Maximum Likelihood Estimation for a Progressively Type II Censored Generalized Inverted Exponential Distribution via EM Algorithm

Karuoya Grace Njeri; Edward Gachangi Njenga

doi:doi:10.11648/j.ajtas.20211001.13

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Maximum Likelihood Estimation for a Progressively Type II Censored Generalized Inverted Exponential Distribution via EM Algorithm

Karuoya Grace Njeri, Edward Gachangi Njenga

Published in American Journal of Theoretical and Applied Statistics (Volume 10, Issue 1)

Received: 20 December 2020 Accepted: 11 January 2021 Published: 22 January 2021

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Abstract

The Generalized Inverted Exponential (GIE) distribution is a mixed lifetime model used in a number of fields such as queuing theory, testing of products or components and modelling the speed of winds. The study aims to focus on the determination of maximum likelihood estimates of GIE distribution when the test units are progressively (type II) censored. The scheme permits the withdrawal of units from the life test at stages during failure. This may be due to cost and time constraints. Both Expectation- Maximization (EM) and Newton-Raphson (NR) methods have been used to obtain the maximum likelihood estimates of the GIE parameters. Also, the variance-covariance matrix of the obtained estimators has been derived. The performance of the obtained MLEs via EM method is compared with those obtained using NR method in terms of bias and root mean squared errors and confidence interval widths for different progressive type II censoring schemes at fixed parameter values of λ and θ Simulation results reveal that estimates obtained via EM approach are more robust compared to those obtained via NR algorithm. It's also noted that the bias, root mean squared errors and confidence interval widths decrease with an increase in the sample size for a fixed number of failures. A similar trend in results is observed with increase in number of failures for a fixed sample size. The results of the obtained estimators are finally illustrated on two real data sets.

Published in	American Journal of Theoretical and Applied Statistics (Volume 10, Issue 1)
DOI	10.11648/j.ajtas.20211001.13
Page(s)	14-21
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

Generalized Inverted Exponential Distribution, Progressive type II Censoring, EM Algorithm, Newton Raphson Algorithm

References

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[2]	Abouammoh A. M and Alshingiti (2009). Reliability estimation of the generalized inverted exponential distribution. Journal of Statistical Computation and Simulation, 79 (11), 1301-1315.
[3]	Aggarwala R. and Balakrishnan N. (1998). Some Properties of Progressive Censored Order Statistics from arbitrary and uniform distributions with applications to Inference and Simulations. Journal of Statistics and Planning Inference, 70, 35-49.
[4]	Bakoban R. A (2012). Estimation in step stress Partially accelerated life tests for the Generalized inverted exponential distribution using type I censoring. American Journal of scientific Research, 3, 25-35.
[5]	Balakrishnan N. and Sandhu A. (1995). A simple simulation algorithm for generating progressive Type II censored sample. American journal of Statistics, 49, 229-230.
[6]	Childs A. and Balakrishnan N. (2000). Conditional inference procedures for the Laplace distribution when the observed samples are progressively censored. Journal of Metrika, 52 (3), 253-265.
[7]	Dempster A. P, Laird N. M, Rubin D. D. (1977). Maximum likelihood from incomplete data via EM algorithm. Journal of the Royal Statistical Society, Series B, 39, 1-38.
[8]	Dey and Pradhan (2014). Generalized exponential distribution under hybrid censoring. Statistical Methodology, 18, 101-114.
[9]	Horst R. (2009). The Weibull Distribution Handbook. CRC Press, Taylor and Francis Group.
[10]	Krishna H and Kumar K (2013). Reliability estimation in generalized inverted exponential distribution with progressively type II censored sample. Journal of statistical computation and simulation, 83, 1007-1019.
[11]	Louis T. A (1982). Finding the observed matrix when using the EM algorithm. Journal of the Royal Statistical Society Series B., 44, 226-233.
[12]	Sanku D. and Tanujit D. (2016). Statistical inference for the Generalized inverted exponential distribution based on upper record values. Journal of Mathematics and Computers in simulation, 4200.
[13]	Salem A. M and Abo-Kasem O. E (2011). International journal of Contemporary Mathematics and Sciences. (6) 35, 1713-1724.
[14]	Singh S. K, Singh U. and Kumar M. (2013). Singh S. K, Singh U. and Kumar M. (2013). Estimation of parameters of generalized inverted exponential distribution for progressive type II censored sample with Binomial removals. Journal of probability and statistics, Article ID 183652 http://dxi.org./10.1155/2013/183652.
[15]	Von Alven (1964 pg 156).
[16]	Wang and Cheng (2009) EM algorithm for estimating the Burr XII parameters with multiple censored data. Quality and reliability Engineering Journal International. 30 (5), 1622-1638.

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Karuoya Grace Njeri, Edward Gachangi Njenga. (2021). Maximum Likelihood Estimation for a Progressively Type II Censored Generalized Inverted Exponential Distribution via EM Algorithm. American Journal of Theoretical and Applied Statistics, 10(1), 14-21. https://doi.org/10.11648/j.ajtas.20211001.13

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ACS Style

Karuoya Grace Njeri; Edward Gachangi Njenga. Maximum Likelihood Estimation for a Progressively Type II Censored Generalized Inverted Exponential Distribution via EM Algorithm. Am. J. Theor. Appl. Stat. 2021, 10(1), 14-21. doi: 10.11648/j.ajtas.20211001.13

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AMA Style

Karuoya Grace Njeri, Edward Gachangi Njenga. Maximum Likelihood Estimation for a Progressively Type II Censored Generalized Inverted Exponential Distribution via EM Algorithm. Am J Theor Appl Stat. 2021;10(1):14-21. doi: 10.11648/j.ajtas.20211001.13

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@article{10.11648/j.ajtas.20211001.13,
  author = {Karuoya Grace Njeri and Edward Gachangi Njenga},
  title = {Maximum Likelihood Estimation for a Progressively Type II Censored Generalized Inverted Exponential Distribution via EM Algorithm},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {10},
  number = {1},
  pages = {14-21},
  doi = {10.11648/j.ajtas.20211001.13},
  url = {https://doi.org/10.11648/j.ajtas.20211001.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20211001.13},
  abstract = {The Generalized Inverted Exponential (GIE) distribution is a mixed lifetime model used in a number of fields such as queuing theory, testing of products or components and modelling the speed of winds. The study aims to focus on the determination of maximum likelihood estimates of GIE distribution when the test units are progressively (type II) censored. The scheme permits the withdrawal of units from the life test at stages during failure. This may be due to cost and time constraints. Both Expectation- Maximization (EM) and Newton-Raphson (NR) methods have been used to obtain the maximum likelihood estimates of the GIE parameters. Also, the variance-covariance matrix of the obtained estimators has been derived. The performance of the obtained MLEs via EM method is compared with those obtained using NR method in terms of bias and root mean squared errors and confidence interval widths for different progressive type II censoring schemes at fixed parameter values of λ and θ Simulation results reveal that estimates obtained via EM approach are more robust compared to those obtained via NR algorithm. It's also noted that the bias, root mean squared errors and confidence interval widths decrease with an increase in the sample size for a fixed number of failures. A similar trend in results is observed with increase in number of failures for a fixed sample size. The results of the obtained estimators are finally illustrated on two real data sets.},
 year = {2021}
}

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TY  - JOUR
T1  - Maximum Likelihood Estimation for a Progressively Type II Censored Generalized Inverted Exponential Distribution via EM Algorithm
AU  - Karuoya Grace Njeri
AU  - Edward Gachangi Njenga
Y1  - 2021/01/22
PY  - 2021
N1  - https://doi.org/10.11648/j.ajtas.20211001.13
DO  - 10.11648/j.ajtas.20211001.13
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  - 21
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20211001.13
AB  - The Generalized Inverted Exponential (GIE) distribution is a mixed lifetime model used in a number of fields such as queuing theory, testing of products or components and modelling the speed of winds. The study aims to focus on the determination of maximum likelihood estimates of GIE distribution when the test units are progressively (type II) censored. The scheme permits the withdrawal of units from the life test at stages during failure. This may be due to cost and time constraints. Both Expectation- Maximization (EM) and Newton-Raphson (NR) methods have been used to obtain the maximum likelihood estimates of the GIE parameters. Also, the variance-covariance matrix of the obtained estimators has been derived. The performance of the obtained MLEs via EM method is compared with those obtained using NR method in terms of bias and root mean squared errors and confidence interval widths for different progressive type II censoring schemes at fixed parameter values of λ and θ Simulation results reveal that estimates obtained via EM approach are more robust compared to those obtained via NR algorithm. It's also noted that the bias, root mean squared errors and confidence interval widths decrease with an increase in the sample size for a fixed number of failures. A similar trend in results is observed with increase in number of failures for a fixed sample size. The results of the obtained estimators are finally illustrated on two real data sets.
VL  - 10
IS  - 1
ER  -

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Author Information

Karuoya Grace Njeri

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

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

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Plain Text BibTeX RIS

APA Style

Karuoya Grace Njeri, Edward Gachangi Njenga. (2021). Maximum Likelihood Estimation for a Progressively Type II Censored Generalized Inverted Exponential Distribution via EM Algorithm. American Journal of Theoretical and Applied Statistics, 10(1), 14-21. https://doi.org/10.11648/j.ajtas.20211001.13

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ACS Style

Karuoya Grace Njeri; Edward Gachangi Njenga. Maximum Likelihood Estimation for a Progressively Type II Censored Generalized Inverted Exponential Distribution via EM Algorithm. Am. J. Theor. Appl. Stat. 2021, 10(1), 14-21. doi: 10.11648/j.ajtas.20211001.13

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AMA Style

Karuoya Grace Njeri, Edward Gachangi Njenga. Maximum Likelihood Estimation for a Progressively Type II Censored Generalized Inverted Exponential Distribution via EM Algorithm. Am J Theor Appl Stat. 2021;10(1):14-21. doi: 10.11648/j.ajtas.20211001.13

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@article{10.11648/j.ajtas.20211001.13,
  author = {Karuoya Grace Njeri and Edward Gachangi Njenga},
  title = {Maximum Likelihood Estimation for a Progressively Type II Censored Generalized Inverted Exponential Distribution via EM Algorithm},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {10},
  number = {1},
  pages = {14-21},
  doi = {10.11648/j.ajtas.20211001.13},
  url = {https://doi.org/10.11648/j.ajtas.20211001.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20211001.13},
  abstract = {The Generalized Inverted Exponential (GIE) distribution is a mixed lifetime model used in a number of fields such as queuing theory, testing of products or components and modelling the speed of winds. The study aims to focus on the determination of maximum likelihood estimates of GIE distribution when the test units are progressively (type II) censored. The scheme permits the withdrawal of units from the life test at stages during failure. This may be due to cost and time constraints. Both Expectation- Maximization (EM) and Newton-Raphson (NR) methods have been used to obtain the maximum likelihood estimates of the GIE parameters. Also, the variance-covariance matrix of the obtained estimators has been derived. The performance of the obtained MLEs via EM method is compared with those obtained using NR method in terms of bias and root mean squared errors and confidence interval widths for different progressive type II censoring schemes at fixed parameter values of λ and θ Simulation results reveal that estimates obtained via EM approach are more robust compared to those obtained via NR algorithm. It's also noted that the bias, root mean squared errors and confidence interval widths decrease with an increase in the sample size for a fixed number of failures. A similar trend in results is observed with increase in number of failures for a fixed sample size. The results of the obtained estimators are finally illustrated on two real data sets.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - Maximum Likelihood Estimation for a Progressively Type II Censored Generalized Inverted Exponential Distribution via EM Algorithm
AU  - Karuoya Grace Njeri
AU  - Edward Gachangi Njenga
Y1  - 2021/01/22
PY  - 2021
N1  - https://doi.org/10.11648/j.ajtas.20211001.13
DO  - 10.11648/j.ajtas.20211001.13
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  - 21
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20211001.13
AB  - The Generalized Inverted Exponential (GIE) distribution is a mixed lifetime model used in a number of fields such as queuing theory, testing of products or components and modelling the speed of winds. The study aims to focus on the determination of maximum likelihood estimates of GIE distribution when the test units are progressively (type II) censored. The scheme permits the withdrawal of units from the life test at stages during failure. This may be due to cost and time constraints. Both Expectation- Maximization (EM) and Newton-Raphson (NR) methods have been used to obtain the maximum likelihood estimates of the GIE parameters. Also, the variance-covariance matrix of the obtained estimators has been derived. The performance of the obtained MLEs via EM method is compared with those obtained using NR method in terms of bias and root mean squared errors and confidence interval widths for different progressive type II censoring schemes at fixed parameter values of λ and θ Simulation results reveal that estimates obtained via EM approach are more robust compared to those obtained via NR algorithm. It's also noted that the bias, root mean squared errors and confidence interval widths decrease with an increase in the sample size for a fixed number of failures. A similar trend in results is observed with increase in number of failures for a fixed sample size. The results of the obtained estimators are finally illustrated on two real data sets.
VL  - 10
IS  - 1
ER  -

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