Estimation of Parameters of the Two-Parameter Rayleigh Distribution Based on Progressive Type-II Censoring Using Maximum Likelihood Method via the NR and the EM Algorithms

Murithi Daniel Fundi; Edward Gachangi Njenga; Kemboi George Keitany

doi:doi:10.11648/j.ajtas.20170601.11

| Peer-Reviewed

Estimation of Parameters of the Two-Parameter Rayleigh Distribution Based on Progressive Type-II Censoring Using Maximum Likelihood Method via the NR and the EM Algorithms

Murithi Daniel Fundi, Edward Gachangi Njenga, Kemboi George Keitany

Published in American Journal of Theoretical and Applied Statistics (Volume 6, Issue 1)

Received: 15 November 2016 Accepted: 30 November 2016 Published: 20 December 2016

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

In this article, Maximum likelihood estimates for the shape and scale parameters of two-parameter Rayleigh distribution are obtained based on progressive type-II censored samples using the Newton-Raphson (NR) method and the Expectation-Maximization (EM) algorithm. A simple algorithm discussed in [2-3] is used for generating progressive type-II censored samples. Based on this censoring scheme, approximate asymptotic variances are derived and used to construct approximate confidence intervals of the parameters. The performance of these two maximum likelihood estimation algorithms is compared in terms of simulation results of root mean squared error (RMSE) and the coverage rates. Simulation results showed that in nearly all the combination of simulation conditions the estimators based on the EM algorithm have less root mean squared error (RMSE) and narrower widths of confidence intervals compared to those obtained using the NR algorithm. Finally, an illustrative example with real-life data sets is provided to illustrate how maximum likelihood estimation using the two algorithms works in practice.

Published in	American Journal of Theoretical and Applied Statistics (Volume 6, Issue 1)
DOI	10.11648/j.ajtas.20170601.11
Page(s)	1-9
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), 2016. Published by Science Publishing Group

Keywords

Two-Parameter Rayleigh Distribution, Maximum Likelihood Estimation, EM Algorithm, NR Method, Progressive Type-II Censoring

References

[1]	Al-Zahrani, Bander M., and Maha S. Gindwan. "Estimating the parameter of the Lindley distribution under progressive type-II censored data." Electronic Journal of Applied Statistical Analysis 8.1 (2015): 100-111.
[2]	Balakrishnan, Narayanaswamy, and Rita Aggarwala. Progressive Censoring: theory, methods, and applications. Springer Science & Business Media, 2000.
[3]	Balakrishnan, N., and R. A. Sandhu. "A simple simulational algorithm for generating progressive Type-II censored samples." The American Statistician 49.2 (1995): 229-230.
[4]	Balakrishnan, N. "Progressive censoring methodology: an appraisal." Test 16.2 (2007): 211-259.
[5]	Bekker, A., J. J. J. Roux, and P. J. Mosteit. "A generalization of the compound Rayleigh distribution: using a Bayesian method on cancer survival times." Communications in Statistics-Theory and Methods 29.7 (2000): 1419-1433.
[6]	Cox, D. R., and D. V. Hinkley. "Theoretical Statistics Chapman and Hall, London." See Also (1974).
[7]	Dempster, Arthur P., Nan M. Laird, and Donald B. Rubin. "Maximum likelihood from incomplete data via the EM algorithm." Journal of the Royal Statistical Society. Series B (Methodological) (1977): 1-38.
[8]	Devore, Jay L., and Kenneth N. Berk. Modern mathematical statistics with applications. Cengage Learning, 2007.
[9]	Dey, Sanku. "Comparison of Bayes Estimators of the parameter and reliability function for Rayleigh distribution under different loss functions." Malaysian Journal of Mathematical Sciences 3 (2009): 247-264.
[10]	Dey, Sanku, and M. K. Das. "A note on prediction interval for a Rayleigh distribution: Bayesian approach." American Journal of Mathematical and Management Sciences 27.1-2 (2007): 43-48.
[11]	Dey, Sanku, Tanujit Dey, and Debasis Kundu. "Two-parameter Rayleigh distribution: different methods of estimation." American Journal of Mathematical and Management Sciences 33.1 (2014): 55-74.
[12]	Dey, Tanujit, Sanku Dey, and Debasis Kundu. "On Progressively Type-II Censored Two-parameter Rayleigh Distribution." Communications in Statistics-Simulation and Computation 45.2 (2016): 438-455.
[13]	Gulhar, Monika, et al. "A comparison of some confidence intervals for estimating the population coefficient of variation: a simulation study." Sort: Statistics and Operations Research Transactions 36.1 (2012): 45-68.
[14]	Helu, Amal, Hani Samawi, and Mohammad Z. Raqab. "Estimation on Lomax progressive censoring using the EM algorithm." Journal of Statistical Computation and Simulation 85.5 (2015): 1035-1052.
[15]	Howlader, H. A., and A. Hossain. "On Bayesian estimation and prediction from Rayleigh based on type II censored data." Communications in Statistics-Theory and Methods 24.9 (1995): 2251-2259.
[16]	Johnson, Norman L., Samuel Kotz, and N. Balakrishnan. Continuous multivariate distributions, volume 1, models and applications. Vol. 59. New York: John Wiley & Sons, 2002.
[17]	Khan, Hafiz MR, Serge B. Provost, and Ashima Singh. "Predictive inference from a two-parameter Rayleigh life model given a doubly censored sample." Communications in Statistics-Theory, and Methods 39.7 (2010): 1237-1246.
[18]	Kundu, Debasis, and Arabin Kumar Dey. "Estimating the parameters of the Marshall–Olkin bivariate Weibull distribution by EM algorithm." Computational Statistics & Data Analysis 53.4 (2009): 956-965.
[19]	Lio, Y. L., Ding-Geng Chen, and Tzong-Ru Tsai. "Parameter Estimations for Generalized RayleighDistribution under Progressively Type-I IntervalCensored Data." Open Journal of Statistics 1.02 (2011): 46.
[20]	Top of Form McLachlan, Geoffrey J, and Thriyambakam Krishnan. The Em Algorithm and Extensions. New York: J. Wiley, 1997. Print.
[21]	Bottom of Form Lee, Jason, Yuekai Sun, and Michael Saunders. "Proximal Newton-type methods for convex optimization." Advances in Neural Information Processing Systems. 2012.
[22]	Ng, H. K. T., P. S. Chan, and N. Balakrishnan. "Estimation of parameters from progressively censored data using EM algorithm." Computational Statistics & Data Analysis 39.4 (2002): 371-386.
[23]	Ortega, James M., and Werner C. Rheinboldt. Iterative solution of nonlinear equations in several variables. Vol. 30. Siam, 1970.
[24]	Pradhan, Biswabrata, and Debasis Kundu. "Bayes estimation and prediction of the two-parameter gamma distribution." Journal of Statistical Computation and Simulation 81.9 (2011): 1187-1198.
[25]	Rayleigh, Lord. "On the capillary phenomena of jets." Proc. R. Soc. London. Vol. 29. No. 196-199. 1879.
[26]	Surles, J. G., and W. J. Padgett. "Inference for P (Y< X) in the Burr type X model." Communications in Statistics-Theory and Methods 7 (1998): 225-238.
[27]	Seo, Eun-Hyung, and Suk-Bok Kang. "AMLEs for Rayleigh distribution based on progressive Type-II censored data." Communications for Statistical Applications and Methods 14.2 (2007): 329-344.
[28]	Smith, Richard L. "Maximum likelihood estimation in a class of nonregular cases." Biometrika 72.1 (1985): 67-90.
[29]	Watanabe, Michiko, and Kazunori Yamaguchi, eds. The EM algorithm and related statistical models. CRC Press, 2003.

Cite This Article

Plain Text BibTeX RIS

APA Style

Murithi Daniel Fundi, Edward Gachangi Njenga, Kemboi George Keitany. (2016). Estimation of Parameters of the Two-Parameter Rayleigh Distribution Based on Progressive Type-II Censoring Using Maximum Likelihood Method via the NR and the EM Algorithms. American Journal of Theoretical and Applied Statistics, 6(1), 1-9. https://doi.org/10.11648/j.ajtas.20170601.11

Copy | Download

ACS Style

Murithi Daniel Fundi; Edward Gachangi Njenga; Kemboi George Keitany. Estimation of Parameters of the Two-Parameter Rayleigh Distribution Based on Progressive Type-II Censoring Using Maximum Likelihood Method via the NR and the EM Algorithms. Am. J. Theor. Appl. Stat. 2016, 6(1), 1-9. doi: 10.11648/j.ajtas.20170601.11

Copy | Download

AMA Style

Murithi Daniel Fundi, Edward Gachangi Njenga, Kemboi George Keitany. Estimation of Parameters of the Two-Parameter Rayleigh Distribution Based on Progressive Type-II Censoring Using Maximum Likelihood Method via the NR and the EM Algorithms. Am J Theor Appl Stat. 2016;6(1):1-9. doi: 10.11648/j.ajtas.20170601.11

Copy | Download

@article{10.11648/j.ajtas.20170601.11,
  author = {Murithi Daniel Fundi and Edward Gachangi Njenga and Kemboi George Keitany},
  title = {Estimation of Parameters of the Two-Parameter Rayleigh Distribution Based on Progressive Type-II Censoring Using Maximum Likelihood Method via the NR and the EM Algorithms},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {6},
  number = {1},
  pages = {1-9},
  doi = {10.11648/j.ajtas.20170601.11},
  url = {https://doi.org/10.11648/j.ajtas.20170601.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20170601.11},
  abstract = {In this article, Maximum likelihood estimates for the shape and scale parameters of two-parameter Rayleigh distribution are obtained based on progressive type-II censored samples using the Newton-Raphson (NR) method and the Expectation-Maximization (EM) algorithm. A simple algorithm discussed in [2-3] is used for generating progressive type-II censored samples. Based on this censoring scheme, approximate asymptotic variances are derived and used to construct approximate confidence intervals of the parameters. The performance of these two maximum likelihood estimation algorithms is compared in terms of simulation results of root mean squared error (RMSE) and the coverage rates. Simulation results showed that in nearly all the combination of simulation conditions the estimators based on the EM algorithm have less root mean squared error (RMSE) and narrower widths of confidence intervals compared to those obtained using the NR algorithm. Finally, an illustrative example with real-life data sets is provided to illustrate how maximum likelihood estimation using the two algorithms works in practice.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - Estimation of Parameters of the Two-Parameter Rayleigh Distribution Based on Progressive Type-II Censoring Using Maximum Likelihood Method via the NR and the EM Algorithms
AU  - Murithi Daniel Fundi
AU  - Edward Gachangi Njenga
AU  - Kemboi George Keitany
Y1  - 2016/12/20
PY  - 2016
N1  - https://doi.org/10.11648/j.ajtas.20170601.11
DO  - 10.11648/j.ajtas.20170601.11
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  - 1
EP  - 9
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20170601.11
AB  - In this article, Maximum likelihood estimates for the shape and scale parameters of two-parameter Rayleigh distribution are obtained based on progressive type-II censored samples using the Newton-Raphson (NR) method and the Expectation-Maximization (EM) algorithm. A simple algorithm discussed in [2-3] is used for generating progressive type-II censored samples. Based on this censoring scheme, approximate asymptotic variances are derived and used to construct approximate confidence intervals of the parameters. The performance of these two maximum likelihood estimation algorithms is compared in terms of simulation results of root mean squared error (RMSE) and the coverage rates. Simulation results showed that in nearly all the combination of simulation conditions the estimators based on the EM algorithm have less root mean squared error (RMSE) and narrower widths of confidence intervals compared to those obtained using the NR algorithm. Finally, an illustrative example with real-life data sets is provided to illustrate how maximum likelihood estimation using the two algorithms works in practice.
VL  - 6
IS  - 1
ER  -

Copy | Download

Author Information

Murithi Daniel Fundi

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

Department of Statistics and Actuarial Science, Kenyatta University (KU), Nairobi, Kenya
Kemboi George Keitany

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

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Murithi Daniel Fundi, Edward Gachangi Njenga, Kemboi George Keitany. (2016). Estimation of Parameters of the Two-Parameter Rayleigh Distribution Based on Progressive Type-II Censoring Using Maximum Likelihood Method via the NR and the EM Algorithms. American Journal of Theoretical and Applied Statistics, 6(1), 1-9. https://doi.org/10.11648/j.ajtas.20170601.11

Copy | Download

ACS Style

Murithi Daniel Fundi; Edward Gachangi Njenga; Kemboi George Keitany. Estimation of Parameters of the Two-Parameter Rayleigh Distribution Based on Progressive Type-II Censoring Using Maximum Likelihood Method via the NR and the EM Algorithms. Am. J. Theor. Appl. Stat. 2016, 6(1), 1-9. doi: 10.11648/j.ajtas.20170601.11

Copy | Download

AMA Style

Murithi Daniel Fundi, Edward Gachangi Njenga, Kemboi George Keitany. Estimation of Parameters of the Two-Parameter Rayleigh Distribution Based on Progressive Type-II Censoring Using Maximum Likelihood Method via the NR and the EM Algorithms. Am J Theor Appl Stat. 2016;6(1):1-9. doi: 10.11648/j.ajtas.20170601.11

Copy | Download

@article{10.11648/j.ajtas.20170601.11,
  author = {Murithi Daniel Fundi and Edward Gachangi Njenga and Kemboi George Keitany},
  title = {Estimation of Parameters of the Two-Parameter Rayleigh Distribution Based on Progressive Type-II Censoring Using Maximum Likelihood Method via the NR and the EM Algorithms},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {6},
  number = {1},
  pages = {1-9},
  doi = {10.11648/j.ajtas.20170601.11},
  url = {https://doi.org/10.11648/j.ajtas.20170601.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20170601.11},
  abstract = {In this article, Maximum likelihood estimates for the shape and scale parameters of two-parameter Rayleigh distribution are obtained based on progressive type-II censored samples using the Newton-Raphson (NR) method and the Expectation-Maximization (EM) algorithm. A simple algorithm discussed in [2-3] is used for generating progressive type-II censored samples. Based on this censoring scheme, approximate asymptotic variances are derived and used to construct approximate confidence intervals of the parameters. The performance of these two maximum likelihood estimation algorithms is compared in terms of simulation results of root mean squared error (RMSE) and the coverage rates. Simulation results showed that in nearly all the combination of simulation conditions the estimators based on the EM algorithm have less root mean squared error (RMSE) and narrower widths of confidence intervals compared to those obtained using the NR algorithm. Finally, an illustrative example with real-life data sets is provided to illustrate how maximum likelihood estimation using the two algorithms works in practice.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - Estimation of Parameters of the Two-Parameter Rayleigh Distribution Based on Progressive Type-II Censoring Using Maximum Likelihood Method via the NR and the EM Algorithms
AU  - Murithi Daniel Fundi
AU  - Edward Gachangi Njenga
AU  - Kemboi George Keitany
Y1  - 2016/12/20
PY  - 2016
N1  - https://doi.org/10.11648/j.ajtas.20170601.11
DO  - 10.11648/j.ajtas.20170601.11
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  - 1
EP  - 9
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20170601.11
AB  - In this article, Maximum likelihood estimates for the shape and scale parameters of two-parameter Rayleigh distribution are obtained based on progressive type-II censored samples using the Newton-Raphson (NR) method and the Expectation-Maximization (EM) algorithm. A simple algorithm discussed in [2-3] is used for generating progressive type-II censored samples. Based on this censoring scheme, approximate asymptotic variances are derived and used to construct approximate confidence intervals of the parameters. The performance of these two maximum likelihood estimation algorithms is compared in terms of simulation results of root mean squared error (RMSE) and the coverage rates. Simulation results showed that in nearly all the combination of simulation conditions the estimators based on the EM algorithm have less root mean squared error (RMSE) and narrower widths of confidence intervals compared to those obtained using the NR algorithm. Finally, an illustrative example with real-life data sets is provided to illustrate how maximum likelihood estimation using the two algorithms works in practice.
VL  - 6
IS  - 1
ER  -

Copy | Download