Simulation of generalized Gamma distribution with maximum likelihood estimation and expectation-maximization algorithm on right censored data type 1

Kurniasari, D.; Warsono, W.; Indryani, N.; Usman, M.; Hadi, S.

Growing Science » Decision Science Letters » Simulation of generalized Gamma distribution with maximum likelihood estimation and expectation-maximization algorithm on right censored data type 1

Journals

DSL Volumes

Volume 1 (10)
- Issue 1 (5)
- Issue 2 (5)

Volume 2 (30)
- Issue 1 (5)
- Issue 2 (6)
- Issue 3 (9)
- Issue 4 (10)

Volume 3 (53)
- Issue 1 (15)
- Issue 2 (10)
- Issue 3 (19)
- Issue 4 (9)

Volume 4 (48)
- Issue 1 (10)
- Issue 2 (12)
- Issue 3 (14)
- Issue 4 (12)

Volume 5 (39)
- Issue 1 (12)
- Issue 2 (10)
- Issue 3 (8)
- Issue 4 (9)

Volume 6 (30)
- Issue 1 (8)
- Issue 2 (6)
- Issue 3 (9)
- Issue 4 (7)

Volume 7 (41)
- Issue 1 (8)
- Issue 2 (8)
- Issue 3 (8)
- Issue 4 (17)

Volume 8 (38)
- Issue 1 (8)
- Issue 2 (6)
- Issue 3 (14)
- Issue 4 (10)

Volume 9 (39)
- Issue 1 (8)
- Issue 2 (9)
- Issue 3 (14)
- Issue 4 (8)

Volume 10 (43)
- Issue 1 (7)
- Issue 2 (8)
- Issue 3 (20)
- Issue 4 (8)

Volume 11 (49)
- Issue 1 (9)
- Issue 2 (9)
- Issue 3 (14)
- Issue 4 (17)

Volume 12 (64)
- Issue 1 (12)
- Issue 2 (24)
- Issue 3 (13)
- Issue 4 (15)

Volume 13 (78)
- Issue 1 (21)
- Issue 2 (18)
- Issue 3 (19)
- Issue 4 (20)

Volume 14 (21)
- Issue 1 (21)

Countries

Decision Science Letters

ISSN 1929-5812 (Online) - ISSN 1929-5804 (Print) Quarterly Publication Volume 10 Issue 3 pp. 415-424 , 2021

Simulation of generalized Gamma distribution with maximum likelihood estimation and expectation-maximization algorithm on right censored data type 1 Pages 415-424 Download PDF

Authors: Dian Kurniasari, Warsono Warsono, Nourma Indryani, Mustofa Usman, Sutopo Hadi

DOI: 10.5267/j.dsl.2021.1.003

Keywords: Generalized Gamma Distribution, Right Censored Data type 1, Expectation Maximization (EM), Maximum Likelihood Estimaton (MLE), MSE

Abstract: The Generalized Gamma distribution is very suitable for modeling data with various forms of hazard (risk) functions, which makes the Generalized Gamma distribution useful in survival analysis. Survival analysis aims are to predict chances of survival, disease recurrence, death, and other events over a period of time. One characteristic of survival data is the possibility of sensors. Let X be the life span of the person being studied and the right censorship time of Cr, X is assumed to be independent with the probability density function f(x), the survival function S(x), and the hazard function h(x). A person's X life span will be known if X is less than or equal to Cr. If X is greater than Cr, the individual X survives or is censored right now. Statistical inference, especially parameter estimation is needed in analyzing empirical data. Obviously the estimation results obtained are expected to be a good estimator, namely to meet the nature of unbiased and minimum variance. This paper will discuss the results of the estimation of Generalized Gamma distribution parameters with type 1 right censored data through simulations using the Expectation Maximization method and the Maximum Likelihood Estimation method. The simulation is conducted by generating data with the sample size: 25, 50, 100, 200, 500, 1000, 1500 and 2000 as well as determining censored data of 10%, 20% and 30% by first setting the parameters used which are obtained from the data of patients with gastric cancer namely α = 1.0649, β = 1,072, θ = 59.766. Based on the results obtained from the simulations on the two estimation methods that the parameter estimation using the Maximum Likelihood Estimation method is better than the Expectation Maximization method because it provides a smaller bias and MSE value where the larger the sample size used, the estimated parameter value will get closer to the parameter in fact. In addition, the Expectation Maximization method can also be used as an alternative estimation of generalized gamma distribution parameters with type 1 right censored data because it has a bias value and MSE approaching the MLE method.

How to cite this paper

 Kurniasari, D., Warsono, W., Indryani, N., Usman, M & Hadi, S. (2021). Simulation of generalized Gamma distribution with maximum likelihood estimation and expectation-maximization algorithm on right censored data type 1.Decision Science Letters , 10(3), 415-424.

Refrences

Agarwal, S.K. & Kalla. S.L. (1996). A generalized gamma distribution and its application in reliability. Communication in Statistics Theory-Methods, 25, 201- 210.
Agarwal, S.K. & Al-Saleh, A.A. (2001). Generalized gamma type distribution and its hazard rate function. Communication in Statistics Theory-Methods, 30, 309-318.
Bain, L.J. & Engelhardt, M. 1992. Introduction to Probability and Mathematical Statistics. Duxbury, United States.
Cordeiro, G.M., Castellares, F., Montenegro, L.C., & Casto, M. (2012). The Beta Generalizatiom of The Gamma Distribution. Statistics, 47, 1-13.
Cox, C., Chu, H., Schneider, M.F., & Muñoz, A. (2007). Parametric survival analysis and taxonomy of hazard functions for the generalized gamma distribution. Statistics in Medicine, 26, 4352–4374.
Cox, C. (2008). The generalized F distribution: an umbrella for parametric survival analysis. Statistics in Medicine, 27, 4301–4312.
Hogg, R.V., & Craig, A.T. (1995). Introduction of Mathematical Statistic. Fifth Edition. John Wiley & Sons, The United States of America.
Hogg, V.H., Mckean, J.W., & Craig, A.T. (2013). Introduction to Mathematical Statistics. Pearson Education Inc., Boston.
Hoq, A., Ali, M. & Templetion. (1974). Estimation of parameters of a generalized life testing models. Journal of Statistical Research, 9, 67-79.
Kalla, S.L. Al-Saqabi, B.N., & Khajah, H.G. 2001. A unified form of gamma type distributions. Applied Mathematics and Computation, 118, 175-187.
Klein, J.P., & Moeschberger, M.L. (1997). Survival Analysis: Techniques for Censored and Truncated Data. Springer-Verlag, New York.
Lee, E., & Wang, J.W. (2003). Statistical Method for Survival Data Analysis. John Wiley & Sons Inc., New York.
Lee, M., & Gross, A. (1991). Lifetime distributions under unknown environment. Journal of Statistical Planning and Inference, 29, 137-143.
Liu, X. (2012). Survival Analysis : Models and Applications. John Wiley & Sons, USA.
Mclachlan, G.J., & Krishnan, T. (2008). The EM Algorithm and Extensions. John Wiley & Sons Inc., Canada.
Millar, R.B. (2011). Maximum Likelihood Estimation and Inference: With Examples in R, SAS, and ADMB. John Wiley & Sons, Ltd., United Kingdom.
Nadarajah, S., & Gupta, A.K. (2007). Moments and cumulants of the skew normal distribution, Kobe Journal of Mathematics, 24, 107-124
Ortega, E.M.M., Cancho, V.G., & Paula, G.A. (2009). Generalized log-gamma regression models with cure fraction. Lifetime Data Analysis, 15, 79–106.
Ruhi, S., Sarker, S., & Karim, M.R. (2015). Mixture Model for Analyzing Product Reliability data: a Case Study. SpringerPlus, 4, 634.
Stacy, E.W. (1962). A Generalization of The Gamma Distribution. The Annals of Mathematical Statistics, 33, 1187-1192.
Wang, F.K. & Cheng, Y.F. (2009). EM Algorithm for Estimating The Burr XII Parameters with Multiple Censored Data. John Wiley & Sons Inc., pp. 615-630.
Warsono. (2009). Moment Properties of the Generalized Gamma Distribution, National Seminar on Sains, Matematika dan Aplikasinya IV, pp. 157-162.
Warsono, Gustavia, E., Kurniasari, D., Amanto, & Antonio, Y. (2019). On The Comparison of The Methods of Parameter Estimation for Pareto Distribution. Journal of Physics: Conf. Series, 1338, 012042.