The aim of this study was to employ Maximum Likelihood (MLE) jointly with a numerical Method (Newton Raphson method) to obtain parameter estimates from the two-parameter Gamma model. The profile likelihood of the two-parameter Gamma model was also put into consideration. The methods were demonstrated using simulation studies and real life data considering data sets generated by R statistical software for different sample sizes. Standard errors were computed and 5 % Wald-confidence interval was constructed for the estimates of the model. The result of the study shows that Maximum Likelihood Estimation (MLE) jointly with Newton Raphson method was more efficient for estimating parameters of the Gamma model in simulation study than real life data. The study recommends that parameter estimates from the two-parameter Gamma model should be obtained by employing Maximum Likelihood Estimation jointly with Newton Raphson Method.
| Published in | International Journal of Statistical Distributions and Applications (Volume 3, Issue 2) |
| DOI | 10.11648/j.ijsd.20170302.11 |
| Page(s) | 13-17 |
| 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), 2017. Published by Science Publishing Group |
Parameter Estimation, Two-Parameter Gamma Model, Profile Likelihood, Maximum Likelihood Estimation, Newton Raphson Method
| [1] | Aldrich J.; R. A. Fisher and the Making of Maximum Likelihood 1912-1922, Statistical |
| [2] | Atkinson, A. C., Pericchi, L. R., & Smith, R. L. (1991). Grouped likelihood for the shifted power transformation. Journal of the Royal Statistical Society: Series B, 53: 473- 482. |
| [3] | Beaumont, M. A., Zhang, W., & Balding, D. J.(2002). Approximate Bayesian computation in population genetics. Genetics, 2 (162), 2025-2035. |
| [4] | Cheng, R. C. H. & Amin, N. A. K. (1983). Estimating parameters in continuous univariate distributions with a shifted origin. Journal of the Royal Statistical Society: Series B, 45: 394-403. |
| [5] | Cheng, R. C. H. & Traylor, L. (1995). Non-regular maximum likelihood problems. Journal of the Royal Statistical Society: Series B, 57: 3-44. |
| [6] | Cox, D. R. & Reid, N. (2004). A note on pseudolikelihood constructed from marginal densities., Biometrika, 2( 91), 729-737. |
| [7] | Cramer H.; Mathematical Methods of Statistics, Princeton University Press, 1946. |
| [8] | Cule, M. L., Samworth, R. J. & Stewart, M. I. (2010). Maximum likelihood estimation of a multi-dimensional log-concave density. Journal Royal Statistical Society B 72: 545600. |
| [9] | Dean, T. A., Singh, S. S., Jasra, A. & Peters G. W. (2011). Parameter estimation for hidden Markov models with intractable likelihoods., Arxiv preprint ar Xiv:1103.5399v1. |
| [10] | Didelot, X., Everitt, R. G., Johansen, A. M. and Lawson, D. J. (2011). Likelihood-free estimation of model evidence. Bayesian Analysis, vol. 6,49-76. |
| [11] | Fukunaga, K. & Hostetler, L. D. (1975). The Estimation of the Gradient of a Density Function, with Applications in Pattern Recognition. IEEE Transactions on Information Theory 21: 3240. |
| [12] | Fearnhead, P. & Prangle, D. (2012).”Constructing Summary Statistics for Approximate Bayesian Computation: Semi-automatic ABC (with discussion). Journal of the Royal Statistical Society Series B (Methodology) in press. |
| [13] | Fisher, R. A. (1922). On the mathematical foundations of theoretical statistics.Philosophical Transactions of the Royal Society of London: Series A, 222: 309-368. |
| [14] | Nemes G.; New asymptotic expansion for the Γ(z) function, Stan’s Library, Volume II, 2007. |
| [15] | Marjoram, P., Molitor, J., Plagnol, V., &Tavare, S. (2003). Markov chain Monte Carlo without likelihoods. Proceedings of the National Academy of Sciences USA: 1532415328. |
| [16] | Pritchard, J. K., Seielstad, M. T., Perez-Lezaun, A., & Feldman, M. T. (1999). Population Growth of Human Y Chromosomes: A Study of Y Chromosome Microsatellites. Molecular Biology and Evolution 16: 1791 1798. |
| [17] | Pyke R.; Spacing, Journal of the Royal Statistical Statistical Society Series B, 27(3), 1965, pp. 395– 499. |
| [18] | Robert, C. P., Cornuet, J., Marin, J. & Pillai, N. S. (2011). Lack of confidence in ABC model choice. Proceedings of the National Academy of Sciences of the United States of America 108: 1511215117. |
| [19] | Rockette H., Antle C., Klimko L. A.; Maximum likelihood model, Journal of the American Statistical Association, 69(345), 1974, pp. 246–249. |
| [20] | Rosin and Rammler (1933) application of Weibull dist- ribution to describe the size distribution of particles |
| [21] | Wilkinson, R. D. (2008). Approximate Bayesian computation (ABC) gives exact results under the assumption of error model. Arxiv preprint ar Xiv:0811.3355. |
APA Style
A. M. Yahaya, N. P. Dibal, H. R. Bakari. (2017). Comparing Parameter Estimates Obtained by Simulation Study and Real Life Data from the Two-Parameter Gamma Model. International Journal of Statistical Distributions and Applications, 3(2), 13-17. https://doi.org/10.11648/j.ijsd.20170302.11
ACS Style
A. M. Yahaya; N. P. Dibal; H. R. Bakari. Comparing Parameter Estimates Obtained by Simulation Study and Real Life Data from the Two-Parameter Gamma Model. Int. J. Stat. Distrib. Appl. 2017, 3(2), 13-17. doi: 10.11648/j.ijsd.20170302.11
AMA Style
A. M. Yahaya, N. P. Dibal, H. R. Bakari. Comparing Parameter Estimates Obtained by Simulation Study and Real Life Data from the Two-Parameter Gamma Model. Int J Stat Distrib Appl. 2017;3(2):13-17. doi: 10.11648/j.ijsd.20170302.11
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Download@article{10.11648/j.ijsd.20170302.11,
author = {A. M. Yahaya and N. P. Dibal and H. R. Bakari},
title = {Comparing Parameter Estimates Obtained by Simulation Study and Real Life Data from the Two-Parameter Gamma Model},
journal = {International Journal of Statistical Distributions and Applications},
volume = {3},
number = {2},
pages = {13-17},
doi = {10.11648/j.ijsd.20170302.11},
url = {https://doi.org/10.11648/j.ijsd.20170302.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijsd.20170302.11},
abstract = {The aim of this study was to employ Maximum Likelihood (MLE) jointly with a numerical Method (Newton Raphson method) to obtain parameter estimates from the two-parameter Gamma model. The profile likelihood of the two-parameter Gamma model was also put into consideration. The methods were demonstrated using simulation studies and real life data considering data sets generated by R statistical software for different sample sizes. Standard errors were computed and 5 % Wald-confidence interval was constructed for the estimates of the model. The result of the study shows that Maximum Likelihood Estimation (MLE) jointly with Newton Raphson method was more efficient for estimating parameters of the Gamma model in simulation study than real life data. The study recommends that parameter estimates from the two-parameter Gamma model should be obtained by employing Maximum Likelihood Estimation jointly with Newton Raphson Method.},
year = {2017}
}
TY - JOUR T1 - Comparing Parameter Estimates Obtained by Simulation Study and Real Life Data from the Two-Parameter Gamma Model AU - A. M. Yahaya AU - N. P. Dibal AU - H. R. Bakari Y1 - 2017/05/06 PY - 2017 N1 - https://doi.org/10.11648/j.ijsd.20170302.11 DO - 10.11648/j.ijsd.20170302.11 T2 - International Journal of Statistical Distributions and Applications JF - International Journal of Statistical Distributions and Applications JO - International Journal of Statistical Distributions and Applications SP - 13 EP - 17 PB - Science Publishing Group SN - 2472-3509 UR - https://doi.org/10.11648/j.ijsd.20170302.11 AB - The aim of this study was to employ Maximum Likelihood (MLE) jointly with a numerical Method (Newton Raphson method) to obtain parameter estimates from the two-parameter Gamma model. The profile likelihood of the two-parameter Gamma model was also put into consideration. The methods were demonstrated using simulation studies and real life data considering data sets generated by R statistical software for different sample sizes. Standard errors were computed and 5 % Wald-confidence interval was constructed for the estimates of the model. The result of the study shows that Maximum Likelihood Estimation (MLE) jointly with Newton Raphson method was more efficient for estimating parameters of the Gamma model in simulation study than real life data. The study recommends that parameter estimates from the two-parameter Gamma model should be obtained by employing Maximum Likelihood Estimation jointly with Newton Raphson Method. VL - 3 IS - 2 ER -
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