scheme. PE is a two-parameter lifetime distribution having an increasing hazard function. It has been applied in complementary risks problems in latent risks, that is in scenarios where maximum lifetime values are observed but information concerning factors accounting for component failure is unavailable. Under progressive type-I interval censoring, observations are known within two consecutively pre-arranged times and items would be withdrawn at pre-scheduled time points. This scheme is most suitable in those cases where continuous examination is impossible. Maximum likelihood estimates of Poisson-Exponential parameters are obtained via Expectation-Maximization (EM) algorithm. The EM algorithm is preferred as it has been confirmed to be a more superior tool when dealing with incomplete data sets having missing values, or models having truncated distributions. Asymptotic properties of the estimates are studied through simulation and compared based on bias and the mean squared error under different censoring schemes and parameter values. It is concluded that for an increasing sample size, the estimated values of the parameters tend to the true value. Among the four censoring schemes considered, the third scheme provides the most precise and accurate results followed by fourth scheme, first scheme and finally the second scheme.
VL  - 9
IS  - 2
ER  -

Copy | Download