Properties and Applications of the Gompertz Distribution

J. A. Adewara; J. S. Adeyeye; C. P. Thron

J. A. Adewara Distance Learning Institute, University of Lagos.
J. S. Adeyeye Department of Mathematics, University of Lagos.
C. P. Thron Department of Science and Mathematics, Texas A & M University-Central Texas.

Keywords: Gompertz distribution;, Skewed data, Maximum likelihood, Parameters, Reliability, Survival function, Hazard function, Quantile function

Abstract

The importance of statistical distributions in describing and predicting real world events cannot be over-emphasized. The Gompertz distribution is one example of a widely-used distribution, with many applications to survival analysis. In this paper, several properties of the Gompertz distribution are studied. The two-parameter Gompertz distribution is shown to be identical to the three-parameter Gompertz exponential distribution. Functions used in reliability analysis related to the Gompertz distribution are reviewed. Properties of maximum likelihood estimate (MLE) parameter estimates for the Gompertz distribution are studied: the bias and root mean squared error of parameter estimates are expressed as a function of sample size and parameter values. When the Gompertz shape parameter is large, MLE parameter estimates may fail to exist because of parameter degeneracy, as the two-parameter Gompertz distribution approaches a 1-parameter exponential distribution. The distribution is fitted to real life data sets from both industrial and biological applications. Compared to several 3-parameter distributions, the Gompertz distribution provides significantly better fits to the industrial data sets chosen, but the 3-parameter generalized Gompertz distribution gives a better fit to guinea pig lifetime data.

References

Pollard, J. H. & Valkovics, E. J. The Gompertz distribution and its applications. Genus, 40(3), pp. 15-28, (1992).

Jafari A.A., Tahmasebi S, & AlizadehM. The Beta Gompertz distribution ,Revista Colombiana de Estadistica, 37(1), pp 141 -158, (2014).

El-Gohary A., Alshamrani A, & Naif Al-Otaibi A. The generalized Gompertz distribution, Applied Mathematics Modelling 37(1-2) pp. 13- 24,(2013).

Oguntunde, P. E., Khaleel, M. A., Ahmed, M. T., Adejumo, A. O. & Odetunmibi, O. A. A New Generalization of the Lomax Distribution with Increasing, Decreasing, and Constant Failure Rate. Hindawi: Modelling and Simulation in Engineering (online journal), Article ID 6043169,

pages, (2017).

Abu-Zinadah H. H. Some Characterizations of the Exponentiated Gompertz Distribution, International Mathematics Forum, Volume 9, no 30, 1427-1439, (2014).

Alizadeh, M., Cordeiro, G.M., Pinho, L.G.B. & Ghosh, I. The Gompertz - G family of Distributions, Journal of Statistics Theory and Practice,11(1), 179-207, DOI:10:1080=15598608:2016:1267668,(2016).

Smith, R. L. & Naylor, J. C. A comparison of maximum likelihood and Bayesian estimators for the three-parameter Weibull distribution. Applied Statistics 36(3) 358-369, (1987).

Bourguignon, M., Silva, R. B. & Cordeiro, G. M. The Weibull-G family of probability distributions. Journal of Data Science, 12(1), 53-68, (2014).

Aarset, M. V. How to identify a bathtub hazard rate. IEEE Transactions on Reliability, 36(1), 106-108, (1987).

Bjerkedal, T. Acquisition of resistance in guinea pigs infected with dierent doses of virulent tubercle bacilli. Am.J. Hygiene, 72:130-148, (1960).

Rama S., Kamlesh K. S., Ravi S. & Tekie A. L. A Three- Parameter Lindley Distribution, American Journal of Mathematics and Statistics 7(1): 15 - 26, DOI: 10.5923/j.ajms.20170701.03, (2017).